Incompatibility of quantum instruments

Leevi Leppäjärvi1 and Michal Sedlák1,2

1RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
2Faculty of Informatics, Masaryk University, Botanická 68a, 602 00 Brno, Czech Republic

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Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.

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Cited by

[1] Stan Gudder, "Multi-Observables and Multi-Instruments", arXiv:2307.11223, (2023).

[2] Stanley Gudder, "A Theory of Quantum Instruments", arXiv:2305.17584, (2023).

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