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arXiv:1402.7311 · Physical Review A

Disturbance trade-off principle for quantum measurements

Prabha Mandayam, M. D. Srinivas

Published Dec 23, 2014• 8 citations
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Abstract

We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy (${T}_{2}$). We also investigate the disturbances associated with the class of nonprojective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.

DOI
10.1103/physreva.90.062128

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We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy (T2{T}_{2}T2​). We also investigate the disturbances associated with the class of nonprojective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.
Information-theoretic definitions for the noise associated with a quantum measurement and the corresponding disturbance to the state of the system have recently been introduced [F. Buscemi et al., Phys. Rev. Lett. 112, 050401 (2014)]. These definitions are invariant under relabeling of measurement outcomes, and lend themselves readily to the formulation of state-independent uncertainty relations both for the joint estimate of observables (noise-noise relations) and the noise-disturbance tradeoff. Here we derive such relations for incompatible qubit observables, which we prove to be tight in the case of joint estimates, and present progress towards fully characterizing the noise-disturbance tradeoff. In doing so, we show that the set of obtainable noise-noise values for such observables is convex, whereas the conjectured form for the set of obtainable noise-disturbance values is not. Furthermore, projective measurements are not optimal with respect to the joint-measurement noise or noise-disturbance tradeoffs. Interestingly, it seems that four-outcome measurements are needed in the former case, whereas three-outcome measurements are optimal in the latter.
Heisenberg's original intuition was that there should be a tradeoff between measuring a particle's position with greater precision and disturbing its momentum. Rigorous formulations of this idea have primarily focused on the question of how well two complementary observables can be jointly measured. Here, we provide an alternative approach based on how enhancing the predictability of one observable necessarily disturbs a complementary one. The tradeoff refers to a clear operational scenario capturing the effect of the measurement process on a single quantum system. Moreover, our relation is expressed by entropic quantities with clear statistical meaning evading recent criticism directed at some previous formulations. We discuss the performance of our tradeoff relation for existing experimental setups involving qubit measurements performed in Vienna and Toronto, and show that our relation is perfectly tight for all measurement strengths in the Toronto setup.
We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables AAA and BBB with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of AAA followed by a measurement of BBB on a given state and the other obtained from a measurement of BBB alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables AAA and BBB, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite-dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important relation between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of nonprojective measurements. In particular, we obtain a nontrivial upper bound on the incompatibility of one L\"uders instrument with another.
Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report various error-disturbance tradeoff relations by decomposing the measurement errors and disturbance into two different components, namely, operator bias and fuzziness. Our uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance.
Obtaining information from a quantum system through a measurement typically disturbs its state. The postmeasurement states for a given measurement, however, are not unique and highly rely on the chosen measurement model, complicating the puzzle of information disturbance. Two distinct questions are then in order. First, what is the minimum disturbance a measurement may induce? Second, when a fixed disturbance occurs, how informative is the possible measurement in the best-case scenario? Here we propose various approaches to tackle these questions and provide explicit solutions for the set of unbiased binary qubit measurements and postmeasurement state spaces that are equivalent to the image of a unital qubit channel. In particular, we show there are different tradeoff relations between the sharpness of this measurement and the average fidelity of the premeasurement and postmeasurement state spaces as well as the sharpness and quantum resources preserved in the postmeasurement states in terms of coherence and discordlike correlation once the measurement is applied locally.
The indeterminacy of quantum mechanics was originally presented by Heisenberg through the tradeoff between the measuring error of the observable A and the consequential disturbance to the value of another observable B. This tradeoff now has become a popular interpretation of the uncertainty principle. However, the historic idea has never been exactly formulated previously and is recently called into question. A theory built upon operational and state-relevant definitions of error and disturbance is called for to rigorously reexamine the relationship. Here by putting forward such natural definitions, we demonstrate both theoretically and experimentally that there is no tradeoff if the outcome of measuring B is more uncertain than that of A. Otherwise, the tradeoff will be switched on and well characterized by the Jensen-Shannon divergence. Our results reveal the hidden effect of the uncertain nature possessed by the measured state, and conclude that the state-relevant relation between error and disturbance is not almosteverywhere a tradeoff as people usually believe.