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arXiv:2510.00064

An operator-based bound on information and disturbance in quantum measurements

Williams, Hollis, Holger Friedrich Hofmann

Published Sep 29, 2025• 0 citations
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Abstract

Quantum measurements can be described by operators that assign conditional probabilities to different outcomes while also describing unavoidable physical changes to the system. Here, we point out that operators describing information gain at minimal disturbance can be expanded into a set of unitary operators representing experimentally distinguishable patterns of disturbance. The observable statistics of disturbance defines a tight upper bound on the information gain of the measurement.

DOI
10.48550/arxiv.2510.00064

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Quantum measurements can be described by operators that assign conditional probabilities to different outcomes while also describing unavoidable physical changes to the system. Here, we point out that operators describing information gain at minimal disturbance can be expanded into a set of unitary operators representing experimentally distinguishable patterns of disturbance. The observable statistics of disturbance defines a tight upper bound on the information gain of the measurement.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
Quantum mechanics predicts the joint probability distributions of the outcomes of simultaneous measurements of commuting observables, but the current formulation lacks the operational definition of simultaneous measurements. In order to provide foundations of joint statistics of local general measurements on entangled systems in a general theoretical framework, the question is answered as to under what condition the outputs of two measuring apparatuses satisfy the joint probability formula for simultaneous measurements of their observables. For this purpose, all the possible state changes caused by measurements of an observable are characterized, and the notion of disturbance in measurement is formalized in terms of operations derived by the measuring interaction.
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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.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.