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Il Nuovo Cimento B

Bounds on the entropic uncertainty in distinguished and successive measurements of spin-(1/2) operators

Mojtaba Jafarpour, Shaaker Hajati

Published Mar 1, 2002• 0 citations
OpenAlex

Abstract

We calculate the exact entropic uncertainties in distinguished and successive measurements of two and three spin-(½) operators. Optimal upper and lower bounds for each measurement are obtained and an already existing upper bound is also improved.

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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.
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We show that for fermion states, measurements of any two finite outcome particle quantum numbers (e.g., spin) are not constrained by a minimum total uncertainty. We begin by defining uncertainties in terms of the outputs of a measurement apparatus. This allows us to compare uncertainties between multiparticle states of distinguishable and indistinguishable particles. Entropic uncertainty relations are derived for both distinguishable and indistinguishable particles. We then derive upper bounds on the minimum total uncertainty for bosons and fermions. These upper bounds apply to any pair of particle quantum numbers and depend only on the number of particles NNN and the number of outcomes nnn for the quantum numbers. For general NNN, these upper bounds necessitate a minimum total uncertainty much lower than that for distinguishable particles. The fermion upper bound on the minimum total uncertainty for NNN an integer multiple of nnn is zero. Our results show that uncertainty limits derived for single particle observables are valid only for particles that can be effectively distinguished. Outside this range of validity, the apparent fundamental uncertainty limits can be overcome.
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.