A thermodynamically consistent approach to the energy costs of quantum measurements

Camille L Latune; Cyril Elouard

doi:10.22331/q-2025-01-28-1614

A thermodynamically consistent approach to the energy costs of quantum measurements

Camille L Latune¹ and Cyril Elouard²

¹Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS UMR 6303, Université Bourgogne Europe, BP 47870, F-21078 Dijon, France
²Université de Lorraine, CNRS, LPCT, F-54000 Nancy, France

Published:	2025-01-28, volume 9, page 1614
Editor:	Paul Erker
Eprint:	arXiv:2402.16037v6
Doi:	https://doi.org/10.22331/q-2025-01-28-1614
Citation:	Quantum 9, 1614 (2025).

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Abstract

Considering a general microscopic model for a quantum measuring apparatus comprising a quantum probe coupled to a thermal bath, we analyze the energetic resources necessary for the realization of a quantum measurement, which includes the creation of system-apparatus correlations, the irreversible transition to a statistical mixture of definite outcomes, and the apparatus resetting. Crucially, we do not resort to another quantum measurement to capture the emergence of objective measurement results, but rather exploit the properties of the thermal bath which redundantly records the measurement result in its degrees of freedom, naturally implementing the paradigm of quantum Darwinism. In practice, this model allows us to perform a quantitative thermodynamic analysis of the measurement process. From the expression of the second law, we show how the minimal required work depends on the energy variation of the system being measured plus information-theoretic quantities characterizing the performance of the measurement – efficiency and completeness. Additionally, we show that it is possible to perform a thermodynamically reversible measurement, thus reaching the minimal work expenditure, and provide the corresponding protocol. Finally, for finite-time measurement protocols, we illustrate the increasing work cost induced by rising entropy production inherent in finite-time thermodynamic processes. This highlights an emerging trade-off between velocity of the measurement and work cost, on top of a trade-off between efficiency of the measurement and work cost. We apply those findings to bring new insights in the thermodynamic balance of the measurement-powered quantum engines.

Featured image: Pictural representation of a quantum measurement:
The quantum system S is measured by a measurement apparatus described as an open quantum system A interacting with a thermal bath B. The interaction with the thermal bath B reproduces the wave-function collapse of S via bath-induced decoherence. The apparatus also interacts with a memory M in which information on S is stored, and a battery which provides the necessary energy (work) to switch on and off the coupling with the bath. This setup is naturally suited to thermodynamic analysis.

Popular summary

Measuring quantum systems is central for quantum technologies, e.g. to read quantum information, estimate parameters, prepare quantum states or implement control protocols. Although quantum measurements are routinely performed in diverse experimental platforms, there is still an active debate around the foundations of the irreversible dynamics it induces on measured systems. Recently, interrogations have concerned the energy transfers occurring during a measurement, and the prediction for its energy cost. Bearing in mind that standard quantum applications like fault-tolerant quantum computing may require millions of measurements, understanding and optimizing such energy cost is crucial for the future development of quantum technologies.

Inspired by experimental setups implementing quantum measurements, we use a general dynamical model comprising all the ingredients relevant to capture the nonequilibrium thermodynamics of quantum measurements. This includes in particular the non-unitary dynamics usually associated with the wave-function collapse but modeled here by bath-induced dissipation.

This model allows us to perform a precise energetic analysis of the measurement process, including realistic scenarios of nonideal measurements. We use it to predict a tight lower bound on the measurement cost which directly depends on the measurement quality – the higher the quality, the larger the energy cost. Moreover, we identify work cost above this bound with entropy production taking place when manipulating the measuring apparatus, and during the system’s wave-function collapse. We deduce a fundamental trade-off between the work cost and the duration of the measurement on top of the trade-off between the work cost and the quality of the measurement.

Our results pave the road towards quantitative energetic optimization of quantum measurements, and their numerous applications. In particular, we draw by the end of the paper some important consequences for measurement-powered engines, a special kind of quantum engines including measurement-induced energy transfers in their cycle.

► BibTeX data

@article{LLatune2025thermodynamically,
  doi = {10.22331/q-2025-01-28-1614},
  url = {https://doi.org/10.22331/q-2025-01-28-1614},
  title = {A thermodynamically consistent approach to the energy costs of quantum measurements},
  author = {L Latune, Camille and Elouard, Cyril},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1614},
  month = jan,
  year = {2025}
}

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