Prime decomposition and entanglement measure of finite quantum systems

Under the name prime decomposition (pd), a unique decomposition of an arbitrary N dimensional density matrix ρ 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 ρ from its pd a trace-norm entaglement index E is introduced and its invariance groups are determined. 03.65 Bz, 42.50 Dv, 89.70.+c ⋄ Email: [email protected] ⊳ Email: [email protected] Quantum correlations, an emblematic notion of quantum theory, remains an open challenge since the early days of Quantum Mechanics [1, 2]. Recent investigations have set important questions concerning classification of various types of quantum correlations and their appropriate quantification. These theoretical activities have parallel developments with, and are partly motivated from recent interesting proposals which engage quantum correlations to such diverge tasks as e.g quantum computation and communication [3, 4], quantum cryptography [5], teleportation [6], and some new frequency standards [7]. Although the classification of quantum correlations is still open to refinements, it appears to include the following cases: for pure states, correlations entail nonlocality and give rise to violation of Bell inequalities [2]. For mixed states, two systems are consided uncorrelated if the composite system density matrix factorizes into a product of reduced density matrices, one for each isolated quantum subsystem viz. ρ = ρ1 ⊗ ρ2, where ρ1,2 = Tr1,2ρ, are determined by partial tracing. Quantification measures for that case include the von Neumann entropy [8] and other invariant indices [9]. On the other hand classical correlations for quantum subsystems imply seperability of the joint system density matrix, which is analysed into a convex sum for products of pure states viz. ρ = ∑ i piρ i 1 ⊗ ρ i 2, 0 ≤ pi ≤ 1, ∑ i pi = 1, [10]. Necessary and sufficient conditions for the existence of such convex decompositions for ρ’s acting on C| 2 ×C| 2 and C| 2 ×C| , became available recently [11, 12]. Upper bounds for the number of terms in such convex expansions of seperable matrices have also been determined, together with construction algorithms for the cases dimH ≤ 6 [13] and dimH ≤ ∞ [14]. Beyond these types of classical correlations we encounter inseperable or entangle quantum states. For their characterization and the quantification of their entanglement some general conditions have been presented that good entanglement measures should satisfy [15]. In this Letter we address the problem of the prime decomposition (pd), of a finite but otherwise arbitrary N -dimensional square density matrix ρ, into a sum of products of elementary density matrices, the number and the respective dimensions of which are determined by the compositeness of the dimension of ρ. This is achieved by means of 1 ) the so called Chinese Remainder Theorem (CRT) [16], that is based on the prime decomposition of N (this also explains the name we have chosen for the decomposition), and 2 ) by the fact that the discrete Heisenberg group HN , provides a projective representation of the abelian cyclic group Z N [17]. More concretely, if N = p1 1 p m2 2 . . . p mt t , is the prime factor decomposition of N , where p’s are distinct primes, then the pd of the density matrix involves square matrices ρ i = 1, . . . , t, with power prime dimension equal to Ni = p mi i . Also the number t of ρ-factors is bounded by the number of coprime factors of N . As a measure of the entanglement of a given mixed state ρ, from a possible prime or other decomposition, we evaluate the trace-norm distance between the two densities, study its unitary invariant symmetries, and interpret it in terms of the quantum covariances between local operators of the subsystems of the decomposition. We start by considering the matrix realization of the discrete Heisenberg group HN generated by the operator set of N elements Jm ≡ Jm1m2 = ω 1 2 12g1h2 , where the