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arXiv:1510.03240 · Physical Review Letters

Verifying the Quantumness of Bipartite Correlations

Claudio Carmeli, Teiko Heinosaari, Antti Karlsson, Jussi Schultz, Alessandro Toigo

Published Jun 7, 2016• 22 citations
arXivHTMLPDF

Abstract

Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question for an arbitrary state necessarily gives enough information to identify the state completely. We also extend our treatment to other classes of correlated states by considering the problem of deciding if a state has negative partial transpose, is discordant, or is fully classically correlated. Remarkably, only the question related to quantum discord can be answered without resorting to full state tomography.

DOI
10.1103/physrevlett.116.230403

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Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question for an arbitrary state necessarily gives enough information to identify the state completely. We also extend our treatment to other classes of correlated states by considering the problem of deciding if a state has negative partial transpose, is discordant, or is fully classically correlated. Remarkably, only the question related to quantum discord can be answered without resorting to full state tomography.
Entanglement does not describe all quantum correlations and several authors have shown the need to go beyond entanglement when dealing with mixed states. Various different measures have sprung up in the literature, for a variety of reasons, to describe bipartite and multipartite quantum correlations; some are known under the collective name {\it quantum discord}. Yet, in the same sprit as the criteria for entanglement measures, there is no general mechanism that determines whether a measure of quantum and classical correlations is a proper measure of correlations. This is partially due to the fact that the answer is a bit muddy. In this article we attempt tackle this muddy topic by writing down several criteria for a ``good" measure of correlations. We breakup our list into \emph{necessary}, \emph{reasonable}, and \emph{debatable} conditions. We then proceed to prove several of these conditions for generalized measures of quantum correlations. However, not all conditions are met by all measures; we show this via several examples. The reasonable conditions are related to continuity of correlations, which has not been previously discussed. Continuity is an important quality if one wants to probe quantum correlations in the laboratory. We show that most types of quantum discord are continuous but none are continuous with respect to the measurement basis used for optimization.
Like entanglement, quantum discord quantifies the quantum correlations. Unlike entanglement, whose detection is extremely difficult, the quantum discord of an arbitrary bipartite state allows itself to be detected perfectly by a single observable, the quantum discord witness. In particular, we report a single observable whose expectation value provides a necessary and sufficient condition for the vanishing quantum discord of an arbitrary bipartite unknown state with four copies. A quantum circuit is designed to measure the quantum discord witness by using only local qubit-measurements.
Abstract Determining whether an unknown quantum state is entangled can be approached through quantum tomography and related algorithms. However, these methods are generally inefficient due to the NP-hardness of the problem. An alternative strategy is to treat it as a decision (or estimation) problem with a quantum data structure (Horodecki and Ekert 2002 Phys. Rev. Lett. 89 127902). Over the years, significant progress has been made: the two-qubit case is now fully understood, but for higher-dimensional systems only entanglement detection via negativity has been established. In this work, we propose schemes for measuring upper bounds on bipartite biconcurrence and concurrence, expressed as functions of a fixed (but arbitrary) unitary operation acting on a Hilbert space that is quadratically larger than the original state space. Independently, we provide a scheme for measuring a lower bound for concurrence. In the case of the states proportional to the projectors the schemes are getting significantly simpler and the upper bound on the biconcurrence, as well as the lower bound on the concurrence, become exact (up to experimental and numerical errors). This framework opens the door to tackling the separability problem using methods inspired by the variational quantum eigensolver.
Quantum catalysts can overcome otherwise impossible quantum state transformations without being consumed, and allowing them to become correlated with the output makes this assistance substantially more powerful. This raises a fundamental question for entanglement theory: which limitations on state manipulation remain when such correlated catalysts are freely available? We answer this question in the positive-partial-transpose (PPT) resource theory, which allows a substantially broader class of operations than local operations and classical communication (LOCC). We identify general conditions under which regularized relative-entropy measures become strongly superadditive, and use them to construct monotones that constrain correlated catalytic PPT transformations without any knowledge of the catalyst. In particular, we prove that the regularized PPT relative entropy is fully additive and strongly superadditive, resolving an open problem in entanglement theory. Most importantly, these constraints show that even arbitrary correlated catalysts cannot restore asymptotic reversibility: for an explicit state, the optimal entanglement distillation rate remains strictly smaller than the entanglement cost. Thus, substantial catalytic assistance does not remove some of the fundamental limitations of mixed-state entanglement manipulation.
Certifying multipartite entanglement is a fundamental task. Since n-qubit state is parameterized by 4n- 1 real numbers, it is interesting to design a measurement setup that detects multipartite entanglement with as little effort as possible, and at a minimum without fully revealing the whole information of the state, the so-called “tomography”. In this paper, we study the relationship between multipartite entanglement certification and tomography, with the constraint that only single-copy measurements are allowed. We show that by using nonadaptive single-copy measurements, universal entanglement detection, among all states, can not be accomplished without full state tomography. Moreover, we show that almost all multipartite correlations, including the genuine entanglement and the entanglement depth, require full state tomography to detect in this measurement setting. We also observe that universal entanglement detection, among pure states, can be accomplished using much fewer measurements than full state tomography even using only local measurements.
Entanglement distillation is a fundamental task in quantum information theory. In this work, we prove that Werner states in arbitrary dimension are 222-copy distillable if and only if they are 111-copy distillable. This answers the longstanding open question of the 222-copy distillability of Werner states. This is an important step on determining whether every non-positive partial transpose (NPT) state is distillable, which remains one of the central open problems in the field of entanglement distillation.