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arXiv:quant-ph/0309091 · Physical Review A

Prior information: How to circumvent the standard joint-measurement uncertainty relation

Michael J. W. Hall

Published May 20, 2004• 140 citations
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Abstract

The principle of complementarity is quantified in two ways: by a universal uncertainty relation valid for arbitrary joint estimates of any two observables from a given measurement setup and by a general uncertainty relation valid for the optimal estimates of the same two observables when the state of the system prior to measurement is known. A formula is given for the optimal estimate of any given observable, based on arbitrary measurement data and prior information about the state of the system, which generalizes and provides a more robust interpretation of previous formulas for ``local expectations'' and ``weak values'' of quantum observables. As an example, the canonical joint measurement of position $X$ and momentum $P$ corresponds to measuring the commuting operators ${X}_{J}=X+{X}^{\ensuremath{'}}$ and ${P}_{J}=P\ensuremath{-}{P}^{\ensuremath{'}}$, where the primed variables refer to an auxiliary system in a minimum-uncertainty state. It is well known that $\ensuremath{\Delta}{X}_{J}\ensuremath{\Delta}{P}_{J}\ensuremath{\geqslant}\ensuremath{\hbar}$. Here it is shown that given the same physical experimental setup and knowledge of the system density operator prior to measurement, one can make improved joint estimates ${X}_{opt}$ and ${P}_{opt}$ of $X$ and $P$. These improved estimates are not only statistically closer to $X$ and $P$ but further satisfy $\ensuremath{\Delta}{X}_{opt}\ensuremath{\Delta}{P}_{opt}\ensuremath{\geqslant}\ensuremath{\hbar}∕4$, where equality can be achieved in certain cases. Thus one can do up to four times better than the standard lower bound (where the latter corresponds to the limit of no prior information). Other applications include the heterodyne detection of orthogonal quadratures of a single-mode optical field and joint measurements based on Einstein-Podolsky-Rosen correlations.

DOI
10.1103/physreva.69.052113

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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.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.