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arXiv:2507.03346

Optimal quantum measurements for additive information and disturbance measures

Hiroaki Terashima

Published Jul 4, 2025• 0 citations
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Abstract

Additive measures for information and disturbance in quantum measurements of a system are defined from well-known multiplicative measures such as estimation and operation fidelities using a logarithm. This is motivated by the fact that information and disturbance are naturally assumed to be additive while performing independent measurements on separable systems. Although the additivity makes no remarkable difference when information and disturbance are separately considered, it can change measurements that only introduce minimal disturbance relative to the amount of information. Such optimal measurements are shown for additive information and disturbance measures with a tradeoff relationship.

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Additive measures for information and disturbance in quantum measurements of a system are defined from well-known multiplicative measures such as estimation and operation fidelities using a logarithm. This is motivated by the fact that information and disturbance are naturally assumed to be additive while performing independent measurements on separable systems. Although the additivity makes no remarkable difference when information and disturbance are separately considered, it can change measurements that only introduce minimal disturbance relative to the amount of information. Such optimal measurements are shown for additive information and disturbance measures with a tradeoff relationship.
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We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under the relabelling of outcomes and allow for the possibility of using quantum or classical operations to "correct" for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also investigated, and the optimality is analyzed by explicit evaluation of fidelities. We found that the estimation fidelity does not depend on the number of users, neither for single-measure inference nor for collective one, whereas the transmission fidelity decreases with N . The resulting trade-off is no longer optimal and degrades with increasing N . We found that optimality can be restored by an effective preparation of the probe states and present explicitly calculations for the 2-user case.
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also investigated, and the optimality is analyzed by explicit evaluation of fidelities. We found that the estimation fidelity does not depend on the number of users, neither for single-measure inference nor for collective one, whereas the transmission fidelity decreases with N . The resulting trade-off is no longer optimal and degrades with increasing N . We found that optimality can be restored by an effective preparation of the probe states and present explicitly calculations for the 2-user case.
We present graphs of information versus disturbance for general quantum measurements of completely unknown states. Each piece of information and disturbance is quantified by two measures: (i) the Shannon entropy and estimation fidelity for the information and (ii) the operation fidelity and physical reversibility for the disturbance. These measures are calculated for a single outcome based on the general formulas derived by the present author (Terashima in Phys Rev A 93:022104, 2016 ) and are plotted on four types of information–disturbance planes to show their allowed regions. In addition, we discuss the graphs of these metrics averaged over all possible outcomes and the optimal measurements when saturating the upper bounds on the information for a given disturbance. The results considerably broaden the perspective of trade-offs between information and disturbances in quantum measurements.