Exact covariance criterion for bipartite Gaussian separability
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- Topics
Problem
What necessary and sufficient condition on a covariance matrix characterizes bipartite Gaussian separability?
Let \(m,n\geq1\) be integers. Let \(\rho_V\) be a zero-mean Gaussian state with \(m\) modes held by Alice and \(n\) modes held by Bob. Its finite real covariance matrix and canonical commutators are
Separability means that \(\rho_V\) belongs to the trace-norm closed convex hull of product density operators. Determine separability from the covariance in Eq. (1).
Source
Werner and Wolf state and prove the complete covariance criterion in Proposition 1 [WW01].
Progress
Werner–Wolf, Proposition 1, proves both directions of the criterion below. The exact criterion is the following finite semidefinite feasibility problem over real symmetric matrices \(V_A\) of size \(2m\) and \(V_B\) of size \(2n\):
\begin{equation} \rho_V\text{ is separable} \quad\Longleftrightarrow\quad \exists V_A,V_B:\quad V_A+i\Omega_m\geq0,\quad V_B+i\Omega_n\geq0,\quad V\geq V_A\oplus V_B. \tag{2} \end{equation}Equation (2) solves the unrestricted Gaussian separability criterion. [WW01]
With \(T_B:=I_{2m}\oplus\bigoplus_{j=1}^{n}\operatorname{diag}(1,-1)\), partial-transpose positivity is equivalent to
\begin{equation} T_BVT_B+i\Omega_{m+n}\geq0. \tag{3} \end{equation}Equation (3) is also sufficient for separability if \(m=1\) or \(n=1\), or if the Gaussian state is invariant under all permutations of the modes on one party; it fails to be sufficient for general \(m=n=2\). [WW01], [LSA18] The permutation-symmetric extension is Theorem 9 of Lami et al. [LSA18]
Comment
The full criterion is solved by peer-reviewed results. Semidefinite feasibility is an exact mathematical characterization. A demand for a particular elementary expression would require a separately specified expression class.