An operational measure for squeezing (Q2834817)

An \(n\)-mode bosonic state \(\varrho\) with covariance matrix \(\gamma_{\varrho}\) is called squeezed if \(\gamma_{\varrho}\) possesses an eigenvalue \(\lambda <1\). The authors defines the measure \(G(\gamma_{\varrho}=\inf\left\{ \left. \sum_{j=1}^n\log s_j^{\downarrow}(S)\;\right|\;\gamma_{\varrho} \geq S^TS, \;S\in Sp(2n)\, \right\}\), where \(s_j^{\downarrow}(S)\) denotes the \(j\)-th singular value of \(S\) ordered decreasingly, and shows that \(G(\gamma_{\varrho})\) quantifies the minimal amount of squeezing needed to prepare the quantum state \(\varrho \). The proposed squeezing measure is convex, subadditive, and can be considered as a squeezing analogue of the entanglement of formation. The given analytical formula does not allow an efficient exact computation in the case \(n>1\), but \(G(\gamma_{\varrho})\) can be approximately computed by using the numerical convex optimization algorithm proposed by the authors.