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.
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.
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.
Summary: \nEntanglement is a strange feature contained in the quantum mechanical framework, first observed theoretically by Erwin Schrödinger and Albert Einstein. An inherent attribute of entanglement is non-locality, and in the continuation of this, several non-reductionist features, which are conceptually challenging and philosophically interesting. \n \nA considerable growth of the interest in quantum entanglement has occurred since the early 1990s, and much of the reason for this is that the tools of quantum optics and modern instrumentation have made it possible to measure and manipulate entangled quantum systems. Today, several applications of entanglement exist, including quantum computers and several areas in quantum communications. \n \nA quantum state is either separable or entangled, it cannot be none or both. A separable state do not exhibit the non-local features found in entangled states, and are in this sense comparable to classic states. An important issue here is the separability problem, which is the problem of determining whether a given state is entangled or not. \n \nWe study these quantum states as Hermitian matrices with positive eigenvalues in convex spaces. The PPT criterion is based on the partial transposition operation, which essentially transposes one of the two subsystems in the quantum system. For some states the result is another quantum (positive) state, and these are PPT (Positive Partial Transpose) states, while others that gain negative eigenvalues are called NPT states. \n \nFor systems of dimensions 2 × 2 and 2 × 3, the PPT states are identical to the separable states, and the NPT states are identical to the entangled states, so the PPT criterion is always conclusive. But for larger dimensions, such as for instance the 3 × 3 system, there always exist PPT states that are entangled, so for this case the PPT criterion can only be used in a contrapositive way. Since entangled PPT states in this sense seem to have great significance, their characteristics are worth studying. \nAs mentioned, the PPT criterion uses the transposition map on one of the two subsystems of a composite quantum system. Transposition is an example of a positive map, in that it maps all quantum states into quantum states. It is possible to use positive maps to formulate a much more general criterion for the separability problem.There is a link (an isomorphism) between positive maps and a subset of Hermitian operators called entanglement witnesses. \n \nThe so-called Horodecki criterion states that a quantum state is separable if and only if the expectation value of all entanglement witnesses in the state are non-negative. If the expectation value of one (or possibly several) entanglement witnesses in a given quantum state is negative, then the state is always entangled. This criterion is both necessary and sufficient, but demands that we can characterize the entire set of entanglement witnesses. \n \nThe approach in this thesis is through geometry. Of fundamental importance is the convex nature of the sets in question. The knowledge and understanding of the extremal points in these sets is very relevant, since they describe the sets completely. Undertaking completely analytic studies of the geometry at hand is in general very hard, especially as the dimension of the systems grow. The development and use of numerical tools is therefore essential.
Werner and Wolf have proven in Phys. Rev. Lett. 86(16) (2001) a very elegant necessary and sufficient condition for a bosonic continuous variable bipartite Gaussian mixed quantum state to be separable. This condition is, however, difficult to implement in practice. In the present Letter, we propose a simpler condition which only involves the calculation of the symplectic matrix in the Williamson diagonalization of the covariance matrix of the state under consideration. The main tool in our construction is the observation, proved in previous work, that the Wigner transform is covariant only under symplectic or antisymplectic linear transformations. We also give a geometric interpretation of our condition in terms of the orthogonal projections of "quantum blobs"..
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.