Completely entangled quantum states are shown to factorize into tensor products of entangled states whose dimensions are powers of prime numbers. The entangled states of each prime-power dimension transform among themselves under a finite Heisenberg group. We are thus led to examine processes in which factors are exchanged between entangled states and so consider canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension.
Completely entangled quantum states are shown to factorize into tensor products of entangled states whose dimensions are powers of prime numbers. The entangled states of each prime-power dimension transform among themselves under a finite Heisenberg group. We are thus led to examine processes in which factors are exchanged between entangled states and so consider canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension.
Under the name prime decomposition (PD), a unique decomposition of an arbitrary N -dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese remainder theorem, and the projective unitary representation of by the discrete Heisenberg group . The PD isomorphism is unitarily implemented and it is shown to be co-associative and to act on as comultiplication. Density matrices with complete PD are interpreted as group-like elements of . To quantify the distance of from its PD a trace-norm correlation index is introduced and its invariance groups are determined.
Under the name prime decomposition (pd), a unique decomposition of an arbitrary N-dimensional density matrix rho into a sum of seperable density matrices with dimensions given by the coprime factors of N is introduced. For a class of density matrices a complete tensor product factorization is achieved. The construction is based on the Chinese Remainder Theorem and the projective unitary representation of ZN by the discrete Heisenberg group HN. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on HN as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of HN. To quantify the distance of rho from its pd a trace-norm entaglement index calE is introduced and its invariance groups are determined.
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.
The Riemann Hypothesis, a long-standing and stubbornly unsolved problem in number theory, has captivated mathematicians for over 150 years. Current computational methods struggle to definitively prove or disprove it, highlighting a significant gap in our understanding of the zeta function. This paper proposes a novel approach to tackle this problem utilizing quantum-enhanced measurement techniques and entangled variables. We outline a new algorithm for exploring the Riemann Hypothesis, leveraging the correlations between entangled particles to identify subtle patterns within the zeta function. The core mechanism centers on creating entangled particles to probe the structure of the zeta function, potentially revealing previously missed connections. This investigation leverages a hybrid quantum-classical approach, combining the precision of quantum mechanics with the analytical power of classical computation. The goal is to develop a framework that can potentially lead to new insights into the Riemann Hypothesis and provide a more robust theoretical foundation.
Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.
There are different techniques that allow us to gain complete knowledge about an unknown quantum state, e.g., to perform full tomography of this state. For instance, quasi-distributions such as the Weyl or Wigner distributions provide complete information about a quantum state which is equivalent to the information contained in the density matrix. In the case of composite systems, of which the subsystems are not necessarily located at the same place, the experimental feasibility of the tomographic process is considerably simplified whenever it can be realized through local operations and classical communications between local observers. This brings us naturally to study the possibility to factorize the (discrete) Wigner distribution of a composite system into the product of local Wigner distributions, which is the subject of the present paper. The discrete Heisenberg-Weyl group is an essential ingredient of our approach.