Uncertainty relations: An operational approach to the error-disturbance tradeoff

Joseph M. Renes1, Volkher B. Scholz1,2, and Stefan Huber1,3

1Institute for Theoretical Physics, ETH Zürich, Switzerland
2Department of Physics, Ghent University, Belgium
3Department of Mathematics, Technische Universität München, Germany

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

Abstract

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.

► BibTeX data

► References

[1] W. Heisenberg ``Über Den Anschaulichen Inhalt Der Quantentheoretischen Kinematik Und Mechanik'' Zeitschrift für Physik 43, 172-198 (1927).
https:/​/​doi.org/​10.1007/​BF01397280

[2] John Archibald Wheelerand Wojciech Hubert Zurek ``Quantum Theory and Measurement'' Princeton University Press (1984).
http:/​/​press.princeton.edu/​titles/​806.html

[3] E. H. Kennard ``Zur Quantenmechanik Einfacher Bewegungstypen'' Zeitschrift für Physik 44, 326–352 (1927).
https:/​/​doi.org/​10.1007/​BF01391200
http:/​/​www.springerlink.com/​content/​p150358420710341/​

[4] H. P. Robertson ``The Uncertainty Principle'' Physical Review 34, 163 (1929).
https:/​/​doi.org/​10.1103/​PhysRev.34.163

[5] Hans Maassenand J. B. M. Uffink ``Generalized Entropic Uncertainty Relations'' Physical Review Letters 60, 1103 (1988).
https:/​/​doi.org/​10.1103/​PhysRevLett.60.1103
http:/​/​link.aps.org/​abstract/​PRL/​v60/​p1103

[6] Mario Berta, Matthias Christandl, Roger Colbeck, Joseph M. Renes, and Renato Renner, ``The Uncertainty Principle in the Presence of Quantum Memory'' Nature Physics 6, 659–662 (2010).
https:/​/​doi.org/​10.1038/​nphys1734
arXiv:0909.0950

[7] Patrick J. Coles, Mario Berta, Marco Tomamichel, and Stephanie Wehner, ``Entropic Uncertainty Relations and Their Applications'' Reviews of Modern Physics 89, 015002 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.015002
arXiv:1511.04857

[8] E. Arthursand J. L. Kelly ``On the Simultaneous Measurement of a Pair of Conjugate Observables'' Bell System Technical Journal 44, 725–729 (1965).
https:/​/​doi.org/​10.1002/​j.1538-7305.1965.tb01684.x

[9] C. Y. Sheand H. Heffner ``Simultaneous Measurement of Noncommuting Observables'' Physical Review 152, 1103–1110 (1966).
https:/​/​doi.org/​10.1103/​PhysRev.152.1103

[10] E. B. Davies ``Quantum Theory of Open Systems'' Academic Press (1976).

[11] S. Twareque Aliand E. Prugovečki ``Systems of Imprimitivity and Representations of Quantum Mechanics on Fuzzy Phase Spaces'' Journal of Mathematical Physics 18, 219–228 (1977).
https:/​/​doi.org/​10.1063/​1.523259

[12] E. Prugovečki ``On Fuzzy Spin Spaces'' Journal of Physics A: Mathematical and General 10, 543 (1977).
https:/​/​doi.org/​10.1088/​0305-4470/​10/​4/​016
http:/​/​iopscience.iop.org/​0305-4470/​10/​4/​016

[13] Paul Busch ``Indeterminacy Relations and Simultaneous Measurements in Quantum Theory'' International Journal of Theoretical Physics 24, 63–92 (1985).
https:/​/​doi.org/​10.1007/​BF00670074

[14] Paul Busch ``Unsharp Reality and Joint Measurements for Spin Observables'' Physical Review D 33, 2253–2261 (1986).
https:/​/​doi.org/​10.1103/​PhysRevD.33.2253

[15] E. Arthursand M. S. Goodman ``Quantum Correlations: A Generalized Heisenberg Uncertainty Relation'' Physical Review Letters 60, 2447–2449 (1988).
https:/​/​doi.org/​10.1103/​PhysRevLett.60.2447

[16] Hans Martensand Willem M. Muynck ``Towards a New Uncertainty Principle: Quantum Measurement Noise'' Physics Letters A 157, 441–448 (1991).
https:/​/​doi.org/​10.1016/​0375-9601(91)91015-6
http:/​/​linkinghub.elsevier.com/​retrieve/​pii/​0375960191910156

[17] Shiro Ishikawa ``Uncertainty Relations in Simultaneous Measurements for Arbitrary Observables'' Reports on Mathematical Physics 29, 257–273 (1991).
https:/​/​doi.org/​10.1016/​0034-4877(91)90046-P
http:/​/​www.sciencedirect.com/​science/​article/​pii/​003448779190046P

[18] M. G. Raymer ``Uncertainty Principle for Joint Measurement of Noncommuting Variables'' American Journal of Physics 62, 986–993 (1994).
https:/​/​doi.org/​10.1119/​1.17657

[19] U. Leonhardt, B. Böhmer, and H. Paul, ``Uncertainty Relations for Realistic Joint Measurements of Position and Momentum in Quantum Optics'' Optics Communications 119, 296–300 (1995).
https:/​/​doi.org/​10.1016/​0030-4018(95)00321-X
http:/​/​linkinghub.elsevier.com/​retrieve/​pii/​003040189500321X

[20] D. M. Appleby ``Concept of Experimental Accuracy and Simultaneous Measurements of Position and Momentum'' International Journal of Theoretical Physics 37, 1491–1509 (1998).
https:/​/​doi.org/​10.1023/​A:1026659601439
arXiv:quant-ph/9803046
http:/​/​link.springer.com/​article/​10.1023/​A%3A1026659601439

[21] Michael J. W. Hall ``Prior Information: How to Circumvent the Standard Joint-Measurement Uncertainty Relation'' Physical Review A 69, 052113 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.052113
arXiv:quant-ph/0309091

[22] R. F. Werner ``The Uncertainty Relation for Joint Measurement of Position and Momentum'' Quantum Information and Computation 4, 546–562 (2004).
arXiv:quant-ph/0405184
http:/​/​www.rintonpress.com/​journals/​qicabstracts/​qicabstracts4-67.html

[23] Masanao Ozawa ``Uncertainty Relations for Joint Measurements of Noncommuting Observables'' Physics Letters A 320, 367–374 (2004).
https:/​/​doi.org/​10.1016/​j.physleta.2003.12.001
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0375960103017560

[24] Yu Watanabe, Takahiro Sagawa, and Masahito Ueda, ``Uncertainty Relation Revisited from Quantum Estimation Theory'' Physical Review A 84, 042121 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.042121
arXiv:1010.3571

[25] Paul Busch, Pekka Lahti, and Reinhard F. Werner, ``Proof of Heisenberg's Error-Disturbance Relation'' Physical Review Letters 111, 160405 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.160405
arXiv:1306.1565

[26] Paul Busch, Pekka Lahti, and Reinhard F. Werner, ``Heisenberg Uncertainty for Qubit Measurements'' Physical Review A 89, 012129 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.012129
arXiv:1311.0837

[27] Paul Busch, Pekka Lahti, and Reinhard F. Werner, ``Measurement Uncertainty Relations'' Journal of Mathematical Physics 55, 042111 (2014).
https:/​/​doi.org/​10.1063/​1.4871444
arXiv:1312.4392

[28] Vladimir B. Braginskyand Farid Ya Khalili ``Quantum Measurement'' Cambridge University Press (1992).
https:/​/​doi.org/​0.1017/​CBO9780511622748

[29] H. Martensand W. M. Muynck ``Disturbance, Conservation Laws and the Uncertainty Principle'' Journal of Physics A: Mathematical and General 25, 4887 (1992).
https:/​/​doi.org/​10.1088/​0305-4470/​25/​18/​021
http:/​/​iopscience.iop.org/​0305-4470/​25/​18/​021

[30] Masanao Ozawa ``Universally Valid Reformulation of the Heisenberg Uncertainty Principle on Noise and Disturbance in Measurement'' Physical Review A 67, 042105 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.042105
arXiv:quant-ph/0207121

[31] Yu Watanabeand Masahito Ueda ``Quantum Estimation Theory of Error and Disturbance in Quantum Measurement'' (2011).
arXiv:1106.2526
http:/​/​arxiv.org/​abs/​1106.2526

[32] Cyril Branciard ``Error-Tradeoff and Error-Disturbance Relations for Incompatible Quantum Measurements'' Proceedings of the National Academy of Sciences 110, 6742–6747 (2013).
https:/​/​doi.org/​10.1073/​pnas.1219331110
arXiv:1304.2071
http:/​/​www.pnas.org/​content/​110/​17/​6742

[33] Francesco Buscemi, Michael J. W. Hall, Masanao Ozawa, and Mark M. Wilde, ``Noise and Disturbance in Quantum Measurements: An Information-Theoretic Approach'' Physical Review Letters 112, 050401 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.050401
arXiv:1310.6603

[34] Asger C. Ipsen ``Error-Disturbance Relations for Finite Dimensional Systems'' (2013).
arXiv:1311.0259
http:/​/​arxiv.org/​abs/​1311.0259

[35] Patrick J. Colesand Fabian Furrer ``State-Dependent Approach to Entropic Measurement–disturbance Relations'' Physics Letters A 379, 105–112 (2015).
https:/​/​doi.org/​10.1016/​j.physleta.2014.11.002
arXiv:1311.7637
http:/​/​linkinghub.elsevier.com/​retrieve/​pii/​S0375960114011098

[36] Masanao Ozawa ``Uncertainty Relations for Noise and Disturbance in Generalized Quantum Measurements'' Annals of Physics 311, 350–416 (2004).
https:/​/​doi.org/​10.1016/​j.aop.2003.12.012
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0003491604000089

[37] Paul Busch, Pekka Lahti, and Reinhard F. Werner, ``Quantum Root-Mean-Square Error and Measurement Uncertainty Relations'' Reviews of Modern Physics 86, 1261–1281 (2014).
https:/​/​doi.org/​10.1103/​RevModPhys.86.1261
arXiv:1312.4393

[38] David Marcus Appleby ``Quantum Errors and Disturbances: Response to Busch, Lahti and Werner'' Entropy 18, 174 (2016).
https:/​/​doi.org/​10.3390/​e18050174
arXiv:1602.09002
http:/​/​www.mdpi.com/​1099-4300/​18/​5/​174

[39] Masanao Ozawa ``Disproving Heisenberg's Error-Disturbance Relation'' (2013).
arXiv:1308.3540
http:/​/​arxiv.org/​abs/​1308.3540

[40] 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, 1708–1717 (2008).
https:/​/​doi.org/​10.1109/​TIT.2008.917696
arXiv:quant-ph/0605009

[41] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner, ``A Continuity Theorem for Stinespring's Dilation'' Journal of Functional Analysis 255, 1889–1904 (2008).
https:/​/​doi.org/​16/​j.jfa.2008.07.023
arXiv:0710.2495
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0022123608002991

[42] Berthold-Georg Englert ``Fringe Visibility and Which-Way Information: An Inequality'' Physical Review Letters 77, 2154 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.2154

[43] Joseph M. Renesand Jean-Christian Boileau ``Conjectured Strong Complementary Information Tradeoff'' Physical Review Letters 103, 020402 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.020402
arXiv:0806.3984
http:/​/​link.aps.org/​abstract/​PRL/​v103/​e020402

[44] Marco Tomamicheland Renato Renner ``Uncertainty Relation for Smooth Entropies'' Physical Review Letters 106, 110506 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.110506
arXiv:1009.2015

[45] Marco Tomamichel, Charles Ci Wen Lim, Nicolas Gisin, and Renato Renner, ``Tight Finite-Key Analysis for Quantum Cryptography'' Nature Communications 3, 634 (2012).
https:/​/​doi.org/​10.1038/​ncomms1631
arXiv:1103.4130

[46] Felipe G. Lacerda, Joseph M. Renes, and Renato Renner, ``Classical Leakage Resilience from Fault-Tolerant Quantum Computation'' (2014).
arXiv:1404.7516
http:/​/​arxiv.org/​abs/​1404.7516

[47] Karl Kraus ``States, Effects, and Operations: Fundamental Notions of Quantum Theory'' Springer-Verlag (1983).
https:/​/​doi.org/​10.1007/​3-540-12732-1

[48] Reinhard F. Werner ``Quantum Information Theory — an Invitation'' Springer Berlin Heidelberg (2001).
https:/​/​doi.org/​10.1007/​3-540-44678-8_2
arXiv:quant-ph/0101061

[49] M. M. Wolf ``Quantum Channels and Operations: A Guided Tour'' (2012).
http:/​/​www-m5.ma.tum.de/​Allgemeines/​MichaelWolf

[50] Cédric Bényand Florian Richter ``Algebraic Approach to Quantum Theory: A Finite-Dimensional Guide'' (2015).
arXiv:1505.03106
http:/​/​arxiv.org/​abs/​1505.03106

[51] A Yu Kitaev ``Quantum Computations: Algorithms and Error Correction'' Russian Mathematical Surveys 52, 1191–1249 (1997).
https:/​/​doi.org/​10.1070/​RM1997v052n06ABEH002155
http:/​/​iopscience.iop.org/​0036-0279/​52/​6/​R02

[52] Vern Paulsen ``Completely Bounded Maps and Operator Algebras'' Cambridge University Press (2003).
http:/​/​www.cambridge.org/​catalogue/​catalogue.asp?isbn=051105470X

[53] Alexei Gilchrist, Nathan K. Langford, and Michael A. Nielsen, ``Distance Measures to Compare Real and Ideal Quantum Processes'' Physical Review A 71, 062310 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.062310
arXiv:quant-ph/0408063

[54] John Watrous ``Semidefinite Programs for Completely Bounded Norms'' Theory of Computing 5, 217–238 (2009).
https:/​/​doi.org/​10.4086/​toc.2009.v005a011
arXiv:0901.4709

[55] John Watrous ``Simpler Semidefinite Programs for Completely Bounded Norms'' Chicago Journal of Theoretical Computer Science 2013, 8 (2013).
https:/​/​doi.org/​10.4086/​cjtcs.2013.008
arXiv:1207.5726

[56] W. Forrest Stinespring ``Positive Functions on C*-Algebras'' Proceedings of the American Mathematical Society 6, 211–216 (1955).
https:/​/​doi.org/​10.1090/​S0002-9939-1955-0069403-4
http:/​/​www.ams.org/​proc/​1955-06-02/​S0002-9939-1955-0069403-4/​

[57] Patrick J. Colesand Marco Piani ``Improved Entropic Uncertainty Relations and Information Exclusion Relations'' Physical Review A 89, 022112 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.022112
arXiv:1307.4265

[58] Peter W. Shorand John Preskill ``Simple Proof of Security of the BB84 Quantum Key Distribution Protocol'' Physical Review Letters 85, 441 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.85.441
arXiv:quant-ph/0003004

[59] Igor Devetak ``The Private Classical Capacity and Quantum Capacity of a Quantum Channel'' IEEE Transactions on Information Theory 51, 44–55 (2005).
https:/​/​doi.org/​10.1109/​TIT.2004.839515
arXiv:quant-ph/0304127

[60] Joseph M. Renes ``Duality of Privacy Amplification against Quantum Adversaries and Data Compression with Quantum Side Information'' Proceedings of the Royal Society A 467, 1604–1623 (2011).
https:/​/​doi.org/​10.1098/​rspa.2010.0445
arXiv:1003.0703
http:/​/​rspa.royalsocietypublishing.org/​content/​467/​2130/​1604

[61] Joseph M. Renes ``The Physics of Quantum Information: Complementarity, Uncertainty, and Entanglement'' thesis (2012).
arXiv:1212.2379
http:/​/​arxiv.org/​abs/​1212.2379

[62] Joseph M. Renes ``Uncertainty Relations and Approximate Quantum Error Correction'' Physical Review A 94, 032314 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032314
arXiv:1605.01420

[63] Patrick J. Coles, Jedrzej Kaniewski, and Stephanie Wehner, ``Equivalence of Wave–particle Duality to Entropic Uncertainty'' Nature Communications 5, 5814 (2014).
https:/​/​doi.org/​10.1038/​ncomms6814
arXiv:1403.4687
http:/​/​www.nature.com/​ncomms/​2014/​141219/​ncomms6814/​full/​ncomms6814.html

[64] Patrick J. Coles ``Entropic Framework for Wave-Particle Duality in Multipath Interferometers'' Physical Review A 93, 062111 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.062111
arXiv:1512.09081

[65] Kamil Korzekwa, David Jennings, and Terry Rudolph, ``Operational Constraints on State-Dependent Formulations of Quantum Error-Disturbance Trade-off Relations'' Physical Review A 89, 052108 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.052108
arXiv:1311.5506

[66] Alberto Barchielli, Matteo Gregoratti, and Alessandro Toigo, ``Measurement Uncertainty Relations for Discrete Observables: Relative Entropy Formulation'' (2016).
arXiv:1608.01986
http:/​/​arxiv.org/​abs/​1608.01986

[67] Massimiliano F. Sacchi ``Entanglement Can Enhance the Distinguishability of Entanglement-Breaking Channels'' Physical Review A 72, 014305 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.014305
arXiv:quant-ph/0505174

[68] V. P. Belavkin ``Optimal Multiple Quantum Statistical Hypothesis Testing'' Stochastics 1, 315 (1975).
https:/​/​doi.org/​10.1080/​17442507508833114

[69] Paul Hausladenand William K. Wootters ``A `Pretty Good' Measurement for Distinguishing Quantum States'' Journal of Modern Optics 41, 2385 (1994).
https:/​/​doi.org/​10.1080/​09500349414552221

Cited by

[1] T Bullock and P Busch, "Measurement uncertainty relations: characterising optimal error bounds for qubits", Journal of Physics A: Mathematical and Theoretical 51 28, 283001 (2018).

[2] Anna-Lena K. Hashagen and Michael M. Wolf, "Universality and Optimality in the Information–Disturbance Tradeoff", Annales Henri Poincaré 20 1, 219 (2019).

[3] Hubert de Guise, Lorenzo Maccone, Barry C. Sanders, and Namrata Shukla, "State-independent uncertainty relations", Physical Review A 98 4, 042121 (2018).

[4] Felipe G. Lacerda, Joseph M. Renes, and Renato Renner, "Classical Leakage Resilience from Fault-Tolerant Quantum Computation", Journal of Cryptology 32 4, 1071 (2019).

[5] Ilya Kull, Philippe Allard Guérin, and Frank Verstraete, "Uncertainty and trade-offs in quantum multiparameter estimation", Journal of Physics A: Mathematical and Theoretical 53 24, 244001 (2020).

[6] René Schwonnek, "Additivity of entropic uncertainty relations", arXiv:1801.04602, Quantum 2, 59 (2018).

The above citations are from Crossref's cited-by service (last updated successfully 2020-10-19 14:34:06) and SAO/NASA ADS (last updated successfully 2020-10-19 14:34:07). The list may be incomplete as not all publishers provide suitable and complete citation data.