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Prime decomposition and correlation measure of finite quantum systems

Demosthenes Ellinas, Emmanuel G. Floratos

Published Jan 1, 1999• 16 citations
OpenAlex

Abstract

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 $Z_N$ by the discrete Heisenberg group $H_N$. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on $H_N$ as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of $H_N$. To quantify the distance of $\\rho$ from its pd a trace-norm entaglement index $\\cal E$ is introduced and its invariance groups are determined.

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Prime decomposition and correlation measure of finite quantum systems

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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.

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Under the name prime decomposition (pd), a unique decomposition of an arbitrary NNN-dimensional density matrix rho\\rhorho into a sum of seperable density matrices with dimensions given by the coprime factors of NNN 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 ZNZ_NZN​ by the discrete Heisenberg group HNH_NHN​. The pd isomorphism is unitarily implemented and it is shown to be coassociative and to act on HNH_NHN​ as comultiplication. Density matrices with complete pd are interpreted as grouplike elements of HNH_NHN​. To quantify the distance of rho\\rhorho from its pd a trace-norm entaglement index calE\\cal EcalE is introduced and its invariance groups are determined.
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.
Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for d-dimensional spaces, and the resulting set of unitary matrices S(d)S(d)S(d) is a basis for $d\ifmmode\times\else\texttimes\fi{}d$ matrices. If ${N=d}_{1}\ifmmode\times\else\texttimes\fi{}{d}_{2}\ifmmode\times\else\texttimes\fi{}\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}\ifmmode\times\else\texttimes\fi{}{d}_{b}$ and ${H}^{[N]}=\ensuremath{\bigotimes}{H}^{[{d}_{k}]},$ we give a sufficient condition for separability of a density matrix $\ensuremath{\rho}$ relative to the H[dk]{H}^{[{d}_{k}]}H[dk​] in terms of the L1{L}_{1}L1​ norm of the spin coefficients of $\ensuremath{\rho}.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space H[N].{H}^{[N]}.H[N]. It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime p and n>1,n>1,n>1, the generalized Werner density matrix W[pn](s){W}^{[{p}^{n}]}(s)W[pn](s) is fully separable if and only if $s<~{(1+p}^{n\ensuremath{-}1}{)}^{\ensuremath{-}1}.$
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.
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.
Abstract We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one‐particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group‐theoretic sense, for the group of unitary transformations of the set of one‐electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle‐number‐conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non‐particle‐number‐conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.
This paper presents self-contained, constructive proofs of two new theorems about stochastic matrices, with direct relevance to quantum theory. The first theorem, herein called the Born Representation Theorem, shows that each entry of any stochastic matrix can be expressed as the trace of a pairwise product of matrices, where the first factor in the pairwise product belongs to a positive-operator-valued measure (POVM) and the second factor belongs to a projection-valued measure (PVM). As its name suggests, this theorem entails that the entries of any stochastic matrix can be expressed in terms of a generalized version of the quantum-theoretic Born rule. It follows as a corollary that if the POVM in this first theorem is a PVM, then the stochastic matrix is unistochastic, meaning that its entries are each the modulus square of the corresponding entry of a unitary matrix of the same size. The second theorem proved in this paper, called the Unistochastic Theorem, then shows that by dilating the underlying vector space by a bounded number of additional dimensions if necessary, each entry of any stochastic matrix can be expressed in terms of the trace of a pairwise product for which both factors belong to PVMs, and can thus be derived via marginalization from a larger unistochastic matrix. This second theorem therefore establishes a kind of primacy of unistochastic matrices over stochastic matrices, and hints at a close connection with unitary time evolution in quantum theory. The paper concludes with a brief discussion of potential applications to discrete-time deterministic processes and Markov chains.