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arXiv:1310.6060 · Physical Review A

Tight bounds for the entanglement of formation of Gaussian states

F. Nicacio, M. C. de Oliveira

Published Jan 30, 2014• 14 citations
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Abstract

We establish tight upper and lower bounds for the entanglement of formation of an arbitrary two-mode Gaussian state employing the necessary properties of Gaussian channels. Both bounds are strictly given by the entanglement of formation of symmetric Gaussian states, which are simply constructed from the reduced states obtained by partial trace of the original one.

DOI
10.1103/physreva.89.012336

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We establish tight upper and lower bounds for the entanglement of formation of an arbitrary two-mode Gaussian state employing the necessary properties of Gaussian channels. Both bounds are strictly given by the entanglement of formation of symmetric Gaussian states, which are simply constructed from the reduced states obtained by partial trace of the original one.
Entanglement of formation quantifies the entanglement of a state in terms of the entropy of entanglement of the least entangled pure state needed to prepare it. An analytical expression for this measure exists only for special cases, and finding a closed formula for an arbitrary state remains an open problem. In this work we focus on two-mode Gaussian states, and we derive narrow upper and lower bounds for the measure that get tight for several special cases. Further, we show that the problem of calculating the actual value of the entanglement of formation for arbitrary two-mode Gaussian states reduces to a trivial single parameter optimization process, and we provide an efficient algorithm for the numerical calculation of the measure.
We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.
Entangled two-mode Gaussian states are a key resource for quantum information technologies such as teleportation, quantum cryptography, and quantum computation, so quantification of Gaussian entanglement is an important problem. Entanglement of formation is unanimously considered a proper measure of quantum correlations, but for arbitrary two-mode Gaussian states no analytical form is currently known. In contrast, logarithmic negativity is a measure that is straightforward to calculate and so has been adopted by most researchers, even though it is a less faithful quantifier. In this work, we derive an analytical lower bound for entanglement of formation of generic two-mode Gaussian states, which becomes tight for symmetric states and for states with balanced correlations. We define simple expressions for entanglement of formation in physically relevant situations and use these to illustrate the problematic behavior of logarithmic negativity, which can lead to spurious conclusions.
We derive two lower bounds on entanglement of formation for arbitrary mixed Gaussian states by two distinct methods. To achieve the first one we use a local measurement procedure that symmetrizes a general Gaussian state and the fact that entanglement cannot increase under local operations and classical communications. The second one is obtained via a generalization to mixed states of an interesting result already known for pure states, which says that squeezed states are those that, for a fixed amount of entanglement, maximize Einstein-Podolsky-Rosen-like correlations.
We find tight lower and upper bounds on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition. Our upper bound is dramatically tighter than the one presented by Linden et al. [Phys. Rev. Lett. 97, 100502 (2006)] and our lower bound can be used to provide lower bounds on different measures of entanglement such as the entanglement of formation and the entanglement of subspaces. We also find that in the case in which the two states are one-sided orthogonal, the entanglement of the superposition state can be expressed explicitly in terms of the entanglement of the two states in the superposition.
Entanglement of formation is a measure that has a clear physically motivated definition, namely, it provides a lower bound over all pure-state entanglement decompositions required to create a given state. We investigate the open system dynamics of entanglement of formation of a two-mode Gaussian state interacting with a common bosonic environment. In the framework of the theory based on completely positive dynamical semigroups, the open system dynamics leads to a master equation in Lindblad form, which preserves the positivity and the Gaussian form of an initially Gaussian state at all times. For an initially separable state we witness the generation of entanglement, and its preservation in the limit of asymptotic times.