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arXiv:quant-ph/0210046

Quantum Limits of Measurement and Computing Induced by Conservation Laws and Uncertainty Relations

Masanao Ozawa

Published Oct 8, 2002• 1 citation
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Abstract

A quantitative extension of the Wigner-Araki-Yanase theorem is obtained on the limitation on precise, non-disturbing measurements of observables which do not commute with additive conserved quantities, and applied to obtaining a limitation on the accuracy of quantum computing with computational bases which do not commute with angular momentum.

DOI
10.48550/arxiv.quant-ph/0210046

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A quantitative extension of the Wigner-Araki-Yanase theorem is obtained on the limitation on precise, non-disturbing measurements of observables which do not commute with additive conserved quantities, and applied to obtaining a limitation on the accuracy of quantum computing with computational bases which do not commute with angular momentum.
The Wigner-Araki-Yanase (WAY) theorem shows that additive conservation laws limit the accuracy of measurements. Recently, various quantitative expressions have been found for quantum limits on measurements induced by additive conservation laws, and have been applied to the study of fundamental limits on quantum-information processing. Here, we investigate generalizations of the WAY theorem to multiplicative conservation laws. The WAY theorem is extended to show that an observable not commuting with the modulus of, or equivalently the square of, a multiplicatively conserved quantity cannot be precisely measured. We also obtain a lower bound for the mean-square noise of a measurement in the presence of a multiplicatively conserved quantity. To overcome this noise it is necessary to make large the coefficient of variation (the so-called relative fluctuation), instead of the variance as is the case for additive conservation laws, of the conserved quantity in the apparatus.
The Wigner-Araki-Yanase theorem shows that conservation laws limit the accuracy of measurement.Here, we generalize the argument to show that conservation laws limit the accuracy of quantum logic operations.A rigorous lower bound is obtained of the error probability of any physical realization of the controlled-NOT gate under the constraint that the computational basis is represented by a component of spin and that physical implementations obey the angular momentum conservation law.The lower bound is shown to be inversely proportional to the number of ancilla qubits or the strength of the external control field.
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 notion of quantum instruments is formalized as statistical equivalence classes of all the possible quantum measurements and mathematically characterized as normalized completely positive map valued measures under naturally acceptable axioms. Recently, universally valid uncertainty relations have been established to set a precision limit for any instruments given a disturbance constraint in a form more general than the one originally proposed by Heisenberg. One of them leads to a quantitative generalization of the Wigner–Araki–Yanase theorem on the precision limit of measurements under conservation laws. Applying this, a rigorous lower bound is obtained for the gate error probability of physical implementations of Hadamard gates on a standard qubit of a spin 1/2 system by interactions with control fields or ancilla systems obeying the angular momentum conservation law.
The relationship between the measurement of an operator and the presence of a conservation law is studied using generalized coherent states. It is shown that an operator which does not commute with a conserved quantity cannot be measured exactly; this is an extension of the theorem by Wigner, Araki and Yanase, expressing the conserved quantity in a more general form. Classical operators play an important role in this approach. We present a simple model of the approximate measurement of a particle-spin which does not commute with the (additive) conserved angular momentum. Also, using two other models we show that the condition of distinguishability of pointer-operators is consistent with the situation where the statistical operator in a physical system tends to just its mixed-part.