Volume of the set of separable states. II

Abstract

The problem of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices ̺ describing N -dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N ≥ 8, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N = 8 and to estimate the volume of the bound entangled states in this case. 03.65.Bz, 42.50.Dv, 89.70.+c

A comparative study of relative entropy of entanglement, concurrence and negativity

“…The problem was raised by Eisert and Plenio [2] on the numerical example of the concurrence and negativity and then studied by others [3,5,6,7,8,9,10,11]. The ordering problem is closely related to existence of the upper and lower bounds of one entanglement measure versus the other [5,7,11,31].…”

“…where subscripts (i, j, k) change over cyclic permutations of (1,2,3). Concluding, the Bell-inequality violation depends on all λ i 's, while the entanglement measures E, C, and N depend solely on the largest λ i .…”

“…In previous works much attention was devoted to the ordering problem for the concurrence versus the negativity [2,3,9,10,11]. Here, we will study analytically the ordering of two qubit-states imposed by the REE in comparison to the other two measures.…”

“…The problem was then analyzed by others [3,4,5,6,7,8,9,10,11]. In particular, Virmani and Plenio [4] proved that all good asymptotic entanglement measures are either identical or fail to impose consistent orderings on the set of all quantum states.…”

Hilbert–Schmidt volume of the set of mixed quantum states

“…For instance, if ̺ describes a composite system, one may ask, what is the volume of the set of separable (entangled) mixed states [4,5]. Furthermore, assume we are given a concrete mixed quantum state ̺.…”