Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations

Abstract

Let us note that Axioms (iii) and (iv) imply that, when n A ≤ n B , D is maximum on maximally entangled pure states, i.e., if ̺ is a maximally entangled pure state then D(̺) = D max [16]. This follows from the facts that a function D on S AB satisfying (iii) is maximal on pure states if n A ≤ n B [33] and that any pure state can be obtained from a maximally entangled pure state via a LOCC [15]. Thus, if Axioms (i-iv) are satisfied, the additional requirement in Axiom (v) is essentially that D(̺) = D max holds only for the maximally entangled states ̺.It has been shown in previous works [11,15] that the geometric discord D G Bu and discord of response D R Bu satisfy Axioms (i)-(iv) for the Bures distance, and hence are bona fide measures of quantum correlations. In this paper, we will prove that this is also the case for the three measures D G He , D M He , and D R He based on the Hellinger distance, as well as for the Bures measurement-induced geometric discord D M Bu and trace discord of response D R Tr . In contrast, it is known that D G HS = D M HS and D R HS do not fulfill Axiom (iii) because of the lack of monotonicity of the Hilbert-Schmidt distance under CPTP maps (an explicit counter-example is given in Ref. [34] for D G HS and applies to D R HS as well, see below). Therefore, the use of the Hilbert-Schmidt distance in the definitions of Eqs. (5)-(7) can and does lead to unphysical predictions. Considering the distances d p associated to the p-norms X p ≡ (Tr |X| p ) 1/p , one has that for p > 1, d p is not contractive under CPTP maps [35] (see also Ref.[36] for a counter-example for p = 2, which also holds for any p > 1). This is why the distances d p cannot be used to define measures of quantumness apart from the case p = 1, corresponding to the contractive trace distance, while for p = 2 the non-contractive Hilbert-Schmidt distance is well tractable and thus used to establish bounds on the bona fide geometric measures.Regarding our last Axiom (v), the only result established so far in the literature concerns the Bures geometric discord [9,15]. We will demonstrate below that all the other measures based on the trace, Bures, and Hellinger distances also satisfy this axiom. Our proofs are valid for arbitrary (finite) space dimensions n A and n B of subsystems A and B, excepted for D G He , for which they are restricted to the special cases n A = 2, 3, and for D M He , D G Tr , and D M Tr , for which they are restricted to n A = 2.The paper is organized as follows. Given its length and the wealth of mathematical relations and bounds that we have determined, we begin by summarizing the main results in Section II. We first give general expressions of the geometric measures for the Bures and Hellinger distances, which are convenient starting points to compare them (see Sec. II A). We then report in some synoptic Tables the various relations and bounds satisfied by D G , D M , and D R for the trace, Hilbert-Schmidt, Bures, and Hellinger distances (see Sec. II B). Closed expressions for the Hellinger geometric ...

Measures and applications of quantum correlations

“…We point again to [56] for a compendium of informative bounds. Finally, the extensions to other distances suggested in Section 3.3.7 can likewise be considered for the measurement induced geometric measures.…”

“…Hellinger distance Similarly, using the squared Hellinger distance D He (see Table 2) one obtains as well valid measures of geometric QCs obeying all the Requirements. An in-depth study of Q G He A (ρ AB ) was carried out in [56], where they simplified the problem of evaluating this measure for arbitrary bipartite states, and showed in particular that for pure states |ψ AB it holds…”

“…(72), it is still a relevant question to ask whether Q Table 2). The one-sided versions, considered in [56], are valid measures of QCs obeying all the Requirements (i)-(iv), while Requirement (v) has not been tested yet. In particular, for pure states |ψ AB it holds…”

“…This requires a modification of the one-sided definition in Eqs. (56), to consider the minimisation of the partial entanglement in the pre-measurement state, Q…”

“…More generally, for arbitrary states ρ AB when A is a qubit, these two measures can be linked analytically to their geometric counterparts by the simple relation [56] …”