Compression of quantum-measurement operations

We generalize the recent work of Massar and Popescu dealing with the amount of classical data that is produced by a quantum measurement on a quantum state ensemble. In the previous work it was shown that quantum measurements generally contain spurious randomness in the outcomes and that this spurious randomness can be eliminated by carrying out collective measurements on many independent copies of the system. In particular it was shown that, without decreasing the amount of knowledge the measurement provides about the quantum state, one can always reduce the amount of data produced by the measurement to the von Neumann entropy $H\left(\rho \right)=-\mathrm{Tr}\rho \log \rho$ of the ensemble. Here we extend this result by giving a more refined description of what constitutes equivalent measurements (that is, measurements which provide the same knowledge about the quantum state) and also by considering incomplete measurements. In particular, we show that one can always associate a positive operator-valued measure (POVM) having elements $a_j$ with an equivalent POVM acting on many independent copies of the system, which produces an amount of data asymptotically equal to the entropy defect of an ensemble canonically associated with the ensemble average state ρ and the initial measurement ${(a}_j).\mathrm{}$ In the case where the measurement is not maximally refined this amount of data is strictly less than the amount $H\left(\rho \right)$ obtained in the previous work. We also show that this is the best achievable, i.e., it is impossible to devise a measurement equivalent to the initial measurement ${(a}_j)$ that produces less data. We discuss the interpretation of these results. In particular, we show how they can be used to provide a precise and model-independent measure of the amount of knowledge that is obtained about a quantum state by a quantum measurement. We also discuss in detail the relation between our results and Holevo’s bound, at the same time providing a self-contained proof of this fundamental inequality.