Conditional work statistics of quantum measurements

M. Hamed Mohammady1,2 and Alessandro Romito1

1Department of Physics, Lancaster University, LA1 4YB, United Kingdom
2RCQI, Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, Bratislava 84511, Slovakia

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In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.

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[3] Leon Loveridge, "A relational perspective on the Wigner-Araki-Yanase theorem", Journal of Physics: Conference Series 1638 1, 012009 (2020).

[4] Giulia Rubino, Gonzalo Manzano, and Časlav Brukner, "Quantum superposition of thermodynamic evolutions with opposing time’s arrows", Communications Physics 4 1, 251 (2021).

[5] Yuxia Zhang, Jian Zou, and Bin Shao, "No-Signaling-in-Time (NSIT) Condition for Energy", Entropy 21 11, 1067 (2019).

[6] M. H. Mohammady, A. Auffèves, and J. Anders, "Energetic footprints of irreversibility in the quantum regime", Communications Physics 3 1, 89 (2020).

[7] Gabriel T. Landi, Mauro Paternostro, and Alessio Belenchia, "Informational Steady States and Conditional Entropy Production in Continuously Monitored Systems", PRX Quantum 3 1, 010303 (2022).

[8] Alessio Belenchia, Mauro Paternostro, and Gabriel T. Landi, "Informational steady states and conditional entropy production in continuously monitored systems: The case of Gaussian systems", Physical Review A 105 2, 022213 (2022).

[9] M. Hamed Mohammady and Alessandro Romito, "Symmetry constrained decoherence of conditional expectation values", arXiv:1901.01460.

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