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

Haruki Emori, Hiroyasu Tajima

TL;DR

The paper presents an irreversibility-based framework to define quantum measurement error and disturbance, mapping these quantities to irreversibility on an ancillary qubit via quantum combs. It unifies prior formalisms (AKG, Ozawa, LT, WSU, BLW) under a single operational perspective and extends the Wigner–Araki–Yanase theorem to arbitrary definitions and processes under conservation laws. By linking irreversibility to the OTOC, it derives a universal bound on quantum chaos metrics and proposes a practical, end-point experimental method to evaluate the OTOC on current hardware. The approach relies on SIQ-trade-offs and quantum Fisher information to quantify coherence costs, enabling a broad applicability across finite-dimensional systems and providing a path from non-equilibrium physics to quantum foundations. Overall, the framework offers a comprehensive, experimentally accessible, and theoretically robust unification of error, disturbance, WAY-type limits, and scrambling diagnostics.

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.

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

TL;DR

The paper presents an irreversibility-based framework to define quantum measurement error and disturbance, mapping these quantities to irreversibility on an ancillary qubit via quantum combs. It unifies prior formalisms (AKG, Ozawa, LT, WSU, BLW) under a single operational perspective and extends the Wigner–Araki–Yanase theorem to arbitrary definitions and processes under conservation laws. By linking irreversibility to the OTOC, it derives a universal bound on quantum chaos metrics and proposes a practical, end-point experimental method to evaluate the OTOC on current hardware. The approach relies on SIQ-trade-offs and quantum Fisher information to quantify coherence costs, enabling a broad applicability across finite-dimensional systems and providing a path from non-equilibrium physics to quantum foundations. Overall, the framework offers a comprehensive, experimentally accessible, and theoretically robust unification of error, disturbance, WAY-type limits, and scrambling diagnostics.

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.
Paper Structure (14 sections, 3 theorems, 147 equations, 4 figures, 2 tables)

This paper contains 14 sections, 3 theorems, 147 equations, 4 figures, 2 tables.

Table of Contents

  1. End Matter
  2. I. Existing errors and disturbances as irreversibility
  3. A. Example 1: Ozawa's error and disturbance
  4. B. Example 2: Arthurs– Kelly– Goodman's error (Ishikawa– Ozawa's error)
  5. C. Example 3: Lee– Tsutsui's error and disturbance
  6. D. Example 4: Watanabe– Sagawa– Ueda's error and disturbance
  7. E. Example 5: Busch– Lahti– Werner's error and disturbance
  8. I I. On the differentiability of our definition with respect to $\theta$
  9. I I I. Wigner– Araki– Yanase theorem for arbitrary errors and disturbances
  10. A. Derivation of Theorem \ref{['th:WAY']} in the main text
  11. B. An improved version of Eq. \ref{['eq:WAY_error']}
  12. C. Relation between our formulation of the WAY theorem and the previous formulation of the WAY theorems for Ozawa's error
  13. D. Discussion: why the quantitative WAY theorem without the Yanase condition is possible
  14. I V. OTOC as irreversibility

Key Result

Theorem 1

Consider a measurement $\mathcal{M}$ of an observable $A$ in $\mathbf{S}$. We realize the measuring process by $\mathcal{P}_{\mathcal{M}}$ for error- and $\mathcal{I}_{\mathcal{M}}$ for disturbance-evaluation under the conservation law of the conserved charges $X_{\bullet}$$(\bullet=\mathbf{S},\math Here, $\mathcal{F}^{\text{cost}}_{\mathcal{N}}$ is the implementation resource cost of a CPTP map $

Figures (4)

  • Figure 1: A schematic of irreversibility evaluation protocols for general definition of error and disturbance, using quantum combs. Each protocol consists of loss process, $\mathcal{L}_{\rho,A,\theta,\mathcal{P}}:=\mathcal{P}_{\mathcal{M}}\circ\mathit{\Lambda}_{\rho,A,\theta}$ for error and $\mathcal{L}_{\rho,B,\theta,\mathcal{I}}:=\mathcal{I}_{\mathcal{M}}\circ\mathit{\Lambda}_{\rho,B,\theta}$ for disturbance, and recovery process $\mathcal{R}$. The measuring processes $\mathcal{M}$ are distinguished between two types, $\mathcal{P}_{\mathcal{M}}$ and $\mathcal{I}_{\mathcal{M}}$, depending on the outputs— either classical outcomes in $\mathbf{P}$ or quantum states in $\mathbf{S'}$.
  • Figure 2: Irreversibility evaluation protocols for Ozawa's (a) error and (b) disturbance.
  • Figure 3: Irreversibility evaluation protocol for OTOC.
  • Figure 4: The experimental result of the measurement of the OTOC using our method. The results present a comparison among the ideal values (solid lines) from matrix calculations, the noiseless values (circle dots) from aer-simulator, and the experimental results (square dots) from the reimei emulator. The error bars represent the standard deviation of the measured values. The results show good agreement with theory, particularly at early and late times, validating our method.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof