Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

Abstract

Heisenberg's uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a 'true' probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.

Entropic Approach to Error-Disturbance Tradeoff in Quantum Measurements

“…Consequently our estimation for the error plus disturbance is an operational one different from the explicit bound as − log c. Now we have finished our textbook-level reproduction of the result by Barchielli et al [3] and are ready to put the formulation by Buscemi et al [2] in the context of Barchielli et al [3] First we generalize the way [9] of calculating the conditional entropy for a single-observable M m to describe general measurement situations. Then we apply it to the bi-observable.…”

“…[3] By the discussion in ref. 3, however, neither of these approximation schemes [6,7] seems to work for the nonorthogonal case.…”

“…The input state of this description is restricted to the uniform ensemble of |x x| or |y y| so that Coles et al [1] classified this description as a "calibration" approach. In the following we discuss the case of general input states after Barchielli et al [3] A significant progress has been made by Barchielli et al [3] after the review by Coles et al [1]. Next we reproduce it in our textbook-level description.…”

“…From the view point of the state-independent error-disturbance tradeoff only the work by Buscemi et al [2] is described in the section VII-C-1 in this review. More recently a significant development has been made by Barchielli et al [3] Here we try to put the formulation by Buscemi et al [2] in the context of Barchielli et al [3] First we reproduce the result by Buscemi et al [2] after Coles et al [1] In the sequential measurement of the observablesX andŶ in the d-dimensional Hilbert space the error is given by N(X, M X ) = H(X|M X ) = H(X) − I(X : M X ), and the "optimized" disturbance by…”

“…The approximated distribution compared with A ρ is the set of M An explicit form of M(x, y) for the case of qubit measurements is as follows. The construction of the bi-observable for qubit experiments by Barchielli et al [3] is done on the basis of a symmetry argument. However, experimental implementations become transparent by the description of approximation schemes such as the stochastic mixing [6] and the misalignment.…”