number theory and quantum mechanics

By far, the most active area of research linking QM and number theory is the work concerning the 'spectral interpretation' of the Riemann zeta zeros, suggesting a possible approach to the Riemann hypothesis involving quantum chaos.

The remainder of this page concerns more general connections between QM and number theory - both the use of number theoretical structures in the modelling of QM phenomena, and the application of QM-related techniques to number theoretical problems:

T. Aoki, S. Kanemitsu, M. Nakahara, and Y. Ohno (Eds.), Zeta Functions, Topology and Quantum Physics, Developments in Mathematics 14 (Springer, 2005)

[publisher's description:] "This volume focuses on various aspects of zeta functions: multiple zeta values, Ohno's relations, the Riemann hypothesis, L-functions, polylogarithms, and their interplay with other disciplines. Eleven articles on recent advances are written by outstanding experts in the above-mentioned fields. Each article starts with an introductory survey leading to the exciting new research developments accomplished by the contributors."

D. Pozdnyakov, "Physical interpretation of the Riemann hypothesis" (preprint 03/2012)

[abstract:] "An equivalent formulation of the Riemann hypothesis is given. The formulation is generalized. The physical interpretation of the Riemann hypothesis generalized formulation is given in the framework of quantum theory terminology. An axiom is laid down on the ground of the interpretation taking into account the observed properties of the surrounding reality. The Riemann hypothesis is true according to the axiom. It is shown that it is unprovable."

Y. Pan, "How to measure the canonical commutation relation $[\hat{x},\hat{p}] = i\hbar$? in quantum mechanics with weak measurement?" (preprint 02/2017)

[abstract:] "The quantum weak value draws many attentions recently from theoretical curiosity to experimental applications. Now we design an unusual weak measuring procedure as the pre-selection, mid-selection and post-selection to study the correlation function of two weak values, which we called the weak correlation function. In this paper, we proposed an weak measurement experiment to measure the canonical commutator $[\hat{x},\hat{p}] = i\hbar$? in quantum mechanics. Furthurmore, we found the intriguing equivalence between the canonical commutation relation and Riemann hypothesis, and then obtained the weak value of nontrivial Riemann zeros. Finally, as an nontrivial example of weak correlations, we also passed successfully a testing on the (anti-)commutators of Pauli operators, which followed the experimental setup of the landmark paper of Aharonov, et al. in 1988. Our proposed experiments could hopefully test the fundamental canonical relationship in quantum worlds and trigger more testing experiments on weak correlations."

G. Cotti, "Coalescence phenomenon of quantum cohomology of Grassmannians and the distribution of prime numbers" (preprint 08/2016)

[abstract:] "The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function."

R. Ramanathan, M. Túlio Quintino, A. Belén Sainz, G. Murta, R. Augusiak, "On the tightness of correlation inequalities with no quantum violation" (preprint 07/2016)

[abstract:] "We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games, illustrating it for the case of three outcomes. We then show as a main result that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the non-local computation games do not constitute tight Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest CHSH Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario.

Finally, we relate the non-contextual polytope of single-party correlation inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the compatibility graph of observables in the contextuality scenario and $\nabla G$ denotes the suspension graph of $G$. We observe that there exist tight non-contextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the Consistent Exclusivity principle."

Y.-L. Wang, " Special unextendible entangled bases with continuous integer cardinality" (preprint 09/2019)

[abstract:] "Special unextendible entangled basis of ``type $k$’’ (SUEBk), a set of incomplete orthonormal special entangled states of ``type $k$’’ whose complementary space has no special entangled state of ``type $k$’’. This concept can be seem as a generalization of the unextendible product basis (UPB) introduced by Bennett et al. in [ Phys. Rev. Lett. \textbf{82}, 5385(1999) ] and the unextendible maximally entangled basis (UMEB) introduced by Bravyi and Smolin in [Phys. Rev. A \textbf{84}, 042306(2011)]. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of one subspace can be easily upper bounded by $k$ while the other one can be generated by two kinds of the special entangled states of type $k$. This method is very effective for those $k = p^m \geq 3$ where $p$ is a prime number. For these cases, we can otain sets of SUEBk with continuous integer cardinality when the local dimensions are large. Moreover, one can find that our method here can be easily extended when there are more than two kinds of the special entangled states of type $k$ at hand."

J. Kaniewski, I. Supić, J. Tura, F. Baccari, A. Salavrakos and R. Augusiak, "Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems (preprint 07/2018)

[abstract:] "Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), defined in the simplest scenario involving two dichotomic measurements, whose all key properties are well understood. While there have been many attempts to generalise it to higher-dimensional quantum systems, quite surprisingly, most of them turn out to be difficult to analyse. In particular, the maximal quantum violation–a key quantity for most device-independent applications–remains unknown except for the simplest cases. Here we propose a new generalisation of the CHSH Bell inequality which preserves several of its attractive features: the maximal quantum value can be computed analytically and can be achieved by the maximally entangled states and mutually unbiased bases. These inequalities involve $d$ measurements settings, each having $d$ outcomes for an arbitrary prime number $d\geq 3$. We then show that in the three-outcome case our Bell inequality is a self-test: it can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which suggests a new weak form of self-testing. The ability to certify high-dimensional MUBs makes them attractive from the device-independent cryptography point of view."

A. Bernal, "On the existence of absolutely maximally entangled states of minimal support II" (preprint 07/2018)

[abstract:] "Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimension $d$ is a power of a prime number. In this paper, an equivalence with latin hypercubes is used to prove that the above sufficient condition fails in the first case in which the local dimension is not a prime power, $d=6$. Results for other values of $d$ are also given."

C.R. de Oliveira and G.Q. Pellegrino, "(De)Localization in the prime Schrödinger operator", J. Phys. A 34(16), L239-L243 (2001)

[abstract:] "It is reported a combined numerical approach to study the localization properties of the one-dimensional tight-binding model with potential modulated along the prime numbers. A localization-delocalization transition was found as function of the potential intensity; it is also argued that there are delocalized states for any value of the potential intensity."

M. Krishna, "xi-zeta relation", Proceedings of the Indian Academy of Sciences 109 (4) (1999) 379-383

[abstract:] "In this note we prove a relation between the Riemann zeta function and the xi function (Krein spectral shift) associated with the Harmonic Oscillator in one dimension. This gives a new integral representation of the zeta function and also a reformulation of the Riemann hypothesis as a question in L1(R)."

M.W. Coffey, "Theta and Riemann xi function representations from harmonic oscillator eigensolutions", Phys. Lett. A 362 (2007) 352-356

[abstract:] "From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula."

A. Córdoba, C.L. Fefferman and L.A. Seco, "A Trigonometric Sum Relevant to the Non-relativistic Theory of Atoms"

[abstract:] "We extend Van der Corput's method for exponential sums to study an oscillatory term appearing in the quantum theory of large atoms. We obtain an interpretation in terms of classical dynamics and we produce sharp asymptotic upper and lower bounds for the oscillations."

C.L. Fefferman and L.A. Seco, "Arithmetic aspects of atomic structures"

C.L. Fefferman and L.A. Seco, "A number-theoretic estimate for the Thomas-Fermi density"

[abstract:] "In this paper we obtain an estimate for the Thomas-Fermi density which plays a role in the analysis of the atomic energy asymptotics. Such estimate has obvious number-theoretic features related to the radial symmetry of a certain Schrödinger operator, and we use number-theoretic methods in our proof. From the technical viewpoint, we also simplify and improve some of the original estimates in the proof of the Dirac-Schwinger correction to the atomic energy asymptotics."

C.L. Fefferman and L.A. Seco, "Interval arithmetic in quantum mechanics"

L.A. Seco, "Number Theory, Classical Mechanics and the Theory of Large Atoms"

C.L. Fefferman and L.A. Seco, "Number Theory and Atomic Densities"

B. Eckhardt, "Eigenvalue statistics in quantum ideal gases"

"The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer $\alpha>1$ one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power $\alpha/(\alpha-1)$. In particular, for $\alpha=3/2$, the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function."

O. Lablée, "Quantum revivals in two degrees of freedom integrable systems : the torus case" (preprint 09/2010)

[abstract:] "The paper deals with the semi-classical behaviour of quantum dynamics for a semi-classical completely integrable system with two degrees of freedom near Liouville regular torus. The phenomomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analysis (limit). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory."

J.F. Clauser, J.P. Dowling, "Factoring integers with Young's $N$-slit interferometer" (preprint 10/2009)

[abstract:] "We show that a Young's $N$ slit interferometer can be used to factor the integer $N$. The device could factor four- or five-digit numbers in a practical fashion. This work shows how number theory may arise in physical problems, and may provide some insight as to how quantum computers can carry out factoring problems by interferometric means."

M.V. Suslov, G.B. Lesovik, G. Blatter, "Quantum abacus for counting and factorizing numbers" (preprint 11/2010)

[abstract:] "We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base 2, base 3, or base d representation. In operating the algorithm, the number n < N = d^K is read into a K-qudit register through its interaction with a stream of n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number n in the base d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semi-classical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where d is a prime number, with a simpler single-shot measurement allows to find the powers of 2, 3, and other primes d in the number n. We show, that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-d systems, d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure a voltage (analog to digital converter) and a simple procedure to entangle multi-particle states."

F. Grosshans, T. Lawson, F. Morain and B. Smith, "Factoring safe semiprimes with a single quantum query" (preprint 11/2015)

[abstract:] "Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a quantum order finding algorithm (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to 1/2, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. We present a new quantum factoring algorithm based on QOFA which is better than SFA at factoring safe semiprimes, an important class of numbers used in RSA encryption (and reputed to be the hardest to factor). With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalising this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm."

J.L. Rosales, "Simulating factorization with a quantum computer" (preprint 05/2015)

[abstract:] "Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would eventually weaken these algorithms. In this paper, one follows Feynman's prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number N into primes. Using Dirac–Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of N in terms of the Euler function f(N), that is quantized. On the other hand, considering N as a parameter of the computer, a Quantum Mechanical Prime Counting Function pQM(x), where x factorizes N, is derived. This function converges to p(x) when N >> x. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone."

S. Tyagi, "Evaluation of exponential sums and Riemann zeta function on quantum computer" (preprint 02/2020)

[abstract:] "We show that exponential sums (ES) of the form \begin{equation*} s(f,N)= \sum_{k=0}^{N-1} \sqrt{w_k} e^{2 \pi i f(k)}, \end{equation*} can be efficiently carried out with a quantum computer (QC). Here $N$ can be exponentially large, $w_k$ are real numbers such that sum $S_w(M)=\sum_{k=0}^{M-1} w_k$ can be calculated in a closed form for any $M$, $S_w(N)=1$ and $f(x)$ is a real function, that is assumed to be easily implementable on a QC. As an application of the technique, we show that Riemann zeta (RZ) function, $\zeta(\sigma+ i t)$ in the critical strip, $\{0 \le \sigma <1, t \in \mathbb{R} \}$, can be obtained in polyLog(t) time. In another setting, we show that RZ function can be obtained with a scaling $t^{1/D}$, where $D \ge 2$ is any integer. These methods provide a vast improvement over the best known classical algorithms; best of which is known to scale as $t^{4/13}$. We present alternative methods to find $\lvert S(f,N) \rvert$ on a QC directly. This method relies on finding the magnitude $A=\lvert \sum_0^{N-1} a_k \rvert$ of a $n$-qubit quantum state with $a_k$ as amplitudes in the computational basis. We present two different ways to do obtain $A$. Finally, a brief discussion of phase/amplitude estimation methods is presented."

A. Sugamoto, "Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy" (preprint 10/2009)

[abstract:] "Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer $n$ is assumed or derived to being proportional to $\ln n$. The one-loop vacuum amplitudes, the free energies and the partition functions at finite temperature of the string-like models are estimated and compared with the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in the free energies is broken down to the additive symmetry of integers, ${\bf Z}_{+}$, after interactions are turned on. In the dynamical model existing behind the zeta function, prepared are the fields labeled by prime numbers. On the other hand the fields in our models are labeled, not by prime numbers but by integers. Nevertheless, we can understand whether a number is prime or not prime by the decay rate, namely by the corresponding particle can decay or can not decay through interactions conserving energy. Among the models proposed, the supersymmetric string-like model has the merit of that the zero point energies are cancelled and the energy levels may be stable against radiative corrections."

J.L. Rosales and V. Martin, "On the quantum simulation of the factorization problem" (preprint 01/2016)

[abstract:] "Feynman's prescription for a quantum computer was to find a Hamitonian for a system that could serve as a computer. Here we concentrate in a system to solve the problem of decomposing a large number $N$ into its prime factors. The spectrum of this computer is exactly calculated obtaining the factors of $N$ from the arithmetic function that represents the energy of the computer.

As a corollary, in the semi-classical large $N$ limit, we compute a new prime counting asymptote $\pi(x|N)$, where $x$ is a candidate to factorize $N$, that has no counterpart in analytic number theory. This rises the conjecture that the quantum solution of factoring obtains prime numbers, thus reaching consistency with Euclid's unique factorization theorem: primes should be quantum numbers of a Feynman's factoring simulator."

H. Mack, M. Bienert, F. Haug, M. Freyberger and W.P. Schleich, "Wave packets can factorize numbers", Phys. Stat. Sol. (B) 233, No. 3 (2002) 408–415.

"We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new, promising and effective method to factorize numbers."

F. Gleisberg, F. Di Pumpo, G. Wolff and W.P. Schleich, "Prime factorization of arbitrary integers with a logarithmic energy spectrum" (preprint 02/2019)

[abstract:] "We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only."

A. Donis-Vela and J.C. Garcia-Escartin, "A quantum primality test with order finding" (preprint 11/2017)

[abstract:] "Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an element of the multiplicative group of integers modulo $N$ with order $N-1$. If one is found, the number is known to be prime. During the test, we can also show most of the times $N$ is composite with certainty (and a witness) or, after $\log\log N$ unsuccessful attempts to find an element of order $N-1$, declare it composite with high probability. The algorithm requires $O((\log n)^2n^3)$ operations for a number $N$ with $n$ bits, which can be reduced to $O(\log\log n(\log n)^3n^2)$ operations in the asymptotic limit if we use fast multiplication."

J.I. Latorre and G. Sierra, "Quantum computation of prime number functions" (preprint 02/2013)

[abstract:] "We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller Rabin primality test. The prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems."

J.I. Latorre and G. Sierra, "There is entanglement in the primes" (preprint 03/2014)

[abstract:] "Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the prime state, which in turn produces new insights in the Hardy–Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a twin prime state shows that this new state is even more entangled than the prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to arithmetics in novel ways."

D. García-Martín, E. Ribas, S. Carrazza, J.I. Latorre and G. Sierra, "The prime state and its quantum relatives" (preprint 05/2020)

[abstract:] "The prime state of $n$ qubits, $|\mathbb{P}_n\rangle$, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than $2^n$. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the quantum Fourier transform of the prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of $n$ qubits, $|\mathbb{P}_n\rangle$ up to $n = 30$ qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision."

J. Ryu, M. Marciniak, M. Wiesniak and M. Zukowski, "Entanglement conditions for integrated-optics multi-port quantum interferometry experiments" (preprint 01/2016)

[abstract:] "Integrated optics allows one to perform interferometric experiments based upon multi-port beam-splitter. To observe entanglement effects one can use multi-mode parametric down-conversion emissions. When the structure of the Hamiltonian governing the emissions has (infinitely) many equivalent Schmidt decompositions into modes (beams), one can have perfect EPR-like correlations of numbers of photons emitted into "conjugate modes" which can be monitored at spatially separated detection stations. We provide series of entanglement conditions for all prime numbers of modes, and show their violations by bright multi-mode squeezed vacuum states. One family of such conditions is given in terms of the usual intensity-related variables. Moreover, we show that an alternative series of conditions expressed in terms averages of observed rates, which is a generalization of the ones given in arXiv:1508.02368, is a much better entanglement indicator. Thus the rates seem to emerge as a powerful concept in quantum optics. Generalizations of the approach are expected."

J. Ryu, M. Marciniak, M. Wieśniak, D. Kaszlikowski and M. Zukowski, "Entanglement conditions involving intensity correlations of optical fields: The case of multi-port interferometry" (preprint 03/2018)

[abstract:] "Normalized quantum Stokes operators introduced in [Phys. Rev. A 95, 042113 (2017)] enable one to better observe non-classical correlations of entangled states of optical fields with undefined photon numbers. For a given run of an experiment the new quantum Stokes operators are defined by the differences of the measured intensities (or photon numbers) at the exits of a polarizer divided by their sum. It is this ratio that is to be averaged, and not the numerator and the denominator separately, as it is in the conventional approach. The new approach allows to construct more robust entanglement indicators against photon-loss noise, which can detect entangled optical states in situations in which witnesses using standard Stokes operators fail. Here we show an extension of this approach beyond phenomena linked with polarization. We discuss EPR-like experiments involving correlations produced by optical beams in a multi-mode bright squeezed vacuum state. EPR-inspired entanglement conditions for all prime numbers of modes are presented. The conditions are much more resistant to noise due to photon loss than similar ones which employ standard Glauber-like intensity, correlations."

T. Olupitan, C. Lei and A. Vourdas, "An analytic function approach to weak mutually unbiased bases" (preprint 07/2016) "Quantum systems with variables in $\mathbb{Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is $1/\sqrt{k}$ (where $k\vert d$) or $0$. The second is maximal lines through the origin in the $\mathbb{Z}(d)\times \mathbb{Z}(d)$ phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where $d = p_1\times p_2$, where $p_1$, $p_2$ are odd prime numbers different from each other, is considered."

M. Asoudeh and V. Karimipour, "Quantum secret sharing and random hopping: Using single states instead of entanglement" (preprint 06/2015)

[abstract:] "Quantum protocols for secret sharing usually rely on multi-party entanglement which with present technology is very difficult to achieve. Recently it has been shown that sequential manipulation and communication of a single $d$-level state can do the same task of secret sharing between $N$ parties, hence alleviating the need for entanglement. However the suggested protocol which is based on using mutually unbiased bases, works only when $d$ is a prime number. We propose a new sequential protocol which is valid for any $d$."

R.V. Ramos, "Quantum physics, algorithmic information theory and the Riemanns hypothesis" (preprint 12/2017)

[abstract:] "In the present work the Riemann hypothesis (RH) is discussed from four different perspectives. In the first case, coherent states and the Stengers approximation to Riemann-zeta function are used to show that RH avoids an indeterminacy of the type 0/0 in the inner product of two coherent states. In the second case, the Hilbert-Pólya conjecture with a quantum circuit is considered. In the third case, randomness, entanglement and the Moebius function are used to discuss the RH. At last, in the fourth case, the RH is discussed by inverting the first derivative of the Chebyshev function. The results obtained reinforce the belief that the RH is true."

F.V. Mendes and R.V. Ramos, "Quantum sequence states" (preprint 08/2014)

[abstract:] "In a recent paper it has been shown how to create a quantum state related to the prime number sequence using Grover's algorithm. Moreover, its multiqubit entanglement was analyzed. In the present work, we compare the multiqubit entanglement of several quantum sequence states as well we study the feasibility of producing such states using Grover's algorithm."

R.V. Ramos and F.V. Mendes, "Riemannian quantum circuit" (preprint 05/2013)

[abstract:] "Number theory is an abstract mathematical field that has found a fertile environment for development in theoretical physics. In particular, several physical systems were related to the zeros of the Riemann-zeta function. In this work we present the theory of a quantum circuit related to a finite number of zeros of the Riemann-zeta function. The existence of such circuit will permit in the future the solution of some number theory problems through the realization of quantum algorithms based on those zeros. "

J.A. Smolin and G. Smith and A. Vargo, "Pretending to factor large numbers on a quantum computer" (preprint 01/2013)

[abstract:] "Shor's algorithm for factoring in polynomial time on a quantum computer gives an enormous advantage over all known classical factoring algorithm. We demonstrate how to factor products of large prime numbers using a compiled version of Shor's quantum factoring algorithm. Our technique can factor all products of $p,q$ such that $p,q$ are unequal primes greater than two, runs in constant time, and requires only two coherent qubits. This illustrates that the correct measure of difficulty when implementing Shor's algorithm is not the size of number factored, but the length of the period found."

J.S. Kim, E. Bae and S. Lee, "Quantum computational algorithm for hidden symmetry subgroup problems on semi-direct product of cyclic groups" (preprint 07/2013)

[abstract:] "We characterize the algebraic structure of semi-direct product of cyclic groups, $\Z_{N}\rtimes\Z_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum computational algorithm solving hidden symmetry subgroup problem of the groups."

M. Marvian and V. Karimipour, "Secure quantum carriers for distributing classical secrets and quantum states for a general threshold scheme" (preprint 07/2010)

[abstract:] "We provide a secure quantum carrier for distributing a secret (classical symbol encoded into a state or a quantum state) among $n$ parties according to a $(k,n)$ threshold scheme, where $2k-1$ is a prime number. The quantum carrier \cite{bk} is an entangled state which is shared between all the participants, and is not measured at any stage. Quantum states are uploaded to the carrier and downloaded from it by the receivers. The quantum carrier is secure against eavesdropping by local Hadamard actions of the participants which leave it invariant. Contrary to measurement-based secret sharing schemes, our protocol can be used for sharing predetermined strings of symbols and quantum states."

H. Bombin and M.A. Martin-Delgado, "Entanglement distillation protocols and number theory" (preprint 03/05)

[abstract:] "We show that the analysis of entanglement distillation protocols for qudits of arbitrary dimension $D$ benefits from applying basic concepts from number theory, since the set $\zdn$ associated to Bell diagonal states is a module rather than a vector space. We find that a partition of $\zdn$ into divisor classes characterizes the invariant properties of mixed Bell diagonal states under local permutations. We construct a very general class of recursion protocols by means of unitary operations implementing these local permutations. We study these distillation protocols depending on whether we use twirling operations in the intermediate steps or not, and we study them both analytically and numerically with Monte Carlo methods. In the absence of twirling operations, we construct extensions of the quantum privacy algorithms valid for secure communications with qudits of any dimension $D$. When $D$ is a prime number, we show that distillation protocols are optimal both qualitatively and quantitatively."

Y. Li and M. Ying, "Debugging quantum processes using monitoring measurements" (preprint 03/2014)

[abstract:] "Since observation on a quantum system may cause the system state collapse, it is usually hard to find a way to monitor a quantum process, which is a quantum system that continuously evolves. We propose a protocol that can debug a quantum process by monitoring, but not disturb the evolution of the system. This protocol consists of an error detector and a debugging strategy. The detector is a projection operator that is orthogonal to the anticipated system state at a sequence of time points, and the strategy is used to specify these time points. As an example, we show how to debug the computational process of quantum search using this protocol. By applying the Skolem–Mahler–Lech theorem in algebraic number theory, we find an algorithm to construct all of the debugging protocols for quantum processes of time independent Hamiltonians."

J. Ossorio-Castillo and J.M. Tornero, "An adiabatic quantum algorithm for the Frobenius problem" (preprint 07/2019)

[abstract:] "The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an adiabatic quantum algorithm which solves it, using the so-called Apéry set of a numerical semigroup, via a translation into a QUBO problem. The algorithm has been specifically designed to run in a D-Wave 2X machine."

Yu.I. Bogdanov, N.A. Bogdanova, D.V. Fastovets and V.F. Lukichev, "Representation of Boolean functions in terms of quantum computation" (preprint 06/2019)

[abstract:] "The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic ($k$-valued logic, where $k$ is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware."

D. Li, "Entanglement classification via integer partitions" (preprint 05/2019)

[abstract:] "In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs."

A. Klappenecker, M. Roetteler, I. Shparlinski and A. Winterhof, "On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states" (preprint 03/05)

[abstract:] "We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory."

A.M. Childs, D. Jao, V. Soukharev, "Constructing elliptic curve isogenies in quantum subexponential time" (preprint 12/2010)

[abstract:] "Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest known classical algorithm takes exponential time, and prior to our work no faster quantum algorithm was known. Recently, public-key cryptosystems based on the presumed hardness of this problem have been proposed as candidates for post-quantum cryptography. In this paper, we give a subexponential-time quantum algorithm for constructing isogenies, assuming the Generalized Riemann Hypothesis (but with no other assumptions). This result suggests that isogeny-based cryptosystems may be uncompetitive with more mainstream quantum-resistant cryptosystems such as lattice-based cryptosystems. As part of our algorithm, we also obtain a second result of independent interest: we provide a new subexponential-time classical algorithm for evaluating a horizontal isogeny given its kernel ideal, assuming (only) GRH, eliminating the heuristic assumptions required by prior algorithms."

Y. Chen, A. Prakash and T.-C. Wei, "Universal quantum computing using $(\mathbb_d)^3$ symmetry-protected topologically ordered states" (preprint 11/2017)

[abstract:] "Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2\times$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2\times$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than $2$. The same construction also holds for other $3$-colorable lattices, including the Union-Jack lattice."

C. Archer, "There is no generalization of known formulas for mutually unbiased bases" (preprint 12/03)

[abstract:] "In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products <ai|bj> have the same modulus N-1/2. This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most N+1 and various constructions of N+1 such bases have been found when N is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings.We then prove that there exists a set of N+1 mutually unbiased bases described by such formulas, if and only if N is a power of a prime number."

A. Fernández-Pérez, A.B. Klimov and C. Saavedra, "Quantum process reconstruction based on mutually unbiased basis" (preprint 04/2011)

[abstract:] "We study a quantum process reconstruction based on the use of mutually unbiased projectors (MUB-projectors) as input states for a D-dimensional quantum system, with D being a power of a prime number. This approach connects the results of quantum-state tomography using mutually unbiased bases (MUB) with the coefficients of a quantum process, expanded in terms of MUB-projectors. We also study the performance of the reconstruction scheme against random errors when measuring probabilities at the MUB-projectors."

I. Perito, A. Roncaglia and A. Bendersky, "Selective and efficient quantum process tomography in arbitrary finite dimension" (preprint 08/2018)

[abstract:] "The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows to acknowledge errors in the implementations of quantum algorithms; on the other, it allows to charcaterize unknown processes occurring in Nature. In previous works [Bendersky, Pastawski and Paz. Phys. Rev. Lett. 100, 190403 (2008) and Phys. Rev. A 80, 032116 (2009)] it was introduced a method to selectively and efficiently measure any given coefficient from the matrix description of a quantum channel. However, this method heavily relies on the construction of maximal sets of mutually unbiased bases (MUBs), which are known to exist only when the dimension of the Hilbert space is the power of a prime number. In this article, we lift the requirement on the dimension by presenting two variations of the method that work on arbitrary finite dimensions: one uses tensor products of maximally sets of MUBs, and the other uses a dimensional cutoff of a higher prime power dimension."

X. F. Liu and C. P. Sun, "On the relative quantum entanglement with respect to tensor product structure" (preprint, 10/04)

[abstract:] "Mathematical foundation of the novel concept of quantum tensor product by Zanardi et. al. is rigorously established. The concept of relative quantum entanglement is naturally introduced and its meaning is made clear both mathematically and physically. For a finite or an infinite dimensional vector space $W$ the so called tensor product partition (TPP) is introduced on $End(W)$, the set of endmorphisms of $W$, and a natural correspondence is constructed between the set of TPP's of $End(W)$ and the set of tensor product structures (TPS's) of $W$. As a byproduct, it is shown that an arbitrarily given wave function belonging to an n-dimensional Hilbert space, n being not a prime number, can be interpreted as a separable state with respect to some man-made TPS, and thus a quantum entangled state of a many-body system with respect to the "God-given" TPS can be regarded as a quantum state without entanglement in some sense. The concept of standard set of observables is also introduced to probe the underlying structure of the object TPP and to establish its connection with practical physical measurement."

M. Revzen and F.C. Khanna, "von Neumann lattices in finite dimensions Hilbert spaces" (preprint 05/2008)

[abstract:] "The prime number decomposition of a finite dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can be viewed as two distinct degrees freedom. These, Schwinger's quantum degrees of freedom, are uniquely related to a von Neumann lattices in the phase space that characterizes the Hilbert space and specifies the simultaneous definitions of both (modular) positions and (modular) momenta. The area in phase space for each quantum state in each of these quantum degrees of freedom, is shown to be exactly $h$, Planck's constant."

I. Bengtsson, "How much complementarity?" (preprint 02/2012)

[abstract:] "Bohr placed complementary bases at the mathematical centre point of his view of quantum mechanics. On the technical side then my question translates into that of classifying complex Hadamard matrices. Recent work (with Barros e Sa) shows that the answer depends heavily on the prime number decomposition of the Hilbert space. By implication so does the geometry of quantum state space."

I. Bengtsson, "SICs: Some explanations" (preprint 04/2020)

[abstract:] "The problem of constructing maximal equiangular tight frames or SICs was raised by Zauner in 1998. Four years ago it was realized that the problem is closely connected to a major open problem in number theory. We discuss why such a connection was perhaps to be expected, and give a simplified sketch of some developments that have taken place in the past four years. The aim, so far unfulfilled, is to prove existence of SICs in an infinite sequence of dimensions."

P. Amore, "A method for classical and quantum mechanics" (preprint 11/04)

[abstract:] "In many physical problems it is not possible to find an exact solution. However, when some parameter in the problem is small, one can obtain an approximate solution by expanding in this parameter. This is the basis of perturbative methods, which have been applied and developed practically in all areas of Physics. Unfortunately many interesting problems in Physics are of non-perturbative nature and it is not possible to gain insight on these problems only on the basis of perturbation theory: as a matter of fact it often happens that the perturbative series are not even convergent.

In this paper we will describe a method which allows to obtain arbitrarily precise analytical approximations for the period of a classical oscillator. The same method is then also applied to obtain an analytical approximation to the spectrum of a quantum anharmonic potential by using it with the WKB method. In all these cases we observe exponential rates of convergence to the exact solutions. An application of the method to obtain a fastly convergent series for the Riemann zeta function is also discussed."

G. Gutin, N.S. Jones, A. Rafiey, S. Severini and A. Yeo, "Mediated digraphs and quantum nonlocality" (preprint 11/04)

[abstract:] "A digraph D=(V,A) is mediated if, for each pair x,y of distinct vertices of D, either xy belongs to A or yx belongs to A or there is a vertex z such that both xz,yz belong to A. For a digraph D, DELTA(D) is the maximum in-degree of a vertex in D. The "nth mediation number" mu(n) is the minimum of DELTA(D) over all mediated digraphs on n vertices. Mediated digraphs and mu(n) are of interest in the study of quantum nonlocality. We obtain a lower bound f(n) for mu(n) and determine infinite sequences of values of n for which mu(n)=f(n) and mu(n)>f(n), respectively. We derive upper bounds for mu(n) and prove that mu(n)=f(n)(1+o(1)). We conjecture that there is a constant c such that mu(n)<f(n)+c. Methods and results of graph theory, design theory and number theory are used."

S. Egger né Endres and F. Steiner, "An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics" (preprint 04/2011)

[abstract:] "We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true."

A. Granville and K. Soundararajan, "An uncertainty principle for arithmetic sequences", Annals of Mathematics, 165 (2007) 593–635

[abstract:] "Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ''well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the ''uncertainty principle''. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples."

There is also another way that a "multiplicative version" of the uncertainty principle is connected with prime numbers, as observed by P. Pollack here.

R.V. Ramos, "Riemann Hypothesis as an uncertainty relation" (preprint 04/2013)

[abstract:] "Physics is a fertile environment for trying to solve some number theory problems. In particular, several tentative of linking the zeros of the Riemann-zeta function with physical phenomena were reported. In this work, the Riemann operator is introduced and used to transform the Riemann's hypothesis in a Heisenberg-type uncertainty relation, offering a new way for studying the zeros of Riemann's function."

W.G. Ritter, "On the number of representations providing noiseless subsystems" (accepted for publication in Physical Review A)

[abstract:] "This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function fd(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory."

P. Benioff, "Space of quantum theory representations of natural numbers, integers, and rational numbers" (preprint 04/2007)

[abstract:] "This paper extends earlier work on quantum theory representations of natural numbers $N$, integers $I$, and rational numbers $Ra$ to describe a space of these representations and transformations on the space. The space is parameterized by $4$-tuple points in a parameter set. Each point, $(k,m,h,g)$, labels a specific representation of $X = N$, $I, $Ra$ as a Fock space $F^{X}_{k,m,h}$ of states of finite length strings of qukits $q$ and a string state basis $B^{X}_{k,m,h,g}$. The pair $(m,h)$ locates the $q$ string in a square integer lattice $I \times I$, $k$ is the $q$ base, and the function $g$ fixes the gauge or basis states for each $q$. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator $W_{k',k}$, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for $N$, $I$, and $Ra$ under any transformation is discussed along with the dependence of the properties of $W_{k',k}$ on the prime factors of $k'$ and $k$. This suggests that one consider prime number $q$'s, $q_2$, $q_3$, $q_5$, etc. as elementary and the base $k$ $q$'s as composites of the prime number $q$'s."

F. Bouzeffour and M. Garayev, "Some aspects of number theory related to phase operators" (preprint 12/2018)

[abstract:] "We first extend the multiplicativity property of arithmetic functions to the setting of operators on the Fock space. Secondly, we use phase operators to get representation of some extended arithmetic functions by operators on the Hardy space. Finally, we show that radial limits to the boundary of the unit disc in the Hardy space is useful in order to go back to the classical arithmetic functions. Our approach can be understudied as a transition from the classical number theory to quantum setting."

M. Planat, F. Anselmi and P. Solé, "Pauli graphs, Riemann hypothesis, Goldbach pairs" (preprint 03/2011)

[abstract:] "Let consider the Pauli group $\mathcal{P}_q=$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure."

S. Egger né Endres and F. Steiner, "An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics" (preprint 04/2011)

[abstract:] "We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true."

A.O. Pittenger and M.H. Rubin, "Wigner functions and separability for finite systems" (preprint 01/05)

[abstract:] "A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension pn where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p2n points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p = 2) and our technique fails to establish the separability property for more than two qubits."

L. Kocia and P. Love, "Stationary phase method in discrete Wigner functions and classical simulation of quantum circuits" (preprint 10/2018)

[abstract:] "We apply the periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension using results from $p$-adic number theory. We derive the Wigner–Weyl–Moyal (WWM) formalism with higher order $\hbar$ corrections representing contextual corrections to non-contextual Clifford operations. We apply this formalism to a subset of unitaries that include diagonal gates such as the $\frac{\pi}{8}$ gates. We characterize the stationary phase critical points as a quantum resource injecting contextuality and show that this resource allows for the replacement of the $p^{2t}$ points that represent $t$ magic state Wigner functions on $p$-dimensional qudits by $\leq p^t$ points. We find that the $\frac{\pi}{8}$ gate introduces the smallest higher order $\hbar$ correction possible, requiring the lowest number of additional critical points compared to the Clifford gates. We then establish a relationship between the stabilizer rank of states and the number of critical points necessary to treat them in the WWM formalism. This allows us to exploit the stabilizer rank decomposition of two qutrit $\frac{\pi}{8}$ gates to develop a classical strong simulation of a single qutrit marginal on $t$ qutrit $\frac{\pi}{8}$ gates that are followed by Clifford evolution, and show that this only requires calculating $3^{\frac{t}{2}+1}$ critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as $\approx 3^{0.8t}$ for $10^{-2}$ precision) for any number of $\frac{\pi}{8}$ gates to full precision."

R.W. Johnson, "Quantum mechanics associated with a finite group", submitted to Intern. J. Theor. Phys.

[abstract:] "I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space containing elements f x f for f an element in the regular representation of a given finite group G. The Hermitian portion of f x f is the Wigner distribution of f whose convolution with a test function leads to a mathematical description of the quantum measurement process. Starting with the Jacobi group that is formed from the semidirect product of the Heisenberg group with its automorphism group SL(2,F{N}) for N an odd prime number I show that the classical phase space is the first order term in a series of subspaces of the Hermitian portion of f x f that are stable under SL(2,F{N}). I define a derivative that is analogous to a pseudodifferential operator to enable a treatment that parallels the continuum case. I give a new derivation of the Schrödinger-Weil representation of the Jacobi group."

M. Marcolli and A. Connes, "Q-lattices: quantum statistical mechanics and Galois theory", Journal of Geometry and Physics 56 no. 1 (2006) 2–23

G. Cornelissen and M. Marcolli, "Quantum Statistical Mechanics, $L$-series and Anabelian Geometry" (preprint 09/2010)

[abstract:] "It is known that two number fields with the same Dedekind zeta function are not necessarily isomorphic. The zeta function of a number field can be interpreted as the partition function of an associated quantum statistical mechanical system, which is a C*-algebra with a one parameter group of automorphisms, built from Artin reciprocity. In the first part of this paper, we prove that isomorphism of number fields is the same as isomorphism of these associated systems. Considering the systems as noncommutative analogues of topological spaces, this result can be seen as another version of Grothendieck's "anabelian" program, much like the Neukirch--Uchida theorem characterizes isomorphism of number fields by topological isomorphism of their associated absolute Galois groups. In the second part of the paper, we use these systems to prove the following. If there is an isomorphism of character groups (viz., Pontrjagin duals) of the abelianized Galois groups of the two number fields that induces an equality of all corresponding $L$-series (not just the zeta function), then the number fields are isomorphic.This is also equivalent to the purely algebraic statement that there exists a topological group isomorphism as a above and a norm-preserving group isomorphism between the ideals of the fields that is compatible with the Artin maps via the other map."

G. Cornelissen, "Number theory and physics, an eternal rusty braid", Eidnhoven Mathematics Colloquiums, 9th November 2011

[abstract:] "I will describe joint work with Matilde Marcolli in which we apply ideas from quantum statistical mechanics and dynamical systems to solve the number theoretical analogue of the problem how to hear the shape of a drum".

G. Mussardo, "The quantum mechanical potential for the prime numbers", preprint ISAS/EP/97/153;see also R. Matthews, New Scientist, 10 January 1998, p.18.

"A simple criterion is derived in order that a number sequence Sn is a permitted spectrum of a quantised system. The sequence of prime numbers fulfils the criterion. The existence of such a potential implies that primality testing can in principle be resolved by the sole use of physical laws".

G. Mussardo, A. Trombettoni and Z. Zhang, "Prime suspects in a quantum ladder" (preprint 05/2020)

[abstract:] "In this paper we set up a suggestive number theory interpretation of a quantum ladder system made of $\mathcal{N}$ coupled chains of spin $1/2$. Using the hard-core boson representation, we associate to the spins $\sigma_a$ along the chains the prime numbers $p_a$ so that the chains become quantum registers for square-free integers. The Hamiltonian of the system consists of a hopping term and a magnetic field along the chains, together with a repulsion rung interaction and a permutation term between next neighborhood chains. The system has various phases, among which there is one whose ground state is a coherent superposition of the first $\mathcal{N}$ prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics."

P.W. Shor, "Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer", SIAM J. Computing 26 (1997) 1484-1509.

P.W. Shor, "Quantum computing", Documenta Mathematica Extra Volume ICM I (1998) 467-486

"...A quantum computer is a hypothetical machine based on quantum mechanics. We explain quantum computing, and give an algorithm for prime factorization on a quantum computer that runs aymptotically much faster than the best known algorithm on a digital computer...

...In 1994, I showed that a quantum computer could factor large numbers in time polynomial in the length of the numbers, a nearly exponential speed-up over classical algorithms...the connection of quantum mechanics with number theory was itself surprising..."

using quantum computation to factorise integers

D.N. Goncalves and R. Portugal, "Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups" (preprint 04/2011)

[abstract:] "The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group $\Z_{p^r} \rtimes \Z_{q^s}$, when $p^r/q= \up{poly}(\log p^r)$, where $p$, $q$ are any odd prime numbers and $r, s$ are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups."

Hong Wang, Zhi Ma, "Quantum algorithms for unit group and principal ideal problem" (preprint 04/2010)

[abstract:] "Computing the unit group and solving the principal ideal problem for a number field are two of the main tasks in computational algebraic number theory. This paper proposes efficient quantum algorithms for these two problems when the number field has constant degree. We improve these algorithms proposed by Hallgren by using a period function which is not one-to-one on its fundamental period. Furthermore, given access to a function which encodes the lattice, a new method to compute the basis of an unknown real-valued lattice is presented. "

B. Grohmann, "On the existence of certain quantum algorithms" (preprint 04/2009)

[abstract:] "We investigate the question if quantum algorithms exist that compute the maximum of a set of conjugated elements of a given number field in quantum polynomial time. We will relate the existence of these algorithms for a certain family of number fields to an open conjecture from elementary number theory."

M. Revzen, F.C. Khanna, A. Mann, J. Zak, "Factorizations and physical representations" (preprint 08/05)

[abstract:] "Hilbert space in M dimensions is shown explicitly to accommodate representations that reflect the prime numbers decomposition of M. Representations that exhibit the factorization of M into two relatively prime numbers: the kq representation (J. Zak, Phys. Today, 23 (2), 51 (1970)), and related representations termed q1q2 representations (together with their conjugates) are analysed, as well as a representation that exhibits the complete factorization of M. In this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up M."

M. Revzen, A. Mann and J. Zak, "Physics of factorization" (preprint 03/05)

[abstract:] "The N distinct prime numbers that make up a composite number M allow 2N-1 bi partioning into two relatively prime factors. Each such pair defines a pair of conjugate representations. These pairs of conjugate representations, each of which spans the M dimensional space are the familiar complete sets of Zak transforms (J. Zak, Phys. Rev. Let. 19, 1385 (1967)) which are the most natural representations for periodic systems. Here we show their relevance to factorizations. An example is provided for the manifestation of the factorization."

J. Maurice Rojas, "A number theoretic interpolation between quantum and classical complexity classes" (preprint 04/2006)

[abstract:] "We reveal a natural algebraic problem whose complexity appears to interpolate between the well-known complexity classes BQP and NP: (*) Decide whether a univariate polynomial with exactly $m$ monomial terms has a $p$-adic rational root. In particular, we show that while (*) is doable in quantum randomized polynomial time when $m=2$ (and no classical randomized polynomial time algorithm is known), (*) is nearly NP-hard for general m: Under a plausible hypothesis involving primes in arithmetic progression (implied by the Generalized Riemann Hypothesis for certain cyclotomic fields), a randomized polynomial time algorithm for (*) would imply the widely disbelieved inclusion $NP \subseteq BPP$. This type of quantum/classical interpolation phenomenon appears to new."

C.M.M. Cosme and R. Portugal, "Quantum algorithm for the hidden subgroup problem on a class of semidirect product groups" (preprint 03/2007)

[abstract:] "We present an efficient quantum algorithm for the hidden subgroup problem (HSP) on the semidirect product of cyclic groups Z_p^r and Z_p^2, where p is any odd prime number and $r$ is any integer such that r>4. This quantum algorithm is exponentially faster than any classical algorithm for the same purpose."

D.N. Goncalves and R. Portugal, "Solution to the Hidden Subgroup Problem for a Class of Noncommutative Groups" (preprint 04/2011)

[abstract:] "The hidden subgroup problem (HSP) plays an important role in quantum computation, because many quantum algorithms that are exponentially faster than classical algorithms can be casted in the HSP structure. In this paper, we present a new polynomial-time quantum algorithm that solves the HSP over the group $\Z_{p^r} \rtimes \Z_{q^s}$, when $p^r/q= \up{poly}(\log p^r)$, where $p$, $q$ are any odd prime numbers and $r, s$ are any positive integers. To find the hidden subgroup, our algorithm uses the abelian quantum Fourier transform and a reduction procedure that simplifies the problem to find cyclic subgroups."

I. Cherednik, "On q-analogues of Riemann's zeta"

[abstract:] "In the paper, we introduce q-deformations of the Riemann zeta function, extend them to the whole complex plane, and establish certain estimates of the number of roots. The construction is based on the recent difference generalization of the Harish-Chandra theory of zonal spherical functions. We also discuss numerical results, which indicate that the location of the zeros of the q-zeta functions is far from random."

M.N. Tran, M.V.N. Murthy, R.K. Bhaduri, "On the quantum density of states and partitioning an integer"

[abstract:] "This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic (N -> infinity) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for ps(n), the latter being the number of partitions of n into a sum of s-th powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain ds(n) for distinct partitions. We find that the distinct square partitions d2(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdös-Lehner formula for restricted partitions for the s = 1 case by our method, we generalize it to obtain a new formula for distinct restricted partitions."

H.C. Rosu, J.M. Moran-Mirabal, M. Planat, "Milne phase for the Coulomb quantum problem related to Riemann's hypothesis" (Group 24: Physical and Mathematical Aspects of Symmetries, Eds. J.-P. Gazeau et. al., IOP Conf. Series No. 173 (2003) 695-697)

[abstract:] "We use the Milne phase function in the continuum part of the spectrum of the particular Coulomb problem that has been employed by Bhaduri, Khare, and Law as an equivalent physical way for calculating the density of zeros of the Riemann's function on the critical line. The Milne function seems to be a promising approximate method to calculate the density of prime numbers."

M. Planat, H.C. Rosu, "Cyclotomy and Ramanujan sums in quantum phase locking", Phys. Lett. A 315 (2003) 1-5

[abstract:] "Phase locking governs the phase noise in classical clocks through effects described in precise mathematical terms. We seek here a quantum counterpart of these effects by working in a finite Hilbert space. We use a coprimality condition to define phase-locked quantum states and the corresponding Pegg-Barnett type phase operator. Cyclotomic symmetries in matrix elements are revealed and related to Ramanujan sums in the theory of prime numbers. The phase-number commutator vanishes as in the classical case, but a new type of quantum phase noise emerges in expectation values of phase and phase variance. The employed mathematical procedures also emphasize the isomorphism between algebraic number theory and the theory of quantum entanglement."

M. Planat, "Huyghens, Bohr, Riemann and Galois: Phase-Locking" (written in relation to the ICSSUR '05 conference held in Besancon, France - to be published at a special issue of IJMPB)

[abstract:] "Several mathematical views of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find them related respectively to the Riemann zeta function and to incomplete Gauss sums."

D. Ellinas and E.G. Floratos, "Prime decomposition and correlation measure of finite quantum systems"

E.G. Floratas, et. al. - work on "finite quantum mechanics"
(explanation and bibliography)

Sze Kui Ng, "A computation of the mass spectrum of mesons and baryons"

[abstract:] "In this paper we give a computation of the mass spectrum of mesons and baryons. By this computation we show that there is a consecutive numbering of the mass spectrum of mesons and baryons. We show that in this numbering many stable mesons and baryons are assigned with a prime number."

Sze Kui Ng, "On a classification of mesons"

[abstract:]" We give a mass formula for computing the mass spectrum of mesons. By this formula we show that there are many mesons with their masses corresponding to a prime number. In particular we show that all strange mesons are with their masses corresponding to a prime number. With these prime numbers indexing the mesons we give a classification of mesons. We set up a knot model of mesons to derive this mass formula. In this knot model mesons and their anti-particles are modeled by knots and their mirror images respectively. Then the amphichiral knots which are equivalent to their mirror images are used to model mesons which are identical with their anti-particles. With this knot model we show that there is a periodic phenomenon in the classification of mesons such that the starting nonet and the ending nonet are nonets of pseudoscalar mesons with the pi meson modelled by an amphichiral knot. From this periodic phenomenon we give a theoretical argument for the existence of charm-anticharm mesons."

Sze Kui Ng, "Knot model of pseudoscalar and vector mesons" (preprint 03/2007)

[author's description:] "In this paper I give a quantum knot model of mesons where prime knots are assigned with prime numbers. These prime numbers are from the masses of the mesons. From this quantum knot model we then have a closed relation between prime numbers, prime knots and mesons. For example the pi meson is modeled by the prime knot 4_1 which is assigned with the prime number 3."

S. Matsutani, Y. Ônishi, "Wave-particle complementarity and reciprocity of Gauss sums on Talbot effects"

[Abstract:] "Berry and Klein (J. Mod. Opt. (1997) 43 2139-2164) showed that the Talbot effects in classical optics are naturally explained by Gauss sums studied in number theory. Their result was based on Helmholtz equation. In this article, we explain the effect based on Fresnel integral also by Gauss sums. These two explanations are shown to agree with by the reciprocity law of Gauss sums. The relation between this agreement and the wave-particle complementarity is also discussed."

S. Matsutani, "Gauss optics and Gauss sum on an optical phenomena" (preprint 03/2008)

[abstract:] "In the previous article (Found Phys. Lett. 16 325-341), we showed that Gauss reciprocity is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for the finite group."

S. Matsutani, "p-adic difference-difference Lotka-Volterra equation and ultra-discrete limit", Int. J. Math. and Math. Sci. 27 (2001) 251-260

[abstract:] "We study the difference-difference Lotka-Volterra equations in p-adic number space and its p-adic valuation version. We point out that the structure of the space given by taking the ultra-discrete limit is the same as that of the p-adic valuation space. Since ultra-discrete limit can be regarded as a classical limit of a quantum object, it implies that a correspondence between classical and quantum objects might be associated with valuation theory."

S. Matsutani, "Lotka-Volterra equation over a finite ring $\mathbb{Z}/p^N \mathbb{Z}$", J. Phys. A 34 (2001) 10737-10744

[abstract:] "The discrete Lotka-Volterra equation over $p$-adic space was constructed since $p$-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking the ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations given in my previous paper (Matsutani, S 2001 Int. J. Math. Math. Sci.). In this paper, using the natural projection from a $p$-adic integer to a ring $\mathbb{Z}/p^N \mathbb{Z}$, a soliton equation is defined over the ring. Numerical computations show that it behaves regularly."

R. De Luca, G.Gargiulo and F. Romeo, "Number theory implications on physical properties of elementary cubic networks of Josephson junctions", Phys. Rev. B 68 (2003) 092511

[abstract:] "Number theory concepts are used to investigate the periodicity properties of the voltage vs applied flux curves of elementary cubic networks of Josephson junctions. It is found that equatorial gaps appearing on the unitary sphere, on which points representing the directions in space for which these curves show periodicity are collected, can be understood by means of Gauss condition on the sum of the squares of three integers."

V. Varadarajan, "Some remarks on arithmetic physics"

(Abstract) "There have been some recent speculations on connections between quantum theory and modern number theory. At the boldest level these suggest that there are two ways of viewing the quantum world, the usual and the arithmetic, which are in some sense complementary. At a more conservative level they suggest that there is much mathematical interest in examining structures which are important in quantum theory and analyze to what extent they make sense when the real and complex fields are replaced by the more unconventional fields and rings, like finite or nonarchimedean fields and adele rings, that arise in number theory. This paper explores some aspects of these questions."

V.S.Varadarajan, "Arithmetic quantum physics: why, what and whither", Proc. Steklov Inst. Math. 245 (2004) 258-265.

Z. Rudnick, "Value distribution for eigenfunctions of desymmetrized quantum maps" (preprint 01/2001)

"We study the value distribution and extreme values of eigenfunctions for the "quantized cat map". This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called "Hecke operators". The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call "Hecke eigenfunctions". In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the "split primes" for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil's theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions."

Y. Abe, "Abelian Chern–Simons theory on the torus and physical views on the Hecke operators" (preprint 04/2018)

[abstract:] "In the previous paper [arXiv:1711.07122], we show that a holomorphic zero-mode wave function in abelian Chern–Simons theory on the torus can be considered as a quantum version of a modular form of weight 2. Motivated by this result, in this paper we consider an action of a Hecke operator on such a wave function from a gauge theoretic perspective. This leads us to obtain some physical views on the Hecke operators in number theory."

M.C. Gutzwiller, "Stochastic behavior in quantum scattering", Physica D: Nonlinear Phenomena 7 (1983) 341-355

[abstract:] "A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincar model of hyperbolic geometry, the solutions of Schrödinger's equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemann's zeta function on the imaginary axis, at a distance of from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimick any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemann's zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned."

K. Bitar, "A study of the Riemann zeta function"

"Using moment integrals over the zeta function we were. . . able to derive expressions for the distribution of the absolute value of the zeta function and its logarithm. These turn out to be expressible as inverse Mellin transforms over well known functions.

Knowing these distributions allows the use of the zeta function in evaluating the path integrals for quantum mechanical systems. We have tested this on simple systems such as the anharmonic oscillator with good results. Further more since the zeta function is an analytic function with known properties its use in these applications may lead to a definition of the path integral in the continuum."

K. Bitar, "Path integrals and Voronin's theorem on the universality of the Riemann zeta function"

J. Twamey and G.J. Milburn, "The quantum Mellin transform", New J. Phys. 8 (2006) 328

[abstract:] "We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to "Hyperbolic phase space". We show that this new unitary change of basis from the position x on the half line to the Hyperbolic momentum $p_\eta$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_\eta$. We utilise this new transform to find quantum wavefunctions whose Hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann Zeta function. We finally give possible physical realisations to perform an indirect measurement of the Hyperbolic momentum of a quantum system on the half-line."

W. Merkel, H. Mack, W.P. Schleich, E. Lutz, G.G. Paulus, B. Girard "Chirping a two-photon transition in a multi-state ladder" (preprint 02/2007)

[abstract:] "We consider a two-photon transition in a specific ladder system driven by a chirped laser pulse. In the weak field limit, we find that the excited state probability amplitude arises due to interference of multiple quantum paths which are weighted by quadratic phase factors. The excited state population has the form of a Gauss sum which plays a prominent role in number theory."

J.C. Phillips, "Microscopic origin of collective exponentially small resistance states" (preprint, 03/03)

[abstract:] "The formation of "zero" (exponentially small) resistance states (ESRS) in high mobility two-dimensional electron systems (2DES) in a static magnetic field B and subjected to strong microwave (MW) radiation has attracted great theoretical interest. These states appear to be associated with a new kind of energy gap $\Delta$. Here I show that the energy gap $\Delta$ is explained by a microscopic quantum model that involves the Prime Number Theorem, hitherto reserved for only mathematical contexts. The model also contains the zeroes of the zeta function, and explains the physical origin of the Riemann hypothesis."

D. Kouzoudis, "Heisenberg s = ring consisting of a prime number of atoms", Journal of Magnetism and Magnetic Materials 173 (1997) 259-265

[abstract:] "In this work it will be shown that the dimensionality of the eigenvalue problem of a Heisenberg s = ring with a prime number N of atoms can be reduced by a factor of N. This makes small systems such as N = 5 and 7 particularly easy to solve analytically for the case of nearest-neighbor interactions, without the use of Bethe's ansatz, as well as for the general case of couplings beyond nearest neighbors. Exact expressions are given for both the magnon dispersion relations and the eigenvectors."

I. Antoniou and Z. Suchanecki, "Quantum systems with fractal spectra", Chaos, Solitons and Fractals 14, (2002) 799-807

[abstract:] "We study Hamiltonians with singular spectra of Cantor type with a constant ratio of dissection and show strict connections between the decay properties of the states in the singular subspace and the algebraic number theory. More specifically, we study the decay properties of free n-particle systems and the computability of decaying and non-decaying states in the singular continuous subspace."

A. Napoli and A. Messina, "An application of the arithmetic Euler function to the construction of nonclassical states of a quantum harmonic oscillator", Reports on Mathematical Physics 48 (2001) 159-166

[abstract:] "All quantum superpositions of two equal intensity coherent states exhibiting infinitely many zeros in their Fock distributions are explicitly constructed and studied. Our approach is based on results from number theory and, in particular, on the properties of arithmetic Euler function. The nonclassical nature of these states is briefly pointed out. Some interesting properties are brought to light."

S. Ouvry, "Random Aharonov–Bohm vortices and some funny families of integrals" (preprint 02/05)

[abstract:] "A review of the random magnetic impurity model, introduced in the context of the integer Quantum Hall effect, is presented. It models an electron moving in a plane and coupled to random Aharonov-Bohm vortices carrying a fraction of the quantum of flux. Recent results on its perturbative expansion are given. In particular, some funny families of integrals show up to be related to the Riemann $\zeta(3)$ and $\zeta(2)$."

R. Bergamin, "$tt^*$ Geometry of modular curves"(preprint 03/2018)

[abstract:] "Motivated by Vafa's model, we study the $tt^{*}$ geometry of a degenerate class of FQHE models with an abelian group of symmetry acting transitively on the classical vacua. Despite it is not relevant for the phenomenology of the FQHE, this class of theories has interesting mathematical properties. We find that these models are parametrized by the family of modular curves $Y_{1}(N)= \mathbb{H}/\Gamma_{1}(N)$, labelled by an integer $N\geq 2$. Each point of the space of level $N$ is in correspondence with a one dimensional $\mathcal{N}=4$ Landau‐Ginzburg theory, which is defined on an elliptic curve with $N$ vacua and $N$ poles in the fundamental cell. The modular curve $Y(N)= \mathbb{H}/\Gamma(N)$ is a cover of degree $N$ of $Y_{1}(N)$ and plays the role of spectral cover for the space of models. The presence of an abelian symmetry allows to diagonalize the Berry's connection of the vacuum bundle and the $tt^{*}$ equations turn out to be the well known $\hat{A}_{N-1}$ Toda equations. The underlying structure of the modular curves and the connection between geometry and number theory emerge clearly when we study the modular properties and classify the critical limits of these models."

B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Novel multi-band quantum soliton states for a derivative nonlinear Schrödinger model" (preprint 07/03)

[abstract:]"We show that localized $N$-body soliton states exist for a quantum integrable derivative nonlinear Schrödinger model for several non-overlapping ranges (called bands) of the coupling constant $\eta$. The number of such distinct bands is given by Euler's $\phi$-function which appears in the context of number theory. The ranges of $\eta$ within each band can also be determined completely using concepts from number theory such as Farey sequences and continued fractions. We observe that $N$-body soliton states appearing within each band can have both positive and negative momentum. Moreover, for all bands lying in the region $\eta > 0$, soliton states with positive momentum have positive binding energy (called bound states), while the states with negative momentum have negative binding energy (anti-bound states)."

B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Multi-band structure of the quantum bound states for a generalized nonlinear Schrödinger model" (preprint 02/05)

[abstract:] "By using the method of coordinate Bethe ansatz, we study N-body bound states of a generalized nonlinear Schrödinger model having two real coupling constants c and \eta. It is found that such bound states exist for all possible values of c and within several nonoverlapping ranges (called bands) of \eta. The ranges of \eta within each band can be determined completely using Farey sequences in number theory. We observe that N-body bound states appearing within each band can have both positive and negative values of the momentum and binding energy."

A.Z. Li and W.G. Harter, "Quantum revivals of Morse oscillators and Farey–Ford geometry" (preprint 08/2013)

[abstract:] "Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time T_min-rev found by Heller is related to a complete quantum revival time T_rev using a quantum deviation parameter that in turn relates Trev to the maximum quantum beat period T_max-beat. Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum information processing and computing."

E Pelantová, Š. Starosta and M. Znojil, "Markov constant and quantum instabilities" (preprint 10/2015)

[abstract:] "For a qualitative analysis of spectra of a rectangular analogue of Pais–Uhlenbeck quantum oscillator several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide an entirely new mathematical insight in the phenomenologically relevant occurrence of spectral instabilities. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of the so called crypto-unitary quantum evolution up to the clarification of the mechanisms of the occurrence of ghosts in quantum cosmology."

R. Jozsa, "Notes on Hallgren's efficient quantum algorithm for solving Pell's equation" (preprint 02/03)

"Pell's equation is x2 - dy2 = 1 where d is a square-free integer and we seek positive integer solutions x, y > 0. Let (x',y') be the smallest solution (i.e. having smallest A = x' + y'd1/2)). Lagrange showed that every solution can easily be constructed from A so given d it suffices to compute A. It is known that A can be exponentially large in d so just to write down A we need exponential time in the input size log d. Hence we introduce the regulator R = ln A and ask for the value of R to n decimal places. The best known classical algorithm has sub-exponential running time O(exp(sqrt(log d)), poly(n)). Hallgren's quantum algorithm gives the result in polynomial time O(poly(log d),poly(n)) with probability 1/poly(log A). The idea of the algorithm falls into two parts: using the formalism of algebraic number theory we convert the problem of solving Pell's equation into the problem of determining R as the period of a function on the real numbers. Then we generalise the quantum Fourier transform period finding algorithm to work in this situation of an irrational period on the (not finitely generated) abelian group of real numbers.

These notes are intended to be accessible to a reader having no prior acquaintance with algebraic number theory; we give a self contained account of all the necessary concepts and we give elementary proofs of all the results needed. Then we go on to describe Hallgren's generalisation of the quantum period finding algorithm, which provides the efficient computational solution of Pell's equation in the above sense."

J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus-fresnel diffraction by a periodic grating", Physica D: Nonlinear Phenomena 1 (1980) 267-290

"Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions q, p requires the Planck's constant take one of the values h = qp/N, where N is an integer. Corresponding to a linear classical map T of points q, p is a unitary operator U mapping quantum states that are periodic in q and p; the construction of U involves techniques from number theory. U has eigenvalues exp(i). The 'eigenangles' must be multiples of 2/n (N), where n (N) is the lowest common multiple of the lengths of the classical 'cycles' mapped under T by those rational points in q, p which are multiples of q/N and p/N (i.e. n (N) is the 'period of T mod N'), at least for odd N. If T is hyperbolic, n is a very erratic function of N, and the classical limit N is very different from the 'Bohr-Sommerfeld' behaviour for parabolic maps. The degeneracy structure of the eigenangle spectrum is related to the distribution of cycle lengths. Computation of the quantal Wigner function shows that eigenstates of U do not correspond to individual cycles."

M.V. Berry and P. Shukla, "Tuck's incompressibility function: statistics for zeta zeros and eigenvalues" (preprint 07/2008)

[abstract:] "For any function that is real for real $x$, positivity of Tuck's function $Q(x)=D'^2(x)/(D'^2(x)-D"(x) D(x))$ is a condition for the absence of the complex zeros close to the real axis. Study of the probability distribution $P(Q)$, for $D(x)$ with $N$ zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of $Q$ are very rare for the Riemann zeros. $P(Q)$ has singularities at $Q=0$, $Q=1$ and $Q=N$. The moments (averages of $Q^m$) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-$N$ limit of $P(Q)$ can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at $Q=1$, while the large-$Q$ decay is determined by the pole closest to the origin. Determining the large-$N$ limit of $P(Q)$ for the GUE seems difficult."

J. Lagarias, "The Schrödinger operator with Morse potential on the right half line" (preprint 12/2007)

[abstract:] "This paper studies the Schr\"{o}dinger operator with Morse potential on a right half line $[u, \infty)$ and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains information on zeros of the Whittaker function $W_{\kappa, \mu}(x)$ for fixed real parameters $\kappa, x$, with $x$ positive, viewed as an entire function of the complex variable $\mu$. In this case all zeros lie on the imaginary axis, with the exception, if $k<0$, of a finite number of real zeros. We obtain an asymptotic formula for the number of zeros of modulus at most $T$ of form $N(T) = (2/\pi) T \log T + f(u) T + O(1)$. Some parallels are noted with zeros of the Riemann zeta function."

T. Okazaki, "AdS2/CFT1, Whittaker vector and Wheeler–De Witt equation" (preprint 10/2015)

[abstract:] "We study the energy representation of conformal quantum mechanics as the Whittaker vector without specifying classical Lagrangian. We show that a generating function of expectation values among two excited states of the dilatation operator in conformal quantum mechanics is a solution to the Wheeler–DeWitt equation and it corresponds to the AdS2 partition function evaluated as the minisuperspace wave function in Liouville field theory. We also show that the dilatation expectation values in conformal quantum mechanics lead to the asymptotic smoothed counting function of the Riemann zeros."

R.L. Monaco and W.A. Rodrigues, Jr., "New integral representation of the solutions of the Schrödinger equation with arbitrary potentials", Physics Letters A 179 (1993) 235-238

[abstract:] "We present a new method for solving the Schrödinger equation with arbitrary potentials. The solution is given in terms of a Fourier-like integral representation which involves a universal function (Rk(z)) for the Schrödinger equation. The integral representation follows from number theory together with some results from the partition theory of operational calculus. The new method can be used to solve any linear differential equation and also can be extended to solve linear partial differential equations."

I.I. Iliev, "Riemann zeta function and hydrogen spectrum", Electronic Journal of Theoretical Physics 10 (2013) 111–134

[abstract:] "Significant analytic and numerical evidence, as well as conjectures and ideas connect the Riemann zeta function with energy-related concepts. The present paper is devoted to further extension of this subject. The problem is analyzed from the point of view of geometry and physics as wavelengths of hydrogen spectrum are found to be in one-to-one correspondence with complex-valued positions. A Zeta Rule for the definition of the hydrogen spectrum is derived from well-known models and experimental evidence concerning the hydrogen atom. The Rydberg formula and Bohr's semiclassical quantization rule are modified. The real and the complex versions of the zeta function are developed on that basis. The real zeta is associated with a set of quantum harmonic oscillators with the help of relational and inversive geometric concepts. The zeta complex version is described to represent continuous rotation and parallel transport of this set within the plane. In both cases we derive the same wavelengths of hydrogen spectral series subject to certain requirements for quantization. The fractal structure of a specific set associated with $\zeta(s)$ is revealed to be represented by a unique box-counting dimension."

A. Sowa, "Encoding spatial data into quantum observables" (preprint 09/2016)

[abstract:] "The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified–in particular, by the results presented here–to pose such a question."

L. Campos Venuti, "The best quasi-free approximation: reconstructing the spectrum from ground state energies" (preprint 01/2011)

[abstract:] "The sequence of ground state energy density at finite size, e_{L}, provides much more information than usually believed. Having at disposal $e_L$ for short lattice sizes, we show how to re-construct an approximate quasi-particle dispersion for any interacting model. The accuracy of this method relies on the best possible quasi-free approximation to the model, consistent with the observed values of the energy $e_L$. We also provide a simple criterion to assess whether such a quasi-free approximation is valid. Perhaps most importantly, our method is able to assess whether the nature of the quasi-particles is fermionic or bosonic together with the effective boundary conditions of the model. The success and some limitations of this procedure are discussed on the hand of the spin-1/2 Heisenberg model with or without explicit dimerization and of a spin-1 chain with single ion anisotropy. A connection with the Riemann Hypothesis is also pointed out."

H. Suchowski and D.B. Uskov, "Complete population transfer in 4-level system via Pythagorean triple coupling" (preprint 11/2009)

[abstract:] "We describe a relation between the requirement of complete population transfer in a four-mode system and the generating function of Pythagorean triples from number theory. We show that complete population transfer will occur if ratios between coupling coefficients exactly match one of the Pythagorean triples $(a; b; c)$ in $Z$, $c^{2} = a^{2} + b^{2}$. For a four-level ladder system this relation takes a simple form $(V_{12}; V_{23}; V_{34}) ~ (c; b; a)$, where coefficients $V_{ij}$ describe the coupling between modes. We find that the structure of the evolution operator and the period of complete population transfer are determined by two distinct frequencies. A combination of these frequencies provides a generalization of the two-mode Rabi frequency for a four-mode system."

J. LaChapelle, "Evidence of a Gamma distribution for prime numbers" (preprint 07/2013)

[abstract:] "If the occurrence of prime numbers is a random process, then analogy with quantum systems suggests that a gamma distribution governs the primes. Consequently, postulating underlying gamma statistics in the context of functional integration, more-or-less standard heuristic arguments from quantum mechanics allows to derive analytic expressions of several average counting functions associated with prime numbers. The expressions are certain sums of incomplete gamma functions that are closely related to logarithmic-type integral functions – which in turn are well-known to give the asymptotic dependence of the various counting functions up to error terms. The relatively broad success of quantum heuristics applied to functional integrals in general along with the excellent agreement of the subsequent analytic expressions obtained for the average counting functions provide strong evidence of a gamma distribution for prime numbers."

M. Hage-Hassan, "A note on quarks and numbers theory" (preprint 02/2013)

[abstract:] "We express the basis vectors of Cartan fundamental representations of unitary groups by binary numbers. We determine the expression of Gel'fand basis of SU(3) based on the usual subatomic quarks notations and we represent it by binary numbers. By analogy with the mesons and quarks we find a new property of prime numbers."

G. Lang, "Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb–Liniger and Yang–Gaudin models” (preprint 07/2019)

[abstract:] "In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb–Liniger Bose gas and the Yang–Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders."

S. Tafazoli, "Divergent integrals, the Riemann zeta function, and the vacuum" (preprint 02/2020)

[abstract:] "This paper presents a new estimate for the vacuum energy density by summing the contributions of all quantum fields vacuum states which turns out to be in the same order of magnitude (but with opposite sign) as the predictions of current cosmological models and all observational data to date. The basis for this estimate is the recent results on the analytical solution to improper integral of divergent power functions using the Riemann Zeta function."

C. Castro, "The Riemann Hypothesis is a consequence of CT-invariant quantum mechanics" (submitted to J. Phys. A, 02/2007)

[abstract:] "The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s_n =1/2 + i lambda_n$. By constructing a continuous family of scaling-like operators involving the Gauss-Jacobi theta series, and by invoking a novel CT-invariant Quantum Mechanics, involving a judicious charge conjugation $C$ and time reversal $T$ operation, we show why the Riemann Hypothesis is true."

This follows earlier attempts:

C. Castro, "On two strategies towards the Riemann Hypothesis: Fractal Supersymmetric QM and a trace formula" (preprint 06/06)

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s_n =1/2+i lambda_n$. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Pólya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions $W(x,p,D)$ and where the summation has to be performed over all primes $p$ in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis."

C. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis", International Journal of Geometric Methods in Modern Physics 1 no. 6 (2004) 751-793

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on Supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model to implement the Hilbert-Pólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the analog of the fractal analog of the CBC (Comtet-Bandrauk-Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form $\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise."

S. Albeverio, R. Cianci, N. De Grande-De Kimpe, A. Khrennikov, "p-Adic probability and an interpretation of negative probabilities in quantum mechanics", Russian J. Math. Phys. 6 (1999) 3-19.

A. Khrennikov, "p-Adic probability interpretation of Bell's inequality paradoxes", Phys. Lett. A 200 (1995) 119-223

A. Khrennikov, "p-Adic probability distribution of hidden variables", Physica A 215 (1995) 577-587

A. Khrennikov, "p-Adic stochastic hidden variable model", J. Math. Phys. 39 No. 3 (1998) 1388-1402

number theory and quantum chaos - spectral interpretation of Riemann zeros

p-adic and adelic quantum mechanics

number theory and quantum field theory

number theory and quantum chromodynamics


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