Ideal Projective Measurements Have Infinite Resource Costs

Yelena Guryanova, Nicolai Friis, and Marcus Huber

Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria

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Abstract

We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of ideal projective measurements and show that when limited by finite resources only one of these features can be salvaged. Our framework is general enough to accommodate any system and measuring device (pointer) models, but for illustration we use an explicit model of an $N$-particle pointer. For a pointer that perfectly reproduces the statistics of the system, we provide tight analytic expressions for the energy cost of performing the measurement. This cost may be broken down into two parts. First, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. Our results show that, even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources for the most general system and pointer Hamiltonians, phrasing our results as fundamental bounds in terms of the dimensions of these systems.

The notion of measurement forms an integral (and often heatedly debated) part of quantum mechanical reasoning. An ideal measurement provides information about the measured system, but also disturbs the latter. Nonetheless, according to the so-called projection postulate of quantum mechanics, the measurement outcome allows one to make precise statements about the system after its interaction with the measuring device. Understanding the process of measurement as an interaction of a system with a pointer elucidates the thermodynamic nature of the process. While previous considerations of the first and second law only implied a modest cost in energy and compensation of entropy, we show that the third law of thermodynamics imposes the most drastic of restrictions. Indeed, ideal measurements are as impossible as cooling a system to the ground state. Impossible because they would require infinite amounts of time or energy, or exact control over infinitely complex measuring devices. Consequently, all practical measurements are non-ideal and to approximate good measurements large amounts of work need to be expended.

Here, we investigate the structure and resource costs of non-ideal measurements and identify three fundamental properties that all ideal measurements possess. We argue that when limited by finite resources the three properties cannot hold simultaneously – one must choose which one of the properties to keep. Under the assumption that at least one property holds, we then derive expressions for the energy cost of performing such an imperfect measurement.

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► References

[1] Sai Vinjanampathy and Janet Anders, Quantum thermodynamics, Contemp. Phys. 57, 545 (2016), arXiv:1508.06099.
https:/​/​doi.org/​10.1080/​00107514.2016.1201896
arXiv:arXiv:1508.06099

[2] James Millen and André Xuereb, Perspective on quantum thermodynamics, New J. Phys. 18, 011002 (2016), arXiv:1509.01086.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​011002
arXiv:arXiv:1509.01086

[3] John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, and Paul Skrzypczyk, The role of quantum information in thermodynamics — a topical review, J. Phys. A: Math. Theor. 49, 143001 (2016), arXiv:1505.07835.
https:/​/​doi.org/​10.1088/​1751-8113/​49/​14/​143001
arXiv:arXiv:1505.07835

[4] Massimiliano Esposito and Christian Van den Broeck, Second law and landauer principle far from equilibrium, Europhys. Lett. 95, 40004 (2011), arXiv:1104.5165.
https:/​/​doi.org/​10.1209/​0295-5075/​95/​40004
arXiv:arXiv:1104.5165

[5] Kurt Jacobs, Quantum measurement and the first law of thermodynamics: the energy cost of measurement is the work value of the acquired information, Phys. Rev. E 86, 040106(R) (2012), arXiv:1208.1561.
https:/​/​doi.org/​10.1103/​PhysRevE.86.040106
arXiv:arXiv:1208.1561

[6] Fernando G. S. L. Brandão, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, and Stephanie Wehner, The second laws of quantum thermodynamics, Proc. Natl. Acad. Sci. U.S.A. 11, 3275 (2015), arXiv:1305.5278.
https:/​/​doi.org/​10.1073/​pnas.1411728112
arXiv:arXiv:1305.5278

[7] Matteo Lostaglio, David Jennings, and Terry Rudolph, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nat. Commun. 6, 6383 (2015), arXiv:1405.2188.
https:/​/​doi.org/​10.1038/​ncomms7383
arXiv:arXiv:1405.2188

[8] Piotr Ć wikliński, Michał Studziński, Michał Horodecki, and Jonathan Oppenheim, Limitations on the Evolution of Quantum Coherences: Towards Fully Quantum Second Laws of Thermodynamics, Phys. Rev. Lett. 115, 210403 (2015), arXiv:1405.5029.
https:/​/​doi.org/​10.1103/​PhysRevLett.115.210403
arXiv:arXiv:1405.5029

[9] Álvaro M. Alhambra, Lluis Masanes, Jonathan Oppenheim, and Christopher Perry, Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality, Phys. Rev. X 6, 041017 (2016), arXiv:1601.05799.
https:/​/​doi.org/​10.1103/​PhysRevX.6.041017
arXiv:arXiv:1601.05799

[10] Henrik Wilming, Rodrigo Gallego, and Jens Eisert, Second law of thermodynamics under control restrictions, Phys. Rev. E 93, 042126 (2016), arXiv:1411.3754.
https:/​/​doi.org/​10.1103/​PhysRevE.93.042126
arXiv:arXiv:1411.3754

[11] Jakob Scharlau and Markus P. Müller, Quantum Horn's lemma, finite heat baths, and the third law of thermodynamics, Quantum 2, 54 (2018), arXiv:1605.06092.
https:/​/​doi.org/​10.22331/​q-2018-02-22-54
arXiv:arXiv:1605.06092

[12] Lluis Masanes and Jonathan Oppenheim, A general derivation and quantification of the third law of thermodynamics, Nat. Commun. 8, 14538 (2017), arXiv:1412.3828.
https:/​/​doi.org/​10.1038/​ncomms14538
arXiv:arXiv:1412.3828

[13] Manabendra Nath Bera, Arnau Riera, Maciej Lewenstein, and Andreas Winter, Generalized Laws of Thermodynamics in the Presence of Correlations, Nat. Commun. 8, 2180 (2017), arXiv:1612.04779.
https:/​/​doi.org/​10.1038/​s41467-017-02370-x
arXiv:arXiv:1612.04779

[14] Leonard J. Schulman, Tal Mor, and Yossi Weinstein, Physical Limits of Heat-Bath Algorithmic Cooling, Phys. Rev. Lett. 94, 120501 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.94.120501

[15] Henrik Wilming and Rodrigo Gallego, Third Law of Thermodynamics as a Single Inequality, Phys. Rev. X 7, 041033 (2017), arXiv:1701.07478.
https:/​/​doi.org/​10.1103/​PhysRevX.7.041033
arXiv:arXiv:1701.07478

[16] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber, Unifying paradigms of quantum refrigeration: fundamental limits of cooling and associated work costs, Phys. Rev. E 100, 042130 (2019a), arXiv:1710.11624.
https:/​/​doi.org/​10.1103/​PhysRevE.100.042130
arXiv:arXiv:1710.11624

[17] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber, Unifying Paradigms of Quantum Refrigeration: A Universal and Attainable Bound on Cooling, Phys. Rev. Lett. 123, 170605 (2019b), arXiv:1903.04970.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.170605
arXiv:arXiv:1903.04970

[18] Tien D. Kieu, Principle of Unattainability of absolute zero temperature, the Third Law of Thermodynamics, and projective quantum measurements, Phys Lett A 383, 125848 (2019), arXiv:1804.04182.
https:/​/​doi.org/​10.1016/​j.physleta.2019.125848
arXiv:arXiv:1804.04182

[19] Jarosław K. Korbicz, Edgar A. Aguilar, Piotr Ć wikliński, and Paweł Horodecki, Generic appearance of objective results in quantum measurements, Phys. Rev. A 96, 032124 (2017), arXiv:1604.02011.
https:/​/​doi.org/​10.1103/​PhysRevA.96.032124
arXiv:arXiv:1604.02011

[20] Wojciech Hubert Zurek, Quantum Darwinism, Nat. Phys. 5, 181 (2009), arXiv:0903.5082.
https:/​/​doi.org/​10.1038/​nphys1202
arXiv:arXiv:0903.5082

[21] Takahiro Sagawa and Masahito Ueda, Minimal Energy Cost for Thermodynamic Information Processing: Measurement and Information Erasure, Phys. Rev. Lett. 102, 250602 (2009), arXiv:0809.4098.
https:/​/​doi.org/​10.1103/​PhysRevLett.102.250602
arXiv:arXiv:0809.4098

[22] Cyril Elouard, David Herrera-Martí, Benjamin Huard, and Alexia Auffèves, Extracting work from quantum measurement in Maxwell demon engines, Phys. Rev. Lett. 118, 260603 (2017), arXiv:1702.01917.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.260603
arXiv:arXiv:1702.01917

[23] Patryk Lipka-Bartosik and Rafal Demkowicz-Dobrzanski, Thermodynamic work cost of quantum estimation protocols, J. Phys. A: Math. Theor. 51, 474001 (2018), arXiv:1805.01477.
https:/​/​doi.org/​10.1088/​1751-8121/​aae664
arXiv:arXiv:1805.01477

[24] Cyril Elouard and Andrew N. Jordan, Efficient Quantum Measurement Engine, Phys. Rev. Lett. 120, 260601 (2018), arXiv:1801.03979.
https:/​/​doi.org/​10.1103/​PhysRevLett.120.260601
arXiv:arXiv:1801.03979

[25] David Reeb and Michael M. Wolf, An improved Landauer Principle with finite-size corrections, New J. Phys. 16, 103011 (2014), arXiv:1306.4352.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​10/​103011
arXiv:arXiv:1306.4352

[26] Kais Abdelkhalek, Yoshifumi Nakata, and David Reeb, Fundamental energy cost for quantum measurement, (2016), arXiv:1609.06981.
arXiv:arXiv:1609.06981

[27] Armen E. Allahverdyan, Karen V. Hovhannisyan, Dominik Janzing, and Guenter Mahler, Thermodynamic limits of dynamic cooling, Phys. Rev. E 84, 041109 (2011), arXiv:1107.1044.
https:/​/​doi.org/​10.1103/​PhysRevE.84.041109
arXiv:arXiv:1107.1044

[28] Gavin E. Crooks, The Entropy Production Fluctuation Theorem and the Nonequilibrium Work Relation for Free Energy Differences, Phys. Rev. E 60, 2721 (1999), arXiv:cond-mat/​9901352.
https:/​/​doi.org/​10.1103/​PhysRevE.60.2721
arXiv:arXiv:cond-mat/9901352

[29] Hal Tasaki, Jarzynski Relations for Quantum Systems and Some Applications, (2000), arXiv:cond-mat/​0009244.
arXiv:arXiv:cond-mat/0009244

[30] Peter Talkner, Eric Lutz, and Peter Hänggi, Fluctuation theorems: Work is not an observable, Phys. Rev. E 75, 050102(R) (2007), arXiv:cond-mat/​0703189.
https:/​/​doi.org/​10.1103/​PhysRevE.75.050102
arXiv:arXiv:cond-mat/0703189

[31] Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis, Work estimation and work fluctuations in the presence of non-ideal measurements, New J. Phys. 21, 113002 (2019), arXiv:1902.08568.
https:/​/​doi.org/​10.1088/​1367-2630/​ab4d9d
arXiv:arXiv:1902.08568

[32] Mihai D. Vidrighin, Oscar Dahlsten, Marco Barbieri, M. S. Kim, Vlatko Vedral, and Ian A. Walmsley, Photonic Maxwell's Demon, Phys. Rev. Lett. 116, 050401 (2016), arXiv:1510.02164.
https:/​/​doi.org/​10.1103/​PhysRevLett.116.050401
arXiv:arXiv:1510.02164

[33] Alhun Aydin, Altug Sisman, and Ronnie Kosloff, Landauer's Principle in a Quantum Szilard Engine Without Maxwell's Demon, (2019), arXiv:1908.04400.
arXiv:arXiv:1908.04400

[34] M. Hamed Mohammady and Janet Anders, A quantum Szilard engine without heat from a thermal reservoir, New J. Phys. 19, 113026 (2017), arXiv:1706.00938.
https:/​/​doi.org/​10.1088/​1367-2630/​aa8ba1
arXiv:arXiv:1706.00938

[35] Michał Oszmaniec, Leonardo Guerini, Peter Wittek, and Antonio Acín, Simulating Positive-Operator-Valued Measures with Projective Measurements, Phys. Rev. Lett. 119, 190501 (2017), arXiv:1609.06139.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.190501
arXiv:arXiv:1609.06139

[36] David E. Bruschi, Martí Perarnau-Llobet, Nicolai Friis, Karen V. Hovhannisyan, and Marcus Huber, The thermodynamics of creating correlations: Limitations and optimal protocols, Phys. Rev. E 91, 032118 (2015), arXiv:1409.4647.
https:/​/​doi.org/​10.1103/​PhysRevE.91.032118
arXiv:arXiv:1409.4647

[37] Marcus Huber, Martí Perarnau-Llobet, Karen V. Hovhannisyan, Paul Skrzypczyk, Claude Klöckl, Nicolas Brunner, and Antonio Ac$\acute{\i}$n, Thermodynamic cost of creating correlations, New J. Phys. 17, 065008 (2015), arXiv:1404.2169.
https:/​/​doi.org/​10.1088/​1367-2630/​17/​6/​065008
arXiv:arXiv:1404.2169

[38] Giuseppe Vitagliano, Claude Klöckl, Marcus Huber, and Nicolai Friis, Trade-off between work and correlations in quantum thermodynamics, in Thermodynamics in the Quantum Regime, edited by Felix Binder, Luis A. Correa, Christian Gogolin, Janet Anders, and Gerardo Adesso (Springer, 2019) Chap. 30, pp. 731–750, arXiv:1803.06884.
https:/​/​doi.org/​10.1007/​978-3-319-99046-0_30
arXiv:arXiv:1803.06884

[39] Faraj Bakhshinezhad, Fabien Clivaz, Giuseppe Vitagliano, Paul Erker, Ali T. Rezakhani, Marcus Huber, and Nicolai Friis, Thermodynamically optimal creation of correlations, J. Phys. A: Math. Theor. 52, 465303 (2019), arXiv:1904.07942.
https:/​/​doi.org/​10.1088/​1751-8121/​ab3932
arXiv:arXiv:1904.07942

[40] Matteo G. A. Paris, The modern tools of quantum mechanics, Eur. Phys. J. S. T. 203, 61 (2012), arXiv:1110.6815.
https:/​/​doi.org/​10.1140/​epjst/​e2012-01535-1
arXiv:arXiv:1110.6815

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[2] Lorenzo Buffoni, Andrea Solfanelli, Paola Verrucchi, Alessandro Cuccoli, and Michele Campisi, "Quantum Measurement Cooling", Physical Review Letters 122 7, 070603 (2019).

[3] Fabien Clivaz, Ralph Silva, Géraldine Haack, Jonatan Bohr Brask, Nicolas Brunner, and Marcus Huber, "Unifying paradigms of quantum refrigeration: A universal and attainable bound on cooling", arXiv:1903.04970, Physical Review Letters 123 17, 170605 (2019).

[4] Mark T. Mitchison, "Quantum thermal absorption machines: refrigerators, engines and clocks", Contemporary Physics 60 2, 164 (2019).

[5] Tiago Debarba, Gonzalo Manzano, Yelena Guryanova, Marcus Huber, and Nicolai Friis, "Work estimation and work fluctuations in the presence of non-ideal measurements", arXiv:1902.08568.

[6] Giuseppe Vitagliano, Claude Klöckl, Marcus Huber, and Nicolai Friis, "Trade-Off Between Work and Correlations in Quantum Thermodynamics", Thermodynamics in the Quantum Regime: Fundamental Aspects and New Directions 195, 731 (2018).

[7] M. Hamed Mohammady and Alessandro Romito, "Conditional work statistics of quantum measurements", arXiv:1809.09010.

[8] Tien D. Kieu, "Principle of Unattainability of absolute zero temperature, the Third Law of Thermodynamics, and projective quantum measurements", Physics Letters A 383, 125848 (2019).

[9] Faraj Bakhshinezhad, Fabien Clivaz, Giuseppe Vitagliano, Paul Erker, Ali Rezakhani, Marcus Huber, and Nicolai Friis, "Thermodynamically optimal creation of correlations", Journal of Physics A Mathematical General 52 46, 465303 (2019).

[10] Mathias R. Jørgensen, Patrick P. Potts, Matteo G. A. Paris, and Jonatan B. Brask, "Tight bound on finite-resolution quantum thermometry at low temperatures", arXiv:2001.04096.

[11] Esteban Calzetta, "The importance of being measurement", arXiv:1909.13178.

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