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Register a new userA cone approach to the quantum separability problem
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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.
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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.
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.
We derive the Casimir force expression from Maxwells stress tensor by means of original quantum-electro-dynamical cavity modes. In contrast with similar calculations, our method is straightforward and does not rely on intricate mathematical extrapolation relations.
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.
In matrix theory, a well established relation $(AB)^{T}=B^{T}A^{T}$ holds for any two matrices $A$ and $B$ for which the product $AB$ is defined. Here $T$ denote the usual transposition. In this work, we explore the possibility of deriving the matrix equality $(AB)^{Gamma}=A^{Gamma}B^{Gamma}$ for any $4 times 4$ matrices $A$ and $B$, where $Gamma$ denote the partial transposition. We found that, in general, $(AB)^{Gamma} eq A^{Gamma}B^{Gamma}$ holds for $4 times 4$ matrices $A$ and $B$ but there exist particular set of $4 times 4$ matrices for which $(AB)^{Gamma}= A^{Gamma}B^{Gamma}$ holds. We have exploited this matrix equality to investigate the separability problem. Since it is possible to decompose the density matrices $rho$ into two positive semi-definite matrices $A$ and $B$ so we are able to derive the separability condition for $rho$ when $rho^{Gamma}=(AB)^{Gamma}=A^{Gamma}B^{Gamma}$ holds. Due to the non-uniqueness property of the decomposition of the density matrix into two positive semi-definte matrices $A$ and $B$, there is a possibility to generalise the matrix equality for density matrices lives in higher dimension. These results may help in studying the separability problem for higher dimensional and multipartite system.
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