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arXiv:1401.1013 · Physical Review A

Optimized entropic uncertainty for successive projective measurements

Kyunghyun Baek, Tristan Farrow, Wonmin Son

Published Mar 10, 2014• 27 citations
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Abstract

We focus here on the uncertainty of an observable $Y$ caused by a precise measurement of $X$. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components $X$ and $Y$. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of $Y$. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.

DOI
10.1103/physreva.89.032108

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We focus here on the uncertainty of an observable YYY caused by a precise measurement of XXX. We illustrate the effect by analyzing the general scenario of two successive measurements of spin components XXX and YYY. We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of YYY. We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.
Information-theoretic uncertainty relations formulate the joint immeasurability of two noncommuting observables in terms of information entropies.The tradeoff of the accuracy in the outcome of two successive measurements manifests in entropic noise-disturbance uncertainty relations.Recent theoretical analysis predicts that projective measurements are not optimal, with respect to the noise-disturbance tradeoffs.Therefore, the results in our previous paper [Phys. Rev. Lett. 115, 030401 (2015)] are outperformed by general quantum measurements.Here, we experimentally test a tight information-theoretic measurement uncertainty relation for three-outcome positive-operator-valued measures, using neutron spin-1 2 qubits.The obtained results violate the lower bound for projective measurements as theoretically predicted.
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.
The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.
Information-theoretic definitions for noise and disturbance in quantum measurements were given in [Phys. Rev. Lett. 112, 050401 (2014)] and a state-independent noise-disturbance uncertainty relation was obtained. Here, we derive a tight noise-disturbance uncertainty relation for complementary qubit observables and carry out an experimental test. Successive projective measurements on the neutron's spin-1/2 system, together with a correction procedure which reduces the disturbance, are performed. Our experimental results saturate the tight noise-disturbance uncertainty relation for qubits when an optimal correction procedure is applied.
Entropic Uncertainty relations are powerful tools, especially in quantum cryptography. They typically bound the amount of uncertainty a third-party adversary may hold on a measurement outcome as a result of the measurement overlap. However, when the two measurement bases are biased towards one another, standard entropic uncertainty relations do not always provide optimal lower bounds on the entropy. Here, we derive a new entropic uncertainty relation, for certain quantum states, which can provide a significantly higher bound even if the two measurement bases are no longer mutually unbiased. We evaluate our bound on two different quantum cryptographic protocols, including BB84 with faulty/biased measurement devices, and show that our new bound can produce substantially higher key-rates under several scenarios when compared with prior work using standard entropic uncertainty relations.
Abstract The uncertainty relation is regarded as a remarkable feature of quantum mechanics differing from the classical counterpart, and it plays a backbone role in the region of quantum information theory. In principle, the uncertainty relation offers a nontrivial limit to predict the outcome of arbitrarily incompatible observed variables. Therefore, to pursue a more general uncertainty relations ought to be considerably important for obtaining accurate predictions of multi-observable measurement results in genuine multipartite systems. In this article, we derive a generalized entropic uncertainty relation (EUR) for multi-measurement in a multipartite framework. It is proved that the bound we proposed is stronger than the one derived from Renes et al. in [Phys. Rev. Lett. 103,020402(2009) ] for the arbitrary multipartite case. As an illustration, we take several typical scenarios that confirm that our proposed bound outperforms that presented by Renes et al. Hence, we believe our findings provide generalized uncertainty relations with regard to multi-measurement setting, and facilitate the EUR’s applications on quantum precision measurement regarding genuine multipartite systems.