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Register a new userThe separability versus entanglement problem
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We present a review of the problem of finding out whether a quantum state of two or more parties is entangled or separable. After a formal definition of entangled states, we present a few criteria for identifying entangled states and introduce some entanglement measures. We also provide a classification of entangled states with respect to their usefulness in quantum dense coding, and present some aspects of multipartite entanglement.
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We show how the separability problem is dual to that of decomposing any given matrix into a conic combination of rank-one partial isometries, thus offering a duality approach different to the positive maps characterization problem. Several inmediate consequences are analyzed: (i) a sufficient criterion for separability for bipartite quantum systems, (ii) a complete solution to the separability problem for pure states also of bipartite systems independent of the classical Schmidt decomposition method and (iii) a natural generalization of these results to multipartite systems.
Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as $rho=(1-lambda)C_{rho}+lambda E_{rho}$, where $C_{rho}$ is a separable matrix whose rank equals that of $rho$ and the rank of $E_{rho}$ is strictly lower than that of $rho$. With the simple choice $C_{rho}=M_{1}otimes M_{2}$ we have a necessary condition of separability in terms of $lambda$, which is also sufficient if the rank of $E_{rho}$ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication (SLOCC). We argue that this approach is not exhausted with the first simple choices included herein.
Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find $82%$ to be free entangled, $2%$ to be certainly separable and as much as $10%$ to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining $6%$ of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong $2$-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.
Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. What determines whether an arbitrary quantum state is entangled or separable is therefore very important for investigating both fundamental physics and practical applications. Here we show that an arbitrary bipartite state can be divided into a unique purely entangled structure and a unique purely separable structure. We show that whether a quantum state is entangled or not is determined by the ratio of its weight of the purely entangled structure and its weight of the purely separable structure. We provide a general algorithm for the purely entangled structure and the purely separable structure, and a general algorithm for the best separable approximation (BSA) decomposition, that has been a long-standing open problem. Our result implies that quantum states exist as families in theory, and that the entanglement (separability) of family members can be determined by referring to a crucial member of the family.
We present a quasipolynomial-time algorithm for solving the weak membership problem for the convex set of separable, i.e. non-entangled, bipartite density matrices. The algorithm decides whether a density matrix is separable or whether it is eps-away from the set of the separable states in time exp(O(eps^-2 log |A| log |B|)), where |A| and |B| are the local dimensions, and the distance is measured with either the Euclidean norm, or with the so-called LOCC norm. The latter is an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by quantum local operations and classical communication (LOCC) between the parties. We also obtain improved algorithms for optimizing over the set of separable states and for computing the ground-state energy of mean-field Hamiltonians. The techniques we develop are also applied to quantum Merlin-Arthur games, where we show that multiple provers are not more powerful than a single prover when the verifier is restricted to LOCC protocols, or when the verification procedure is formed by a measurement of small Euclidean norm. This answers a question posed by Aaronson et al (Theory of Computing 5, 1, 2009) and provides two new characterizations of the complexity class QMA, a quantum analog of NP. Our algorithm uses semidefinite programming to search for a symmetric extension, as first proposed by Doherty, Parrilo and Spedialieri (Phys. Rev. A, 69, 022308, 2004). The bound on the runtime follows from an improved de Finetti-type bound quantifying the monogamy of quantum entanglement, proved in (arXiv:1010.1750). This result, in turn, follows from a new lower bound on the quantum conditional mutual information and the entanglement measure squashed entanglement.
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