Compression of Pure and Mixed States in Quantum Detection

Quantum detection in an N-dimensional Hilbert space H involves quantum states and corresponding measurement operators which span an r-dimensional subspace U of H, with r<=N. Quantum detection could be restricted to this subspace, but the detection operations performed in U are still redundant, since the kets have N components. By applying the singular-value decomposition to the state matrix, it is possible to perform a compression from the subspace U onto a "compressed" space $overline{U}$, where the redundancy is removed and kets are represented by r components. The quantum detection can be perfectly reformulated in the "compressed" space, without loss of information, with a greatly reduced complexity. The compression is particularly attractive when r<<N, as shown with an example of application to quantum optical communications.