Quantum limits of measurements induced by multiplicative conservation laws: Extension of the Wigner-Araki-Yanase theorem

Abstract

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.

Wigner-Araki-Yanase theorem beyond conservation laws

“…implies [A, L 1 ] = 0; a similar result was previously found for multiplicative conserved quantities in [6].…”

“…[6] that the WAYtheorem will set further restrictions for the programmable quantum multimeters. It is the purpose of this section to validate this expectation.…”

“…Moreover, different quantitative generalisations of the WAYtheorem which relax the assumption of perfect precision have been studied in Refs. [4][5][6][7][8][9]. With all its extensions the current form of the WAY-theorem covers a large class of physically important scenarios and consequently has applications, not solely in quantum measurement theory, but also in the fields of quantum information processing and quantum control.…”

“…In particular, ways to omit the assumptions of the repeatability of the measurement [3][4][5] and the additivity of the conserved quantity [6] have been reported. Moreover, different quantitative generalisations of the WAYtheorem which relax the assumption of perfect precision have been studied in Refs.…”

Measurement disturbance and conservation laws in quantum mechanics

“…For instance, there are obstacles to the implementation of quantum logic gates [6][7][8]. The theorem continues to inspire research in a variety of directions (some recent examples are [9][10][11][12][13][14][15]), and connections have been made to other fields of research; for instance in the resource theories of asymmetry [16] and coherence [17,18], the theory of quantum reference frames [19], quantum clocks [20], and quantum thermodynamics [21][22][23][24][25]. Despite this ongoing development, the full scope of the WAY theorem is not known; particularly, the role of disturbance has not yet been carefully examined.…”