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arXiv:1601.02365 · Reviews in Mathematical Physics

On the equivalence of separability and extendability of quantum states

B. V. Rajarama Bhat, Kalyanapuram Rangachari Parthasarathy, Ritabrata Sengupta

Published Mar 20, 2017• 12 citations
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Abstract

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

DOI
10.1142/s0129055x1750012x

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Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.
Motivated by the notions of kkk-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al (Phys. Rev. A, 69:022308), we introduce parallel definitions in the context of Gaussian states and derive necessary and sufficient conditions for their extendability. It is shown that every separable Gaussian state is completely extendable but the converse is still a conjecture. However, the converse is proved for two mode Gaussian states.
We investigate (k_{1},k_{2}) extendibility of fermionic Gaussian states, a property central to quantum correlations and approximations of separability. We show that these states are (k_{1},k_{2})-extendible if and only if they admit a fermionic Gaussian extension, yielding a complete covariance-matrix characterization and a simple semidefinite program (SDP) whose size scales linearly with the number of modes. This provides necessary conditions for arbitrary fermionic states and is sufficient within the Gaussian setting. Our main result is a finite de Finetti-type theorem: we derive trace-norm bounds between (k_{1},k_{2})-extendible fermionic Gaussian states and separable states, improving previous exponential scaling to linear in the number of modes, with complementary relative entropy and squashed entanglement bounds. For two modes, upper and lower bounds match at order 1/sqrt[k_{1}k_{2}]. Extendibility also provides operational support for one of the different notions of separability in fermionic systems. Finally, for fermionic Gaussian channels, we provide a simple semidefinite program criterion for antidegradability and show that entanglement-breaking channels coincide with replacement channels, implying no nontrivial entanglement-breaking fermionic Gaussian channels exist.
The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty et al. The complete family of separability tests it generates, often referred to as the hierarchy of [Formula: see text]-extendibility tests, has already proved to be most promising. The goal of this paper is to try and quantify the efficiency of this separability criterion in typical scenarios. For that, we essentially take two approaches. First, we compute the average width of the set of [Formula: see text]-extendible states, in order to see how it scales with the one of separable states. And second, we characterize when random-induced states are, depending on the ancilla dimension, with high probability violating or not the [Formula: see text]-extendibility test, and compare the obtained result with the corresponding one for entanglement vs separability. The main results can be precisely phrased as follows: on [Formula: see text], when [Formula: see text] grows, the average width of the set of [Formula: see text]-extendible states is equivalent to [Formula: see text], while random states obtained as partial traces over an environment [Formula: see text] of uniformly distributed pure states are violating the [Formula: see text]-extendibility test with probability going to [Formula: see text] if [Formula: see text]. Both statements converge to the conclusion that, if [Formula: see text] is fixed, [Formula: see text]-extendibility is asymptotically a weak approximation of separability, even though any of the other well-studied separability relaxations is outperformed by [Formula: see text]-extendibility as soon as [Formula: see text] is above a certain (dimension independent) value.
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of m versus n modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for: (i) Gaussian states of 1 versus n modes; and (ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently established equivalence between separability of a Gaussian state and and its complete Gaussian extendability. Our techniques are then applied to progress beyond the state of the art. We prove that: (iv) Gaussian states that are invariant under partial transposition are necessarily separable; (v) the PPT criterion is necessary and sufficient for separability for Gaussian states of m versus n modes that are symmetric under the exchange of any two modes belonging to one of the parties; and (vi) Gaussian states which remain PPT under passive optical operations can not be entangled by them either. This is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.
We present a novel approach to the separability problem for Gaussian quantum states of bosonic continuous variable systems. We derive a simplified necessary and sufficient separability criterion for arbitrary Gaussian states of m versus n modes, which relies on convex optimisation over marginal covariance matrices on one subsystem only. We further revisit the currently known results stating the equivalence between separability and positive partial transposition (PPT) for specific classes of Gaussian states. Using techniques based on matrix analysis, such as Schur complements and matrix means, we then provide a unified treatment and compact proofs of all these results. In particular, we recover the PPT-separability equivalence for: (i) Gaussian states of 1 versus n modes; and (ii) isotropic Gaussian states. In passing, we also retrieve (iii) the recently established equivalence between separability of a Gaussian state and and its complete Gaussian extendability. Our techniques are then applied to progress beyond the state of the art. We prove that: (iv) Gaussian states that are invariant under partial transposition are necessarily separable; (v) the PPT criterion is necessary and sufficient for separability for Gaussian states of m versus n modes that are symmetric under the exchange of any two modes belonging to one of the parties; and (vi) Gaussian states which remain PPT under passive optical operations can not be entangled by them either. This is not a foregone conclusion per se (since Gaussian bound entangled states do exist) and settles a question that had been left unanswered in the existing literature on the subject. This paper, enjoyable by both the quantum optics and the matrix analysis communities, overall delivers technical and conceptual advances which are likely to be useful for further applications in continuous variable quantum information theory, beyond the separability problem.
Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is k-extendible if and only if it has a Gaussian k-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide k-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a k-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and Rényi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian k-extendible states, which has no analogue in the finite-dimensional setting.