We propose an error-disturbance relation for general observables on finite dimensional Hilbert spaces based on operational notions of error and disturbance. For two-dimensional systems we derive tight inequalities expressing the trade-off between accuracy and disturbance.
Another relation is due to Hofmann @cite_9 . Consider a POVM (the @math is not an index) F_ H,m = , where @math ranges over some set of measurement outcomes (the number of outcomes does have to be related to the dimension @math of @math ), and introduce the errors H,m ^2 = (A- A )^2 , H,m ^2 = ( A- A )^2 . Here @math is the retrodictive'' state corresponding to the measurement outcome @math , explicitly = F_ H,m [F_ H,m ] . We then have the relation @cite_9 (see also @cite_3 ) H,m H,m 1 2 | [A, A] |.
@cite_9 @cite_3
2027707976 1514676086
The effects of any quantum measurement can be described by a collection of measurement operators l_brace M sub m r_brace acting on the quantum state of the measured system. However, the Hilbert space formalism tends to obscure the relationship between the measurement results and the physical properties of the measured system. In this paper, a characterization of measurement operators in terms of measurement resolution and disturbance is developed. It is then possible to formulate uncertainty relations for the measurement process that are valid for arbitrary input states. The motivation of these concepts is explained from a quantum communication viewpoint. It is shown that the intuitive interpretation of uncertainty as a relation between measurement resolution and disturbance provides a valid description of measurement back action. Possible applications to quantum cryptography, quantum cloning, and teleportation are discussed. We critically revisit the definitions of mean-squared estimation error and disturbance recently used in error-disturbance inequalities derived by Ozawa, Hall, Branciard, and by expressing them in the reduced system space. The interpretation of the definitions as mean-squared deviations relies on a hidden assumption that is incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-definite joint quasiprobability distribution. For unbiased measurements, the estimation error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect estimations can have nonzero estimation error according to this measure. We conclude that the inequalities using these definitions do not capture the spirit of Heisenberg's eponymous inequality, but do indicate a qualitatively different relationship between dispersion and disturbance that applies to ensembles of measurements. To reconnect with the discussion of Heisenberg, we suggest alternate definitions of error and disturbance that are intrinsic to a single apparatus outcome. These definitions naturally involve the retrodictive and interdictive states for that outcome, and produce complementarity and error-disturbance inequalities that have the same form as the traditional Heisenberg relation.