Entropic Entanglement Criteria for Continuous Variables

Abstract

We derive several entanglement criteria for bipartite continuous variable quantum systems based on the Shannon entropy. These criteria are more sensitive than those involving only second-order moments, and are equivalent to well-known variance product tests in the case of Gaussian states. Furthermore, they involve only a pair of quadrature measurements, and will thus prove extremely useful in the experimental identification of entanglement.

Systematic construction of genuine-multipartite-entanglement criteria in continuous-variable systems using uncertainty relations

“…they are related by a π/2 rotation therefore the wavefunctions corresponding to the eigenstates |u and |v are related by a Fourier Transform, we recover the bipartite entanglement criteria in Eq. (13) of [22] that used the old entropic uncertainty relation for conjugate pairs of operators derived in [43] (see Table I for the identification of the non-local operators within our notation). Nevertheless, here we have extended the result in [22] because the Shannon-entropic entanglement criterion given in our Eq.…”

“…(13) of [22] that used the old entropic uncertainty relation for conjugate pairs of operators derived in [43] (see Table I for the identification of the non-local operators within our notation). Nevertheless, here we have extended the result in [22] because the Shannon-entropic entanglement criterion given in our Eq. (10) is valid for generic linear non-local operatorsμ andν not necessarily conjugate pairs.…”

“…Some of these entanglement criteria [15][16][17][18][22][23][24] are constructed using quantum-mechanical uncertainty relations that are based on quantities associated with operators of the formû…”

“…This is rapidly becoming an important experimental concern, since possibly genuine multipartite CV entanglement has been produced for three degenerate [10] and non-degenerate modes [11], and more recently for a large number of temporal [12] or spectral [13,14] modes. Even for the specific case of two-modes, the difficulty of state tomography has also led to several criteria involving second-order [15][16][17][18] and higher-order moments [19][20][21][22][23][24] of the canonical variables. Most of these criteria are based on the positive partial transpose (PPT) argument [25,26] and uncertainty relations involving the variance [15][16][17][18] or entropy [22,23] of marginal distributions.…”

“…Even for the specific case of two-modes, the difficulty of state tomography has also led to several criteria involving second-order [15][16][17][18] and higher-order moments [19][20][21][22][23][24] of the canonical variables. Most of these criteria are based on the positive partial transpose (PPT) argument [25,26] and uncertainty relations involving the variance [15][16][17][18] or entropy [22,23] of marginal distributions. Motivated by the impossibility of PT based entanglement criteria to detect bound entanglement [27], in [28] the authors construct bipartite entanglement witnesses solving the so-called separability eigenvalue equation.…”