Information-theoretical aspects of quantum measurement

Abstract

We present criteria for comparing measurements on a given system from the point of view of the information they provide. These criteria lead to a concept of informational completeness of a set of observables, which generalizes the conventional concept of completeness. The entropy of a state with respect to an arbitrary sample space of potential measurement outcomes is defined, and then studied in the context of configuration space and fuzzy stochastic phase space.

Informationally complete measurements and group representation

“…The problem then arises on how to achieve a kind of quantum measurement that is "complete" [1,2], in the sense that it can be used to infer information on all possible (also exclusive) observables. The main idea is to perform a generalized "unsharp" measurement, described by a so-called POVM (positive-operator valued measure), from which a specific type of information-i.e.…”

Least-bias state estimation with incomplete unbiased measurements

“…The largest value of the von Neumann entropy is 0.6370, however; we obtain it for the (2) other with q j +k z (2) 3 = q j −k+1 z (2) 4 = −0.094 66. It is, of course, hardly surprising that the Shannon and the von Neumann entropy are usually maximized by different (M) other 's.…”

“…ULIN = (4) ULIN have eigenvalues 0, 1 6 (3 ± √ 3) and von Neumann entropy S( (2) ULIN ) = 0.5157. The largest value of the von Neumann entropy is 0.6370, however; we obtain it for the (2) other with q j +k z (2) 3 = q j −k+1 z (2) 4 = −0.094 66.…”

“…When the measurement outcomes form an informationally complete set [2][3][4][5][6], the data obtained contain maximal information and a unique state estimator can be inferred. Unfortunately, as we probe more intricate quantum systems [7,8], such an informationally complete set of measurements becomes extremely difficult to implement.…”

Pure-state informationally complete and “really” complete measurements

“…If the statistics of the outcome of these measurements are sufficient to uniquely identify the state, the measurements are called "informationally complete" [1] (I-complete). In this note I will present some results for a special case of this situation, in which you know that the system is in a pure state, but you don't know in which pure state.…”