An universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results concerning the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of
π-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyse continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
A universal approximation technique for analysis of different characteristics of states of composite infinite-dimensional quantum systems is proposed and used to prove general results on the properties of correlation and entanglement measures in such systems. Then these results are applied to the study of three important characteristics: the relative entropy of π-entanglement, the Rains bound (the unregularized and regularized versions of both characteristics are considered) and the conditional entanglement of mutual information. In particular, we analyze continuity and convexity properties of the above entanglement measures, prove several results simplifying their definitions and establish a finite-dimensional approximation property for these characteristics that allows us to generalize to the infinite-dimensional case the results proved in the finite-dimensional settings.
We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods, we obtain continuity bounds for several important characteristics of a multipartite quantum state: the quantum (conditional) mutual information, the squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information. The continuity bounds for the multipartite quantum mutual information are asymptotically tight for large dimension/energy. The obtained results are used to prove the asymptotic continuity of the n-partite squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information under the energy constraints.
Abstract The relative entropy of entanglement
ER E R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also,
ER E R is everywhere lower semi-continuous. We use this to derive a dual variational expression for
ER E R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative
λ λ -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for
ER E R and closely related quantities in the presence of an energy constraint.
In this paper, we investigate the characterization of multipartite entanglement based on the unified
(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure
Eq,sA∣B(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures
Aq,sk(ρ) and
Gq,sk(ρ) for quantifying
k-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters
q and
s are in certain ranges,
Aq,sk(ρ) and
Gq,sk(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures
Aq,sk(ρ) and
Gq,sk(ρ) with respect to the parameters
q and
s for fixed quantum states. In addition, for fixed
q and
s, we also give their order relations for any two pure states.
Abstract General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.