Einstein–Podolsky–Rosen-Like Correlation on a Coherent-State Basis and Inseparability of Two-Mode Gaussian States

Abstract

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.

Quantum Benchmark via an Uncertainty Product of Canonical Variables

“…We start with the product separable condition [35] in a normalized form [36]: Any separable state JAB satisfies (1) where (u, v) is a real vector with u2 + v2 = 1 and the canonical variables satisfy [icA, pA\ = [xB, pB\ = i. The first inequality is due to the property of variances, (d2) > (A2o).…”

“…Our goal is to derive a bound from the first and second moments of canonical variables for output states of a given channel E by assuming input of coherent states. We start with the product separable condition [35] in a normalized form [36]: Any separable state…”

Amplification uncertainty relation for probabilistic amplifiers

“…For η ∈ (1, 1 + λ), by substituting g = √ η into Eqs. (39) and (40) we obtain the form of the MSDs for the probabilistic amplifier Q g as…”

“…It thus effectively enhances the two-mode squeezed interaction as ξ → gξ (See section IV for a specific statement on the strength of entanglement). On the other hand, it has been known that the two-mode squeezed state minimizes the uncertainty product of Einstein-Podolsky-Rosen-like operators ∆ 2 (x A − g xxB ) ∆ 2 (p A + g ppB ) [39]. This quantity appears in Eq.…”

Bayesian uncertainty relation for a joint measurement of canonical variables

“…Similar setting enables us to derive Eq. ( 8) where a separable inequality for Einstein-Podolsky-Rosen like operators [28,29] is employed instead of Eq. ( 13) [7].…”