Exact covariance criterion for bipartite Gaussian separability
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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
Reports do not certify correctness or automatically change the problem's status. Progress policy.
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.