Covariant catalysis requires correlations and good quantum reference frames degrade little

Lauritz van Luijk, Reinhard F. Werner, and Henrik Wilming

Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little.

In quantum mechanics, transitions between quantum states can be made possible in the presence of a third quantum system, which returns to its initial state at the end — just like a catalyst in Chemistry. We show that if all transitions have to respect conservation laws, then a catalyst has to build up correlations to the system, the environment, or both. Otherwise, it cannot enable new transitions. Therefore, pure states are useless for catalysis because they cannot be correlated to other systems.

► BibTeX data

► References

[1] M. Ahmadi, D. Jennings, and T. Rudolph. Dynamics of a quantum reference frame undergoing selective measurements and coherent interactions. Phys. Rev. A, 82 (3): 032320, sep 2010. 10.1103/​physreva.82.032320.
https:/​/​doi.org/​10.1103/​physreva.82.032320

[2] M. Ahmadi, D. Jennings, and T. Rudolph. The Wigner-Araki-Yanase theorem and the quantum resource theory of asymmetry. New J. Phys., 15 (1): 013057, jan 2013. 10.1088/​1367-2630/​15/​1/​013057.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​1/​013057

[3] R. Alexander, S. Gvirtz-Chen, and D. Jennings. Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system. New J. Phys., 24 (5): 053023, may 2022. 10.1088/​1367-2630/​ac688b.
https:/​/​doi.org/​10.1088/​1367-2630/​ac688b

[4] H. Araki and M. M. Yanase. Measurement of quantum mechanical operators. Phys Rev, 120 (2): 622–626, oct 1960. 10.1103/​physrev.120.622.
https:/​/​doi.org/​10.1103/​physrev.120.622

[5] articleha P. Woods and M. Horodecki. Autonomous quantum devices: When are they realizable without additional thermodynamic costs? Physical Review X, 13 (1), feb 2023. 10.1103/​physrevx.13.011016.
https:/​/​doi.org/​10.1103/​physrevx.13.011016

[6] V. Bargmann. On unitary ray representations of continuous groups. Annals of Mathematics, pages 1–46, 1954. 10.2307/​1969831.
https:/​/​doi.org/​10.2307/​1969831

[7] S. D. Bartlett, T. Rudolph, R. W. Spekkens, and P. S. Turner. Degradation of a quantum reference frame. New J. Phys., 8 (4): 58–58, apr 2006. 10.1088/​1367-2630/​8/​4/​058.
https:/​/​doi.org/​10.1088/​1367-2630/​8/​4/​058

[8] S. D. Bartlett, T. Rudolph, B. C. Sanders, and P. S. Turner. Degradation of a quantum directional reference frame as a random walk. J. Modern Opt., 54 (13-15): 2211–2221, sep 2007a. 10.1080/​09500340701289254.
https:/​/​doi.org/​10.1080/​09500340701289254

[9] S. D. Bartlett, T. Rudolph, and R. W. Spekkens. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys., 79: 555–609, Apr 2007b. 10.1103/​RevModPhys.79.555.
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[10] P. Boes, J. Eisert, R. Gallego, M. P. Mueller, and H. Wilming. Von Neumann entropy from unitarity. Phys. Rev. Lett., 122 (21): 210402, May 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.122.210402.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.210402

[11] F. G. S. L. Brandao, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens. The resource theory of quantum states out of thermal equilibrium. Phys. Rev. Lett., 111: 250404, 2013. 10.1103/​PhysRevLett.111.250404.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.250404

[12] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner. The second laws of quantum thermodynamics. PNAS, 112: 3275–3279, 2015. 10.1073/​pnas.1411728112.
https:/​/​doi.org/​10.1073/​pnas.1411728112

[13] P. Busch and L. Loveridge. Position measurements obeying momentum conservation. Phys. Rev. Lett., 106 (11): 110406, mar 2011. 10.1103/​physrevlett.106.110406.
https:/​/​doi.org/​10.1103/​physrevlett.106.110406

[14] G. Chiribella, Y. Yang, and R. Renner. Fundamental energy requirement of reversible quantum operations. Physical Review X, 11 (2), apr 2021. 10.1103/​physrevx.11.021014.
https:/​/​doi.org/​10.1103/​physrevx.11.021014

[15] F. Ding, X. Hu, and H. Fan. Amplifying asymmetry with correlating catalysts. Phys. Rev. A, 103 (2): 022403, Feb. 2021. ISSN 2469-9926, 2469-9934. 10.1103/​PhysRevA.103.022403.
https:/​/​doi.org/​10.1103/​PhysRevA.103.022403

[16] J. Eisert and M. Wilkens. Catalysis of Entanglement Manipulation for Mixed States. Phys. Rev. Lett., 85 (2): 437–440, July 2000. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.85.437.
https:/​/​doi.org/​10.1103/​PhysRevLett.85.437

[17] P. Faist, F. Dupuis, J. Oppenheim, and R. Renner. The minimal work cost of information processing. Nature Comm., 6: 7669, 2015. 10.1038/​ncomms8669.
https:/​/​doi.org/​10.1038/​ncomms8669

[18] C. Fuchs and J. van de Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Transactions on Information Theory, 45 (4): 1216–1227, may 1999. 10.1109/​18.761271.
https:/​/​doi.org/​10.1109/​18.761271

[19] C. A. Fuchs. Information gain vs. state disturbance in quantum theory. Fortschr. Phys., 46 (4-5): 535–565, 1998. 10.1002/​(SICI)1521-3978(199806)46:4/​5<535::AID-PROP535>3.0.CO;2-0.
https:/​/​doi.org/​10.1002/​(SICI)1521-3978(199806)46:4/​5<535::AID-PROP535>3.0.CO;2-0

[20] C. A. Fuchs and A. Peres. Quantum-state disturbance versus information gain: Uncertainty relations for quantum information. Phys. Rev. A, 53 (4): 2038–2045, apr 1996. 10.1103/​physreva.53.2038.
https:/​/​doi.org/​10.1103/​physreva.53.2038

[21] R. Gallego, J. Eisert, and H. Wilming. Thermodynamic work from operational principles. New J. Phys., 18 (10): 103017, 2016. 10.1088/​1367-2630/​18/​10/​103017.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​10/​103017

[22] G. Gour and R. W. Spekkens. The resource theory of quantum reference frames: manipulations and monotones. New J. Phys., 10 (3): 033023, mar 2008. 10.1088/​1367-2630/​10/​3/​033023.
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033023

[23] G. Gour, I. Marvian, and R. W. Spekkens. Measuring the quality of a quantum reference frame: The relative entropy of frameness. Phys. Rev. A, 80 (1): 012307, jul 2009. 10.1103/​physreva.80.012307.
https:/​/​doi.org/​10.1103/​physreva.80.012307

[24] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Y. Halpern. The resource theory of informational nonequilibrium in thermodynamics. Phys. Rep., 583: 1–58, jul 2015. 10.1016/​j.physrep.2015.04.003.
https:/​/​doi.org/​10.1016/​j.physrep.2015.04.003

[25] G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian. Quantum majorization and a complete set of entropic conditions for quantum thermodynamics. Nat Commun, 9 (1): 5352, Dec. 2018. ISSN 2041-1723. 10.1038/​s41467-018-06261-7.
https:/​/​doi.org/​10.1038/​s41467-018-06261-7

[26] M. Gschwendtner, A. Bluhm, and A. Winter. Programmability of covariant quantum channels. Quantum, 5: 488, jun 2021. 10.22331/​q-2021-06-29-488.
https:/​/​doi.org/​10.22331/​q-2021-06-29-488

[27] M. Horodecki and J. Oppenheim. Fundamental limitations for quantum and nanoscale thermodynamics. Nature Comm., 4: 2059, 2013. 10.1038/​ncomms3059.
https:/​/​doi.org/​10.1038/​ncomms3059

[28] D. Janzing. Quantum thermodynamics with missing reference frames: Decompositions of free energy into non-increasing components. J. Stat. Phys., 125 (3): 761–776, nov 2006. 10.1007/​s10955-006-9220-x.
https:/​/​doi.org/​10.1007/​s10955-006-9220-x

[29] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth. Thermodynamic cost of reliability and low temperatures: Tightening landauer's principle and the second law. Int. J. Th. Phys., 39: 2717, 2000. 10.1023/​A:1026422630734.
https:/​/​doi.org/​10.1023/​A:1026422630734

[30] D. Jonathan and M. B. Plenio. Entanglement-Assisted Local Manipulation of Pure Quantum States. Phys. Rev. Lett., 83 (17): 3566–3569, Oct. 1999. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.83.3566.
https:/​/​doi.org/​10.1103/​PhysRevLett.83.3566

[31] M. Keyl and R. F. Werner. Optimal cloning of pure states, testing single clones. J. Math. Phys., 40 (7): 3283–3299, jul 1999. 10.1063/​1.532887.
https:/​/​doi.org/​10.1063/​1.532887

[32] T. V. Kondra, C. Datta, and A. Streltsov. Catalytic transformations of pure entangled states. Physical Review Letters, 127 (15): 150503, oct 2021. 10.1103/​physrevlett.127.150503.
https:/​/​doi.org/​10.1103/​physrevlett.127.150503

[33] D. Kretschmann, D. Schlingemann, and R. F. Werner. The information-disturbance tradeoff and the continuity of stinespring's representation. IEEE Transactions on Information Theory, 54 (4): 1708–1717, apr 2008. 10.1109/​tit.2008.917696.
https:/​/​doi.org/​10.1109/​tit.2008.917696

[34] Y. Kuramochi and H. Tajima. Wigner-araki-yanase theorem for continuous and unbounded conserved observables. 2022. 10.48550/​arxiv.2208.13494.
https:/​/​doi.org/​10.48550/​arxiv.2208.13494

[35] P. Lipka-Bartosik and P. Skrzypczyk. Catalytic quantum teleportation. Physical Review Letters, 127: 080502, Feb. 2021. 10.1103/​PhysRevLett.127.080502.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.080502

[36] P. Lipka-Bartosik, M. Perarnau-Llobet, and N. Brunner. Operational definition of the temperature of a quantum state. Physical Review Letters, 130 (4), jan 2023a. 10.1103/​physrevlett.130.040401.
https:/​/​doi.org/​10.1103/​physrevlett.130.040401

[37] P. Lipka-Bartosik, H. Wilming, and N. H. Y. Ng. Catalysis in quantum information theory. 2023b. 10.48550/​arXiv.2306.00798.
https:/​/​doi.org/​10.48550/​arXiv.2306.00798

[38] M. Lostaglio and M. P. Müller. Coherence and Asymmetry Cannot be Broadcast. Phys. Rev. Lett., 123 (2): 020403, July 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.123.020403.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.020403

[39] I. Marvian. Operational interpretation of quantum fisher information in quantum thermodynamics. Physical Review Letters, 129 (19), oct 2022. 10.1103/​physrevlett.129.190502.
https:/​/​doi.org/​10.1103/​physrevlett.129.190502

[40] I. Marvian and R. W. Spekkens. An information-theoretic account of the wigner-araki-yanase theorem. 2012. 10.48550/​arxiv.1212.3378.
https:/​/​doi.org/​10.48550/​arxiv.1212.3378

[41] I. Marvian and R. W. Spekkens. The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations. New J. Phys., 15 (3): 033001, Mar. 2013. ISSN 1367-2630. 10.1088/​1367-2630/​15/​3/​033001.
https:/​/​doi.org/​10.1088/​1367-2630/​15/​3/​033001

[42] I. Marvian and R. W. Spekkens. How to quantify coherence: Distinguishing speakable and unspeakable notions. Phys. Rev. A, 94: 052324, Nov 2016. 10.1103/​PhysRevA.94.052324.
https:/​/​doi.org/​10.1103/​PhysRevA.94.052324

[43] I. Marvian and R. W. Spekkens. A no-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting. Phys. Rev. Lett., 123 (2): 020404, July 2019. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.123.020404.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.020404

[44] I. M. Marvian. Symmetry, Asymmetry and Quantum Information. PhD thesis, University of Waterloo, 2012. URL http:/​/​hdl.handle.net/​10012/​7088.
http:/​/​hdl.handle.net/​10012/​7088

[45] T. Miyadera and L. Loveridge. A quantum reference frame size-accuracy trade-off for quantum channels. J. Phys.: Conf. Ser., 1638 (1): 012008, oct 2020. 10.1088/​1742-6596/​1638/​1/​012008.
https:/​/​doi.org/​10.1088/​1742-6596/​1638/​1/​012008

[46] T. Miyadera, L. Loveridge, and P. Busch. Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off. J. Phys. A: Math. Theor., 49 (18): 185301, mar 2016. 10.1088/​1751-8113/​49/​18/​185301.
https:/​/​doi.org/​10.1088/​1751-8113/​49/​18/​185301

[47] M. H. Mohammady, T. Miyadera, and L. Loveridge. Measurement disturbance and conservation laws in quantum mechanics. Quantum, 7: 1033, jun 2023. 10.22331/​q-2023-06-05-1033.
https:/​/​doi.org/​10.22331/​q-2023-06-05-1033

[48] M. P. Müller. Correlating thermal machines and the second law at the nanoscale. Phys. Rev. X, 8 (4): 041051, dec 2018. 10.1103/​physrevx.8.041051.
https:/​/​doi.org/​10.1103/​physrevx.8.041051

[49] M. Ozawa. Conservative quantum computing. Phys. Rev. Lett., 89 (5): 057902, jul 2002a. 10.1103/​physrevlett.89.057902.
https:/​/​doi.org/​10.1103/​physrevlett.89.057902

[50] M. Ozawa. Conservation laws, uncertainty relations, and quantum limits of measurements. Phys. Rev. Lett., 88 (5): 050402, jan 2002b. 10.1103/​physrevlett.88.050402.
https:/​/​doi.org/​10.1103/​physrevlett.88.050402

[51] D. Poulin and J. Yard. Dynamics of a quantum reference frame. New J. Phys., 9 (5): 156–156, may 2007. 10.1088/​1367-2630/​9/​5/​156.
https:/​/​doi.org/​10.1088/​1367-2630/​9/​5/​156

[52] S. Rethinasamy and M. M. Wilde. Relative entropy and catalytic relative majorization. Phys. Rev. Research, 2 (3): 033455, sep 2020. 10.1103/​physrevresearch.2.033455.
https:/​/​doi.org/​10.1103/​physrevresearch.2.033455

[53] H. Shapiro. A survey of canonical forms and invariants for unitary similarity. Linear Algebra Appl., 147: 101–167, mar 1991. 10.1016/​0024-3795(91)90232-l.
https:/​/​doi.org/​10.1016/​0024-3795(91)90232-l

[54] N. Shiraishi and T. Sagawa. Quantum thermodynamics of correlated-catalytic state conversion at small scale. Phys. Rev. Lett., 126 (15): 150502, apr 2021. 10.1103/​physrevlett.126.150502.
https:/​/​doi.org/​10.1103/​physrevlett.126.150502

[55] W. Specht. Zur theorie der matrizen. ii. Jahresber. Dtsch. Math.-Ver., 50: 19–23, 1940. URL http:/​/​eudml.org/​doc/​146243.
http:/​/​eudml.org/​doc/​146243

[56] H. Tajima and K. Saito. Universal limitation of quantum information recovery: symmetry versus coherence. 2021. https:/​/​doi.org/​10.48550/​arXiv.2103.01876.
https:/​/​doi.org/​10.48550/​arXiv.2103.01876

[57] H. Tajima, N. Shiraishi, and K. Saito. Uncertainty relations in implementation of unitary operations. Phys. Rev. Lett., 121 (11): 110403, sep 2018. 10.1103/​physrevlett.121.110403.
https:/​/​doi.org/​10.1103/​physrevlett.121.110403

[58] H. Tajima, N. Shiraishi, and K. Saito. Coherence cost for violating conservation laws. Phys. Rev. Research, 2 (4): 043374, dec 2020. 10.1103/​physrevresearch.2.043374.
https:/​/​doi.org/​10.1103/​physrevresearch.2.043374

[59] H. Tajima, R. Takagi, and Y. Kuramochi. Universal trade-off structure between symmetry, irreversibility, and quantum coherence in quantum processes. 2022. 10.48550/​arxiv.2206.11086.
https:/​/​doi.org/​10.48550/​arxiv.2206.11086

[60] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Jacobs. Tradeoff between extractable mechanical work, accessible entanglement, and ability to act as a reference system, under arbitrary superselection rules. Phys. Rev. A, 77: 032114, Mar 2008. 10.1103/​PhysRevA.77.032114.
https:/​/​doi.org/​10.1103/​PhysRevA.77.032114

[61] J. A. Vaccaro, S. Croke, and S. M. Barnett. Is coherence catalytic? J. Phys. A: Math. Theor., 51 (41): 414008, Oct. 2018. ISSN 1751-8113, 1751-8121. 10.1088/​1751-8121/​aac112.
https:/​/​doi.org/​10.1088/​1751-8121/​aac112

[62] W. van Dam and P. Hayden. Universal entanglement transformations without communication. Phys. Rev. A, 67 (6): 060302, June 2003a. ISSN 1050-2947, 1094-1622. 10.1103/​PhysRevA.67.060302.
https:/​/​doi.org/​10.1103/​PhysRevA.67.060302

[63] W. van Dam and P. Hayden. Universal entanglement transformations without communication. Physical Review A, 67 (6): 060302, June 2003b. 10.1103/​PhysRevA.67.060302. Publisher: American Physical Society.
https:/​/​doi.org/​10.1103/​PhysRevA.67.060302

[64] F. vom Ende. Progress on the kretschmann-schlingemann-werner conjecture. 2023. 10.48550/​arXiv.2308.15389.
https:/​/​doi.org/​10.48550/​arXiv.2308.15389

[65] N. A. Wiegmann. Necessary and sufficient conditions for unitary similarity. J. Aust. Math. Soc., 2 (1): 122–126, apr 1961. 10.1017/​s1446788700026422.
https:/​/​doi.org/​10.1017/​s1446788700026422

[66] E. P. Wigner. Die messung quantenmechanischer operatoren. Zeitschrift für Physik A Hadrons and nuclei, 133 (1-2): 101–108, sep 1952. 10.1007/​bf01948686.
https:/​/​doi.org/​10.1007/​bf01948686

[67] H. Wilming. Entropy and reversible catalysis. Phys. Rev. Lett., 127: 260402, Dec. 2021. 10.1103/​PhysRevLett.127.260402.
https:/​/​doi.org/​10.1103/​PhysRevLett.127.260402

[68] H. Wilming. Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture. Quantum, 6: 858, nov 2022. 10.22331/​q-2022-11-10-858.
https:/​/​doi.org/​10.22331/​q-2022-11-10-858

[69] H. Wilming, R. Gallego, and J. Eisert. Axiomatic characterization of the quantum relative entropy and free energy. Entropy, 19 (6): 241, 2017. 10.3390/​e19060241.
https:/​/​doi.org/​10.3390/​e19060241

[70] M. M. Yanase. Optimal measuring apparatus. Phys Rev, 123 (2): 666–668, jul 1961. 10.1103/​physrev.123.666.
https:/​/​doi.org/​10.1103/​physrev.123.666

[71] Y. Yang, R. Renner, and G. Chiribella. Optimal universal programming of unitary gates. Physical Review Letters, 125 (21), nov 2020. 10.1103/​physrevlett.125.210501.
https:/​/​doi.org/​10.1103/​physrevlett.125.210501

[72] Y. Yang, R. Renner, and G. Chiribella. Energy requirement for implementing unitary gates on energy-unbounded systems. Journal of Physics A: Mathematical and Theoretical, 55 (49): 494003, dec 2022. 10.1088/​1751-8121/​ac717e.
https:/​/​doi.org/​10.1088/​1751-8121/​ac717e

[73] N. Yunger Halpern and J. M. Renes. Beyond heat baths: Generalized resource theories for small-scale thermodynamics. Phys. Rev. E, 93 (2), Feb. 2016. ISSN 2470-0053. 10.1103/​physreve.93.022126.
https:/​/​doi.org/​10.1103/​physreve.93.022126

[74] J. Åberg. Catalytic Coherence. Phys. Rev. Lett., 113 (15): 150402, Oct. 2014. ISSN 0031-9007, 1079-7114. 10.1103/​PhysRevLett.113.150402.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.150402

Cited by

[1] Patryk Lipka-Bartosik, Giovanni Francesco Diotallevi, and Pharnam Bakhshinezhad, "Fundamental Limits on Anomalous Energy Flows in Correlated Quantum Systems", Physical Review Letters 132 14, 140402 (2024).

[2] A. de Oliveira Junior, Martí Perarnau-Llobet, Nicolas Brunner, and Patryk Lipka-Bartosik, "Quantum catalysis in cavity quantum electrodynamics", Physical Review Research 6 2, 023127 (2024).

[3] Patryk Lipka-Bartosik and Kamil Korzekwa, "Finite-size catalysis in quantum resource theories", arXiv:2405.08914, (2024).

[4] Patryk Lipka-Bartosik, Henrik Wilming, and Nelly H. Y. Ng, "Catalysis in Quantum Information Theory", arXiv:2306.00798, (2023).

[5] Elia Zanoni, Thomas Theurer, and Gilad Gour, "Complete Characterization of Entanglement Embezzlement", arXiv:2303.17749, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-24 19:25:41) and SAO/NASA ADS (last updated successfully 2024-05-24 19:25:42). The list may be incomplete as not all publishers provide suitable and complete citation data.