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arXiv:2309.14172

Error and Disturbance as Irreversibility with Applications: Unified Definition, Wigner--Araki--Yanase Theorem and Out-of-Time-Order Correlator

Haruki Emori, Hiroyasu Tajima

Published Sep 25, 2023• 0 citations
arXivHTMLPDF

Abstract

Defining an error of measurement has long been a foundational problem in science: even in classical experiments, data are statistical and admit no single universally optimal definition of error. In quantum mechanics, the challenge deepens: observed entities often lack preexisting definite values, and the act of measurement unavoidably disturbs the system of interest. Consequently, both error and disturbance must be quantified, and various definitions have been proposed to date. However, a unified perspective for understanding the differences and similarities among these diverse definitions of error and disturbance, and an operational framework for distinguishing between them, remain elusive. In this Letter, we propose a novel framework for defining error and disturbance using irreversibility. Our framework converts the error and disturbance of a quantum measurement of a system under consideration into the irreversibility of an ancillary qubit system, using a quantum comb composed of loss and recovery processes. The mechanism enables us to make the operational distinction that error uses the classical outputs, while disturbance uses the quantum outputs of the measurement in the recovery process. Furthermore, our framework yields several key consequences: (i) it encompasses existing definitions, (ii) it establishes a universal constraint on error and disturbance defined by any measure of an arbitrary quantum process under a conservation law, and (iii) it reveals an operational connection between irreversibility and the out-of-time-ordered correlator (OTOC), a metric of quantum chaos. It also provides a constraint on the OTOC under a conservation law and a method for its experimental evaluation, which is demonstrated on a quantum processor.

DOI
10.48550/arxiv.2309.14172

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Defining an error of measurement has long been a foundational problem in science: even in classical experiments, data are statistical and admit no single universally optimal definition of error. In quantum mechanics, the challenge deepens: observed entities often lack preexisting definite values, and the act of measurement unavoidably disturbs the system of interest. Consequently, both error and disturbance must be quantified, and various definitions have been proposed to date. However, a unified perspective for understanding the differences and similarities among these diverse definitions of error and disturbance, and an operational framework for distinguishing between them, remain elusive. In this Letter, we propose a novel framework for defining error and disturbance using irreversibility. Our framework converts the error and disturbance of a quantum measurement of a system under consideration into the irreversibility of an ancillary qubit system, using a quantum comb composed of loss and recovery processes. The mechanism enables us to make the operational distinction that error uses the classical outputs, while disturbance uses the quantum outputs of the measurement in the recovery process. Furthermore, our framework yields several key consequences: (i) it encompasses existing definitions, (ii) it establishes a universal constraint on error and disturbance defined by any measure of an arbitrary quantum process under a conservation law, and (iii) it reveals an operational connection between irreversibility and the out-of-time-ordered correlator (OTOC), a metric of quantum chaos. It also provides a constraint on the OTOC under a conservation law and a method for its experimental evaluation, which is demonstrated on a quantum processor.
Measurement error and disturbance, in the presence of conservation laws, are analysed in general operational terms. We provide novel quantitative bounds demonstrating necessary conditions under which accurate or non-disturbing measurements can be achieved, highlighting an interesting interplay between incompatibility, unsharpness, and coherence. From here we obtain a substantial generalisation of the Wigner-Araki-Yanase (WAY) theorem. Our findings are further refined through the analysis of the fixed-point set of the measurement channel, some extra structure of which is characterised here for the first time.
Quantum technologies offer exceptional -- sometimes almost magical -- speed and performance, yet every quantum process costs physical resources. Designing next-generation quantum devices, therefore, depends on solving the following question: which resources, and in what amount, are required to implement a desired quantum process? Casting the problem in the language of quantum resource theories, we prove a universal cost-irreversibility tradeoff: the lower the irreversibility of a quantum process, the greater the required resource cost for its realization. The trade-off law holds for a broad range of resources -- energy, magic, asymmetry, coherence, athermality, and others -- yielding lower bounds on resource cost of any quantum channel. Its broad scope positions this result as a foundation for deriving the following key results: (1) we show a universal relation between the energetic cost and the irreversibility for arbitrary channels, encompassing the energy-error tradeoff for any measurement or unitary gate; (2) we extend the energy-error tradeoff to free energy and work costs; (3) we extend the Wigner-Araki-Yanase theorem, which is the universal limitation on measurements under conservation laws, to a wide class of resource theories: the probability of failure in distinguishing resourceful states via a measurement is inversely proportional to its resource cost; (4) we prove that infinitely many resource-non-increasing operations in fact require an infinite implementation cost. These findings reveal a universal relationship between quantumness and irreversibility, providing a first step toward a general theory that explains when -- and how -- quantumness can suppress irreversibility.
The out-of-time-ordered correlator (OTOC) is a powerful tool for probing quantum information scrambling, a fundamental process by which local information spreads irreversibly throughout a quantum many-body system. Experimentally measuring the OTOC, however, is notoriously challenging due to the need for time-reversed evolution. Here, we present an experimental evaluation of the OTOC on a quantum computer, using three distinct protocols to address this challenge: the rewinding time method (RTM), the weak-measurement method (WMM), and the irreversibility-susceptibility method (ISM). Our experiments investigate the quantum dynamics of an XXZ spin-1/2 chain prepared in a thermal Gibbs state. As a key contribution, we provide the first experimental demonstration of the ISM, using the trapped-ion quantum computer, reimei. We also conduct a detailed comparative analysis of all three methods, revealing method-dependent behaviors in the measured OTOC. This work not only validates these protocols as practical tools for exploring quantum chaos on near-term hardware but also offers crucial insights into their respective advantages and limitations, providing a practical framework for future experimental investigations.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.
Entanglement and state distinguishability have long been central topics in quantum foundations. While each has developed into a rich subject in its own right, they can be connected through measurements in complementary bases. This connection provides insights into quantum and classical correlations and underlies many information-processing tasks, most notably quantum error correction (QEC). However, it has remained largely open whether the connection between entanglement and distinguishability extends to general measurements without assuming complementarity. We answer this question in the affirmative: the key is irreducibility induced by measurements, a much more relaxed condition than complementarity. Specifically, for POVMs satisfying an irreducibility condition, we establish a quantitative relation between the infidelity of one-sided local transformation into a maximally entangled state and the failure probability in state discrimination. We then characterize when both errors can vanish simultaneously. Our results have direct applications to entanglement distillation and QEC. Their errors can be certified by estimating classical input-output correlations from experimentally accessible local measurements, without requiring complementarity. The results also extend to quantum measurement theory. We derive novel trade-off relations that bridge two historically distinct approaches: information gain--irreversibility and observable noise--disturbance. Our connection between entanglement and distinguishability via irreducibility thus offers a common framework for entanglement distillation, QEC, and measurement trade-offs.