Decoupling cross-quadrature correlations using passive operations

Quadrature correlations between subsystems of a Gaussian quantum state are fully characterized by its covariance matrix. For example, the covariance matrix determines the amount of entanglement or decoherence of the state. Here, we establish when it is possible to remove correlations between conjugate quadratures using only passive operations. Such correlations are usually undesired and arise due to experimental cross-quadrature contamination. Using the Autonne-Takagi factorization, we present necessary and sufficient conditions to determine when such removal is possible. Our proof is constructive, and whenever it is possible we obtain an explicit expression for the required passive operation.