Compression of quantum-measurement operations - art. no. 012311

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 s tate ensemble. In the previous work it was shown that quantum measurements generally contain spurious randomness in the outcomes and that this spuriou s randomness can be eliminated by carrying out collective measurements on m any independent copies of the system. In particular it was shown that, with out decreasing the amount of knowledge the measurement provides about the q uantum state, one can always reduce the amount of data produced by the meas urement to the von Neumann entropy H(rho) = -Tr rho log rho of the ensemble . Here we extend this result by giving a more refined description of what c onstitutes 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 positi ve operator-valued measure (POVM) having elements a(j) with an equivalent P OVM acting on many independent copies of the system, which produces an amou nt of data asymptotically equal to the entropy defect of an ensemble canoni cally associated with the ensemble average state rho and the initial measur ement (a(j)). In the case where the measurement is not maximally refined th is amount of data is strictly less than the amount H(rho) obtained in the p revious work. We also show that this is the best achievable, i.e., it is im possible 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 qu antum state by a quantum measurement. We also discuss in detail the relatio n between our results and Holevo's bound, at the same time providing a self -contained proof of this fundamental inequality.