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A Unified Inequality Linking Error–Disturbance and Order Effects for Sequential Quantum Instruments

Yoshida, Kazuyuki

Published Dec 11, 2025• 0 citations
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Abstract

We relate two operational facets of noncommutativity in sequential quantum measurements: error-disturbance (how much one instrument perturbs another) and order effects (how outcome statistics change under permutation of instruments). Fixing the l1 distance on outcome distributions, we prove a per-state inequality that lower bounds the order-effect magnitude by the sum of two directional disturbances. A minimal qubit family shows tightness for all rotation angles. The inequality is directly testable from finite counts and also connects to discussions of order effects beyond physics.

DOI
10.5281/zenodo.17894830

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We relate two operational facets of noncommutativity in sequential quantum measurements: error-disturbance (how much one instrument perturbs another) and order effects (how outcome statistics change under permutation of instruments). Fixing the l1 distance on outcome distributions, we prove a per-state inequality that lower bounds the order-effect magnitude by the sum of two directional disturbances. A minimal qubit family shows tightness for all rotation angles. The inequality is directly testable from finite counts and also connects to discussions of order effects beyond physics.
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The focus of the present investigation is uncertainty relations for quantum particles, which quantify the fundamental limitations on some of their properties due to their incompatibility. In the first, longer part, we are concerned with preparational uncertainty \nrelations, while in the second we touch upon measurement inequalities through a generalisation of a model of joint measurement of position and momentum. \nSpecifically, starting from a triple of canonical operators, we prove product and sum inequalities for their variances with bounds larger than those following from combining the pairwise ones. We extend these results to N observables for a quantum particle and prove uncertainty relations for the sums and products of their variances \nin terms of the commutators. \nFurthermore, we present a general theory of preparational uncertainty relations for a quantum particle in one dimension and derive conditions for a smooth function of the second moments to assume a lower bound. The Robertson-Schroedinger inequality \nis found to be of special significance and we geometrically study the space of second moments. We prove new uncertainty relations for various functions of the variances and covariance of position and momentum of a quantum particle. Some of our findings are shown to extend to more than one spatial degree of freedom and we derive \nvarious types of inequalities. \nFinally, we propose a generalisation of the Arthurs-Kelly model of joint measurement of position and momentum to incorporate the case of more than two observables and derive joint-measurement inequalities for the statistics of the probes. For the case of \nthree canonical observables and suitable definitions of error and disturbance we obtain a number of error-error-error and error-error-disturbance inequalities and show that the lower bound is identical to the preparational one.
We formulate a new error-disturbance relation, which is free from explicit dependence upon variances in observables. This error-disturbance relation shows improvement over the one provided by the Branciard inequality and the Ozawa inequality for some initial states and for a particular class of joint measurements under consideration. We also prove a modified form of Ozawa's error-disturbance relation. The latter relation provides a tighter bound compared to the Ozawa and the Branciard inequalities for a small number of states.