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arXiv:1610.04607

Quantum measurements need not conserve energy: relation to the Wigner-Araki-Yanase theorem

Stephen Parrott

Published Oct 14, 2016• 0 citations
arXivHTMLPDF

Abstract

The paper focuses on the fact that quantum projective measurements do not necessarily conserve energy. On the other hand the Wigner-Araki-Yanase (WAY) theorem states that assuming a "standard" von Neumann measurement model and "additivity" of the total energy operator, projective measurements of a system must conserve energy as defined by the system's energy operator. This paper explores the ideas behind the WAY theorem in hopes of uncovering the origin of the contradiction. After Araki and Yanase published their proof of the WAY theorem, Yanase appended a new condition now known as the Yanase condition. Under the simplifying assumption that the observable being measured has discrete and non-degenerate eigenvalues, we prove that the Yanase condition actually follows from the hypotheses of the original WAY theorem. The paper also proves that the hypotheses of the WAY theorem, together with the simplifying assumption, imply that the energy operator for the measuring apparatus must be a multiple of the identity, which seems physically unlikely. It seems probable that this surprising conclusion, along with the Yanase condition, also holds without the simplifying assumption.

DOI
10.48550/arxiv.1610.04607

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The paper focuses on the fact that quantum projective measurements do not necessarily conserve energy. On the other hand the Wigner-Araki-Yanase (WAY) theorem states that assuming a "standard" von Neumann measurement model and "additivity" of the total energy operator, projective measurements of a system must conserve energy as defined by the system's energy operator. This paper explores the ideas behind the WAY theorem in hopes of uncovering the origin of the contradiction. After Araki and Yanase published their proof of the WAY theorem, Yanase appended a new condition now known as the Yanase condition. Under the simplifying assumption that the observable being measured has discrete and non-degenerate eigenvalues, we prove that the Yanase condition actually follows from the hypotheses of the original WAY theorem. The paper also proves that the hypotheses of the WAY theorem, together with the simplifying assumption, imply that the energy operator for the measuring apparatus must be a multiple of the identity, which seems physically unlikely. It seems probable that this surprising conclusion, along with the Yanase condition, also holds without the simplifying assumption.
The Wigner-Araki-Yanase (WAY) theorem is a well-known theorem regarding limitations of quantum measurement in the presence of additive conservation laws. Under the assumptions of the von Neumann measurement model, for which the system conserved quantity LSL_{S}LS​ is bounded, given a conserved total additive system plus apparatus quantity LSAL_{SA}LSA​, the measurement operator ESE_{S}ES​ must commute with LSL_{S}LS​. Prior proofs have exploited the properties of unitary evolution constrained by momentum conserving operations that tend to obscure the physical nature of the WAY theorem and as well lead to bounds on performance. As it is generally agreed that momentum is always exactly conserved in measurement, we instead develop a general angular momentum conserving model of measurement. This model is shown to lead to a simple explanation of the major implications of the WAY theorem and provides exact results of the effects of measurement based on the apparatus model. This is shown by both tracing the apparatus from the density matrix and also via a system-only channel model based on Kraus operators.
The Wigner-Araki-Yanase (WAY) theorem states that additive conservation laws imply the commutativity of exactly implementable projective measurements and the conserved observables of the system. Known proofs of this theorem are only restricted to bounded or discrete-spectrum conserved observables of the system and are not applicable to unbounded and continuous observables like a momentum operator. In this Letter, we present the WAY theorem for possibly unbounded and continuous conserved observables under the Yanase condition, which requires that the probe positive operator-valued measure should commute with the conserved observable of the probe system. As a result of this WAY theorem, we show that exact implementations of the projective measurement of the position under momentum conservation and of the quadrature amplitude using linear optical instruments and photon counters are impossible. We also consider implementations of unitary channels under conservation laws and find that the conserved observable L_{S} of the system commutes with the implemented unitary U_{S} if L_{S} is semibounded, while U_{S}^{†}L_{S}U_{S} can shift up to possibly nonzero constant factor if the spectrum of L_{S} is upper and lower unbounded. We give simple examples of the latter case, where L_{S} is a momentum operator.
The Wigner-Araki-Yanase (WAY) theorem states a remarkable limitation to quantum mechanical measurements in the presence of additive conserved quantities. Discovered by Wigner in 1952, this limitation is known to induce constraints on the control of individual quantum systems in the context of information processing. It is therefore important to understand the precise conditions and scope of the WAY theorem. Here we elucidate its crucial assumptions, briefly review some generalizations, and show how a particular extension can already be obtained by a simple modification of the original proofs. We also describe the evolution of the WAY theorem from a strict no-go verdict for certain, highly idealized, precise measurements into a quantitative constraint on the accuracy and approximate repeatability of imprecise measurements.
The Wigner–Araki–Yanase (WAY) theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems.
Symmetry imposes fundamental constraints on quantum measurement and control. The Wigner-Araki-Yanase theorem and its quantitative extensions capture this restriction for continuous symmetries, in terms of fluctuations of conserved generators. Such generator-based bounds, however, do not provide quantitative limitations for discrete unitary symmetries or for antiunitary symmetries. Here we establish quantitative WAY-type theorems for symmetry-breaking projective measurements and unitary gates under arbitrary unitary and antiunitary symmetries. Our approach is based on a two-target no-programming inequality: if a single processor approximately implements two operations that amplify distinguishability, then the corresponding program states must themselves be distinguishable. Applied to symmetric implementations, this converts the error of an asymmetric measurement or gate directly into a lower bound on the asymmetry of the apparatus state, quantified by its fidelity with its symmetry-transformed copy. Our results apply to discrete and antiunitary symmetries, thereby providing a fundamental limit for symmetry-limited quantum measurement and control beyond the continuous-symmetry regime.
The ability to measure every quantum observable is ensured by a fundamental result in quantum measurement theory. Nevertheless, additive conservation laws associated with physical symmetries, such as the angular momentum conservation, may lead to restrictions on the measurability of the observables. Such limitations are imposed by the theorem of Wigner, Araki, and Yanase (WAY). In this paper a formulation of the WAY theorem is presented rephrasing the measurability limitations in terms of quantum incompatibility. This broader mathematical basis enables us to both capture and generalize the WAY theorem by allowing us to drop the assumptions of additivity and even conservation of the involved quantities. Moreover, we extend the WAY theorem to the general level of positive operator-valued measures.