The Quantum Separability Problem for Gaussian States (original) (raw)
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Reviews in Mathematical Physics, 2021
The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper, we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary.
Physical Review A, 2007
The concept of entanglement has gradually developed from the status of a puzzling interpretational problem to that of a crucial operational resource for quantum information tasks and, even more remarkably, to the status of a founding property of quantum theory, whose implications and applications extend into many diverse areas of research ranging from quantum optics and atomic and molecular physics to condensed matter physics and quantum critical phenomena 1, 2. While many open questions, even on defining grounds, ...
On Entanglement and Separability
We propose two necessary sufficient (NS) criteria to decide the separability of quantum states. They follow from two independent ideas: i) the Bloch-sphere-like-representation of states and ii) the proportionality of lines (rows, columns etc.) of certain multimatrix [1] associated with states. The second criterion proposes a natural way to determine the possible partial (or total, when possible) factorization of given multipartite state and in a sense can be used to determine the structure of the entanglement. We also introduce three entanglement measures based on the proposed new characterizations of entanglement. At last we discuss the second criterion mentioned above in the language of density matrix which is an inevitable language especially for mixed states.
An analytic approach to the problem of separability of quantum states based upon the theory of cones
Quantum Information Processing, 2011
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 ρ = (1 − λ)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ 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. We argue that this approach is not exhausted with the first simple choices included herein.
A Cone Approach to the Quantum Separability Problem
Arxiv preprint arXiv: …, 2010
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 ρ = (1 − λ)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ 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.
Einstein-Podolsky-Rosen-like separability indicators for two-mode Gaussian states
Journal of Physics A: Mathematical and Theoretical, 2018
We investigate the separability of the two-mode Gaussian states by using the variances of a pair of Einstein-Podolsky-Rosen (EPR)-like observables. Our starting point is inspired by the general necessary condition of separability introduced by Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)]. We evaluate the minima of the normalized forms of both the product and sum of such variances, as well as that of a regularized sum. Making use of Simon's separability criterion, which is based on the condition of positivity of the partial transpose (PPT) of the density matrix [Phys. Rev. Lett. 84, 2726 (2000)], we prove that these minima are separability indicators in their own right. They appear to quantify the greatest amount of EPR-like correlations that can be created in a two-mode Gaussian state by means of local operations. Furthermore, we reconsider the EPR-like approach to the separability of two-mode Gaussian states which was developed by Duan et al. with no reference to the PPT condition. By optimizing the regularized form of their EPR-like uncertainty sum, we derive a separability indicator for any two-mode Gaussian state. We prove that the corresponding EPR-like condition of separability is manifestly equivalent to Simon's PPT one. The consistency of these two distinct approaches (EPR-like and PPT) affords a better understanding of the examined separability problem, whose explicit solution found long ago by Simon covers all situations of interest.
Improved algorithm for quantum separability and entanglement detection
Physical Review A, 2004
Determining whether a quantum state is separable or entangled is a problem of fundamental importance in quantum information science. It has recently been shown that this problem is NPhard. There is a highly inefficient 'basic algorithm' for solving the quantum separability problem which follows from the definition of a separable state. By exploiting specific properties of the set of separable states, we introduce a new classical algorithm that solves the problem significantly faster than the 'basic algorithm', allowing a feasible separability test where none previously existed e.g. in 3-by-3-dimensional systems. Our algorithm also provides a novel tool in the experimental detection of entanglement.
Entanglement properties of Gaussian states
Fortschritte der Physik, 2003
On the one hand we present necessary and sufficient conditions for both the separability and the distillability of bipartite Gaussian states of an arbitrary number of modes. On the other hand we classify tripartite three-mode Gaussian states according to their separability properties and provide necessary and sufficient conditions for separability of those states. All the conditions presented here can be easily checked by direct calculations, thus providing operational criteria. This solves the separability problem for all bipartite as well as tripartite three-mode Gaussian states and the distillability problem for all bipartite Gaussian states.