Universal freezing of quantum correlations within the geometric approach

Abstract

Quantum correlations in a composite system can be measured by resorting to a geometric approach, according to which the distance from the state of the system to a suitable set of classically correlated states is considered. Here we show that all distance functions, which respect natural assumptions of invariance under transposition, convexity, and contractivity under quantum channels, give rise to geometric quantifiers of quantum correlations which exhibit the peculiar freezing phenomenon, i.e., remain constant during the evolution of a paradigmatic class of states of two qubits each independently interacting with a non-dissipative decohering environment. Our results demonstrate from first principles that freezing of geometric quantum correlations is independent of the adopted distance and therefore universal. This finding paves the way to a deeper physical interpretation and future practical exploitation of the phenomenon for noisy quantum technologies.

Measures and applications of quantum correlations

“…In certain cases, such a resilience can be extreme, as QCs in bipartite and multipartite systems can remain constant (frozen) in time under local decohering maps even though the global state is evolving [326]; this happens for particular classes of states and dynamical conditions. Within the geometric approach to QCs (Section 3.2.1), it has been shown in [55] that, under the allowed conditions, such features are universal and occur for all measures Q G δ independently of the specific choice of the distance D δ in their definition. More details can be found in [55] and references therein, as well as [12], to which the reader is referred for a collection of pertinent literature (including a number of experimental demonstrations).…”

“…Moreover, as already discussed, all forms of non-classical correlations reduce to entanglement for pure states, hence one expects any valid measure of QCs to reduce to a corresponding valid measure of entanglement in such a special case. This is imposed by Requirement (iii), which has appeared frequently in recent literature on QCs [27,54,57,58,59,41,55,56] Another important consideration is how a measure of QCs should behave under the action of local quantum channels. For example, any valid entanglement measure must be nonincreasing on average under the action of LOCC [3], so what are the analogous constraints for a measure of general QCs?…”

“…Various proposals for mandatory, optional and desirable requirements for a measure of QCs have been discussed e.g. in [35,53,12,54,27,33,55,56], to which the reader is referred for additional details.…”

“…Here, following [55], we postulate a strong monotonicity desideratum for measures of QCs under the maximal set of local operations preserving classical states, given by A similar desideratum is postulated for a two-sided measure of QCs Q AB (ρ AB ), where one then considers LCPO Λ LCPO AB acting on both subsystems A and B. Notice that Requirement (v) is only a proposed one, and should not be regarded yet as mandatory, unlike the previous four Requirements.…”

Maximal coherence in a generic basis

“…The mathematical subtlety does not prevent us from guessing the structure of the unitary matrix U. The first thing coming into our sight is the universal freezing phenomenon that occurs for quantum correlation or quantum coherence measures [30][31][32][33][34]. Here the word universal means that under certain initial conditions this phenomenon will inevitably occur independently of the adopted measures, e.g., it is a common feature of all known bona fide measures.…”