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 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.
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) 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] in terms of the 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]. 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, the generalized Werner density matrix 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 p particles for each value of p. 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 N-representability problem is given, and two sufficient conditions and one necessary condition for N representability are given in terms of distances and norms.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
The notion of quantum group replaces ordinary matrices with matrices with non-commutative elements. In TGD framework I have proposed that the notion should relate to the inclusions of von Neumann algebras allowing one to describe mathematically the notion of finite measurement resolution. In this article I will consider the notion of quantum matrix inspired by the recent view about quantum TGD relying on the notion of finite measurement resolution. Complex matrix elements are replaced with operators expressible as products of non-negative Hermitian operators and unitary operators analogous to the products of modulus and phase as a representation for complex numbers. The condition that determinant and sub-determinants exist is crucial for the well-defined eigenvalue problem in the generalized sense. Strong/weak permutation symmetry of determinant requires its invariance under permutations of rows and/or columns. Weak permutation symmetry means development of determinant with respect to a fixed row or column and does not pose additional conditions. For weak permutation symmetry the permutation of rows/columns would however has a natural interpretation as braiding for the Hermitian operators defined by the moduli of operator valued matrix elements and here quantum group structure emerges. The commutativity of all sub-determinants is essential for the replacement of eigenvalues with eigenvalue spectra of Hermitian operators and sub-determinants define mutually commuting set of operators. Quantum matrices define a more general structure than quantum group but provide a concrete representation and interpretation for quantum group in terms of finite measurement resolution, in particular when q is a root of unity. One can also understand the fractal structure of inclusion sequences of hyper-finite factors resulting by replacing operators appearing as matrix elements with quantum matrices.
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.