Gaussian entanglement of symmetric two-mode Gaussian states

Abstract

A Gaussian degree of entanglement for a symmetric two-mode Gaussian state can be defined as its distance to the set of all separable two-mode Gaussian states. The principal property that enables us to evaluate both Bures distance and relative entropy between symmetric two-mode Gaussian states is the diagonalization of their covariance matrices under the same beam-splitter transformation. The multiplicativity property of the Uhlmann fidelity and the additivity of the relative entropy allow one to finally deal with a single-mode optimization problem in both cases. We find that only the Bures-distance Gaussian entanglement is consistent with the exact entanglement of formation.

Gaussification through decoherence

“…Application of this measure for M -photon-added thermal states showed us its good agreement with previously defined distance-type measures [14,15,17]. Note that fidelity-based metrics have proven to be fruitful in quantum optics and quantum information as measures of nonclassicality [28] and entanglement [29][30][31]. Second, we use the three abovementioned distance-type measures to investigate the loss of non-Gaussianity for M -photon-added thermal states of a field coupled to a heat bath.…”

Measures of non-Gaussianity for one-mode field states

“…In other words, our definition is of the type (1.1) and uses a well-known metric related to the fidelity between two quantum states [14]. Fidelity-based metrics have proven to be fruitful in quantum optics and quantum information as measures of nonclassicality [15], entanglement [16,17,18], and polarization [19,20,21,22,23]. We also intend to compare the three abovementioned distance-type measures in analyzing the non-Gaussianity for a definite class of one-mode states.…”

“…In other words, our definition is of the type (1.1) and uses a well-known metric related to the fidelity between two quantum states [14]. Fidelity-based metrics have proven to be fruitful in quantum optics and quantum information as measures of nonclassicality [15], entanglement [16,17,18],…”

Entanglement of Formation for an Arbitrary Two-Mode Gaussian State

“…This implies the existence of a convenient root p m = w 1 w 2 ≥ 1 for any inseparable mixed TMGS. Had we got p m , it could be used to obtain the optimal y m as the smallest root of a quadratic trinomial B 2 (p)y 2 + B 1 (p)y + B 0 (p) whose coefficients, B 0 (p) = −Dp ≥ 0, (16) are evaluated at p = p m . We mention that in four significant particular cases (defined by special relations between standard-form parameters) we have found simple solutions by direct use of Eqs.…”

“…We have then recovered them by exploiting Eqs. (15) and (16). As a first salient example, we consider an entangled symmetric TMGS, whose standard-form parameters are b 1 = b 2 =: b, c ≥ |d| = −d > 0.…”