Measurement disturbance and conservation laws in quantum mechanics

M. Hamed Mohammady1,2, Takayuki Miyadera3, and Leon Loveridge4

1QuIC, École Polytechnique de Bruxelles, CP 165/59, Université Libre de Bruxelles, 1050 Brussels, Belgium
2RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia
3Department of Nuclear Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan
4Quantum Technology Group, Department of Science and Industry Systems, University of South-Eastern Norway, 3616 Kongsberg, Norway

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Abstract

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 measurement is a physical process, resulting from an interaction between a system under investigation and a measuring apparatus. While the formal framework of quantum measurement theory allows for any measurement to be realised, if the interaction is constrained by a conservation law then some measurements may be ruled out.

In the presence of additive conserved quantities such as energy, charge, or angular momentum, there are restrictions on both accurate and non-disturbing measurements of some observables. A classic result on this topic is the Wigner-Araki-Yanase (WAY) theorem which dates back to the $50$s/$60$s, and states that when the measurement interaction is unitary, then the only sharp observables (corresponding to self-adjoint operators) that admit accurate or non-disturbing measurements are those that commute with the conserved quantity.

In this paper, we generalize the WAY theorem by addressing the question of accurate or non-disturbing measurements (in the presence of conservation laws) for observables represented by POVMs (positive operator valued measures) and measurement interactions represented by quantum channels. We find that in order to achieve accurate or non-disturbing measurements for observables that do not commute with the conserved quantity, the observables cannot be sharp, and the measuring apparatus must be prepared in a state with a large coherence in the conserved quantity. In the spirit of the original WAY theorem, we therefore find both a no-go result which prohibits precise measurement and manipulation of individual quantum objects, and a positive counterpart which delineates conditions under which good measurements can be achieved.

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