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Determining the entanglement of formation for arbitrary two-mode Gaussian states using Einstein-Podolsky-Rosen-like uncertainty

Y. Akbari-Kourbolagh, Hoda Alijanzadeh-Boora

Published Feb 13, 2014• 0 citations
OpenAlex

Abstract

In this article, we have derived a compact form for the entanglement of formation of arbitrary two-mode Gaussian states by using the Duan et al. inequality which gives a neccessary and sufficient separability condition for any two-mode Gaussian state.

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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.
The entanglement of general asymmetric pure Gaussian two-mode states is examined in terms of the coefficients of the quadrature components of the wave function. The entanglement criterion and the entanglement of formation are directly evaluated as a function of these coefficients, without the need for local unitary transformations. These reproduce results for the special case of symmetric pure states which employ a relation between squeezed states and Einstein-Podolsky-Rosen correlations. The modification of the quadrature coefficients and the corresponding entanglement due to application of various optical elements is also derived.
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.
The strange property of the Einstein-Podolsky-Rosen (EPR) correlation between two remote physical systems is a primitive object on the study of quantum entanglement. In order to understand the entanglement in canonical continuous-variable systems, a pair of the EPR-like uncertainties is an essential tool. Here, we consider a normalized pair of the EPR-like uncertainties and introduce a state-overlap to a classically correlated mixture of coherent states. The separable condition associated with this state-overlap determines the strength of the EPR-like correlation on a coherent-state basis in order that the state is entangled. We show that the coherent-state-based condition is capable of detecting the class of two-mode Gaussian entangled states. We also present an experimental measurement scheme for estimation of the state-overlap by a heterodyne measurement and a photon detection with a feedforward operation.
Abstract We investigate the separability of the two-mode Gaussian states (TMGSs) 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 (2000 Phys. Rev. Lett . 84 2722). 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 (Simon 2000 Phys. Rev. Lett . 84 2726), 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 TMGS by means of local operations. Furthermore, we reconsider the EPR-like approach to the separability of TMGSs 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 TMGS. 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.
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.