Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages

Beata Zjawin1, David Schmid1,2,3, Matty J. Hoban4,5, and Ana Belén Sainz1

1International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
3Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
4Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge, CB2 1UB, United Kingdom
5Department of Computing, Goldsmiths, University of London, New Cross, London SE14 6NW, United Kingdom

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

Abstract

Einstein-Podolsky-Rosen (EPR) steering is often (implicitly or explicitly) taken to be evidence for spooky action-at-a-distance. An alternative perspective on steering is that Alice has no causal influence on the physical state of Bob's system; rather, Alice merely updates her knowledge of the state of Bob's system by performing a measurement on a system correlated with his. In this work, we elaborate on this perspective (from which the very term 'steering' is seen to be inappropriate), and we are led to a resource-theoretic treatment of correlations in EPR scenarios. For both bipartite and multipartite scenarios, we develop the resulting resource theory, wherein the free operations are local operations and shared randomness (LOSR). We show that resource conversion under free operations in this paradigm can be evaluated with a single instance of a semidefinite program, making the problem numerically tractable. Moreover, we find that the structure of the pre-order of resources features interesting properties, such as infinite families of incomparable resources. In showing this, we derive new EPR resource monotones. We also discuss advantages of our approach over a pre-existing proposal for a resource theory of 'steering', and discuss how our approach sheds light on basic questions, such as which multipartite assemblages are classically explainable.

► BibTeX data

► References

[1] Albert Einstein, Boris Podolsky, and Nathan Rosen. ``Can quantum-mechanical description of physical reality be considered complete?''. Physical review 47, 777 (1935).
https:/​/​doi.org/​10.1103/​PhysRev.47.777

[2] Erwin Schrödinger. ``Discussion of probability relations between separated systems''. Mathematical Proceedings of the Cambridge Philosophical Society 31, 555–563 (1935).
https:/​/​doi.org/​10.1017/​S0305004100013554

[3] Eric Gama Cavalcanti, Steve J Jones, Howard M Wiseman, and Margaret D Reid. ``Experimental criteria for steering and the einstein-podolsky-rosen paradox''. Physical Review A 80, 032112 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.032112

[4] Howard M Wiseman, Steve James Jones, and Andrew C Doherty. ``Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox''. Physical review letters 98, 140402 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.140402

[5] Roope Uola, Ana CS Costa, H Chau Nguyen, and Otfried Gühne. ``Quantum steering''. Reviews of Modern Physics 92, 015001 (2020).
https:/​/​doi.org/​10.1103/​RevModPhys.92.015001

[6] Daniel Cavalcanti, Paul Skrzypczyk, Gabriel Aguilar, Ranieri V Nery, PH Souto Ribeiro, and Stephen Walborn. ``Detection of entanglement in asymmetric quantum networks and multipartite quantum steering''. Nature communications 6, 1–6 (2015).
https:/​/​doi.org/​10.1038/​ncomms8941

[7] Alejandro Máttar, Paul Skrzypczyk, Gabriel Aguilar, Ranieri V Nery, PH Souto Ribeiro, Stephen Walborn, and Daniel Cavalcanti. ``Experimental multipartite entanglement and randomness certification of the w state in the quantum steering scenario''. Quantum Science and Technology 2, 015011 (2017).
https:/​/​doi.org/​10.1088/​2058-9565/​aa629b

[8] Cyril Branciard, Eric G Cavalcanti, Stephen P Walborn, Valerio Scarani, and Howard M Wiseman. ``One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering''. Physical Review A 85, 010301 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.010301

[9] Yu Xiang, Ioannis Kogias, Gerardo Adesso, and Qiongyi He. ``Multipartite gaussian steering: Monogamy constraints and quantum cryptography applications''. Phys. Rev. A 95, 010101 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.010101

[10] Matthew F Pusey. ``Negativity and steering: A stronger Peres conjecture''. Physical Review A 88, 032313 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.032313

[11] Paul Skrzypczyk, Miguel Navascués, and Daniel Cavalcanti. ``Quantifying Einstein-Podolsky-Rosen steering''. Physical review letters 112, 180404 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.112.180404

[12] Marco Piani and John Watrous. ``Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering''. Physical review letters 114, 060404 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.060404

[13] Rodrigo Gallego and Leandro Aolita. ``Resource theory of steering''. Physical Review X 5, 041008 (2015).
https:/​/​doi.org/​10.1103/​PhysRevX.5.041008

[14] Judea Pearl. ``Causality''. Cambridge University Press. (2009).
https:/​/​doi.org/​10.1017/​CBO9780511803161

[15] Christopher J Wood and Robert W Spekkens. ``The lesson of causal discovery algorithms for quantum correlations: causal explanations of bell-inequality violations require fine-tuning''. New Journal of Physics 17, 033002 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​3/​033002

[16] Julio I De Vicente. ``On nonlocality as a resource theory and nonlocality measures''. Journal of Physics A: Mathematical and Theoretical 47, 424017 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424017

[17] Joshua Geller and Marco Piani. ``Quantifying non-classical and beyond-quantum correlations in the unified operator formalism''. Journal of Physics A: Mathematical and Theoretical 47, 424030 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424030

[18] Rodrigo Gallego and Leandro Aolita. ``Nonlocality free wirings and the distinguishability between bell boxes''. Physical Review A 95, 032118 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.032118

[19] Antonio Acín, Serge Massar, and Stefano Pironio. ``Randomness versus nonlocality and entanglement''. Physical review letters 108, 100402 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.100402

[20] Elie Wolfe, David Schmid, Ana Belén Sainz, Ravi Kunjwal, and Robert W Spekkens. ``Quantifying Bell: The resource theory of nonclassicality of common-cause boxes''. Quantum 4, 280 (2020).
https:/​/​doi.org/​10.22331/​q-2020-06-08-280

[21] David Schmid, Thomas C Fraser, Ravi Kunjwal, Ana Belén Sainz, Elie Wolfe, and Robert W Spekkens. ``Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory'' (2020).
arXiv:2004.09194

[22] David Schmid, Denis Rosset, and Francesco Buscemi. ``The type-independent resource theory of local operations and shared randomness''. Quantum 4, 262 (2020).
https:/​/​doi.org/​10.22331/​q-2020-04-30-262

[23] Denis Rosset, David Schmid, and Francesco Buscemi. ``Type-independent characterization of spacelike separated resources''. Physical Review Letters 125, 210402 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.210402

[24] David Schmid, John H Selby, and Robert W Spekkens. ``Unscrambling the omelette of causation and inference: The framework of causal-inferential theories'' (2020).
arXiv:2009.03297

[25] E.T. Jaynes. ``Probability in quantum theory''. in Complexity, Entropy, and the Physics of Information edited by W. H. ZurekPage 381 (1990).
https:/​/​doi.org/​10.1201/​9780429502880

[26] Stephen D. Bartlett, Terry Rudolph, and Robert W. Spekkens. ``Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction''. Phys. Rev. A 86, 012103 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.012103

[27] Christopher A Fuchs. ``Quantum mechanics as quantum information (and only a little more)'' (2002).
arXiv:quant-ph/0205039

[28] David Schmid, Katja Ried, and Robert W. Spekkens. ``Why initial system-environment correlations do not imply the failure of complete positivity: A causal perspective''. Phys. Rev. A 100, 022112 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.022112

[29] David Schmid. ``Guiding our interpretation of quantum theory by principles of causation and inference''. PhD thesis, University of Waterloo (2021).

[30] Bob Coecke, Tobias Fritz, and Robert W Spekkens. ``A mathematical theory of resources''. Information and Computation 250, 59–86 (2016).
https:/​/​doi.org/​10.1016/​j.ic.2016.02.008

[31] Eric Chitambar and Gilad Gour. ``Quantum resource theories''. Reviews of Modern Physics 91, 025001 (2019).
https:/​/​doi.org/​10.1103/​RevModPhys.91.025001

[32] Michael A Nielsen. ``Conditions for a class of entanglement transformations''. Physical Review Letters 83, 436 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.436

[33] Charles H Bennett, Herbert J Bernstein, Sandu Popescu, and Benjamin Schumacher. ``Concentrating partial entanglement by local operations''. Physical Review A 53, 2046 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.53.2046

[34] Yuval Rishu Sanders and Gilad Gour. ``Necessary conditions for entanglement catalysts''. Physical Review A 79, 054302 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.054302

[35] Francesco Buscemi. ``All entangled quantum states are nonlocal''. Physical review letters 108, 200401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.200401

[36] David Schmid, Haoxing Du, Maryam Mudassar, Ghi Coulter-de Wit, Denis Rosset, and Matty J Hoban. ``Postquantum common-cause channels: the resource theory of local operations and shared entanglement''. Quantum 5, 419 (2021).
https:/​/​doi.org/​10.22331/​q-2021-03-23-419

[37] Jonathan Barrett, Noah Linden, Serge Massar, Stefano Pironio, Sandu Popescu, and David Roberts. ``Nonlocal correlations as an information-theoretic resource''. Physical review A 71, 022101 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.022101

[38] Nicolas Brunner and Paul Skrzypczyk. ``Nonlocality distillation and postquantum theories with trivial communication complexity''. Physical review letters 102, 160403 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.160403

[39] Iman Marvian and Robert W Spekkens. ``How to quantify coherence: Distinguishing speakable and unspeakable notions''. Physical Review A 94, 052324 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.052324

[40] Iman Marvian, Robert W Spekkens, and Paolo Zanardi. ``Quantum speed limits, coherence, and asymmetry''. Physical Review A 93, 052331 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.052331

[41] Andreas Winter and Dong Yang. ``Operational resource theory of coherence''. Physical review letters 116, 120404 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.120404

[42] Fernando GSL Brandao, Michał Horodecki, Jonathan Oppenheim, Joseph M Renes, and Robert W Spekkens. ``Resource theory of quantum states out of thermal equilibrium''. Physical review letters 111, 250404 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.250404

[43] Paul Skrzypczyk, Anthony J Short, and Sandu Popescu. ``Work extraction and thermodynamics for individual quantum systems''. Nature communications 5, 1–8 (2014).
https:/​/​doi.org/​10.1038/​ncomms5185

[44] Dominik Janzing, Pawel Wocjan, Robert Zeier, Rubino Geiss, and Th Beth. ``Thermodynamic cost of reliability and low temperatures: tightening landauer's principle and the second law''. International Journal of Theoretical Physics 39, 2717–2753 (2000).
https:/​/​doi.org/​10.1023/​A:1026422630734

[45] Michał Horodecki and Jonathan Oppenheim. ``Fundamental limitations for quantum and nanoscale thermodynamics''. Nature communications 4, 1–6 (2013).
https:/​/​doi.org/​10.1038/​ncomms3059

[46] Gilad Gour, Markus P Müller, Varun Narasimhachar, Robert W Spekkens, and Nicole Yunger Halpern. ``The resource theory of informational nonequilibrium in thermodynamics''. Physics Reports 583, 1–58 (2015).
https:/​/​doi.org/​10.1016/​j.physrep.2015.04.003

[47] Zoë Holmes, Erick Hinds Mingo, Calvin Y-R Chen, and Florian Mintert. ``Quantifying athermality and quantum induced deviations from classical fluctuation relations''. Entropy 22, 111 (2020).
https:/​/​doi.org/​10.3390/​e22010111

[48] Francesco Buscemi, Eric Chitambar, and Wenbin Zhou. ``Complete resource theory of quantum incompatibility as quantum programmability''. Physical review letters 124, 120401 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.120401

[49] Ana Belén Sainz, Nicolas Brunner, Daniel Cavalcanti, Paul Skrzypczyk, and Tamás Vértesi. ``Postquantum steering''. Physical review letters 115, 190403 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.190403

[50] Ana Belén Sainz, Matty J Hoban, Paul Skrzypczyk, and Leandro Aolita. ``Bipartite postquantum steering in generalized scenarios''. Physical Review Letters 125, 050404 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.050404

[51] Matthew S Leifer and Robert W Spekkens. ``Towards a formulation of quantum theory as a causally neutral theory of bayesian inference''. Physical Review A 88, 052130 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.052130

[52] Marco Piani. ``Channel steering''. JOSA B 32, A1–A7 (2015).
https:/​/​doi.org/​10.1364/​JOSAB.32.0000A1

[53] Nicolas Gisin. ``Stochastic quantum dynamics and relativity''. Helvetica Physica Acta 62, 363–371 (1989).
https:/​/​doi.org/​10.5169/​seals-116034

[54] Lane P Hughston, Richard Jozsa, and William K Wootters. ``A complete classification of quantum ensembles having a given density matrix''. Physics Letters A 183, 14–18 (1993).
https:/​/​doi.org/​10.1016/​0375-9601(93)90880-9

[55] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Physical Review A 80, 022339 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.022339

[56] Michael A. Nielsen and Isaac L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''. Cambridge University Press. (2011).
https:/​/​doi.org/​10.1017/​CBO9780511976667

[57] Arthur Fine. ``Hidden Variables, Joint Probability, and the Bell Inequalities''. Phys. Rev. Lett. 48, 291–295 (1982).
https:/​/​doi.org/​10.1103/​PhysRevLett.48.291

[58] Man-Duen Choi. ``Completely positive linear maps on complex matrices''. Linear algebra and its applications 10, 285–290 (1975).
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

[59] Andrzej Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Reports on Mathematical Physics 3, 275–278 (1972).
https:/​/​doi.org/​10.1016/​0034-4877(72)90011-0

[60] ``Matlab''. url: https:/​/​www.mathworks.com/​.
https:/​/​www.mathworks.com/​

[61] Michael Grant and Stephen Boyd. ``Cvx: MATLAB software for disciplined convex programming''. url: http:/​/​cvxr.com/​cvx.
http:/​/​cvxr.com/​cvx

[62] Michael Grant and Stephen Boyd. ``Graph implementations for nonsmooth convex programs''. In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control. Pages 95–110. Lecture Notes in Control and Information Sciences. Springer-Verlag Limited (2008).

[63] K. C. Toh, M. J. Todd, and R. H. Tütüncü. ``SDPT3 — a matlab software package for semidefinite programming, version 1.3''. Optimization Methods and Software 11, 545–581 (1999).
https:/​/​doi.org/​10.1080/​10556789908805762

[64] Nathaniel Johnston. ``QETLAB: a MATLAB toolbox for quantum entanglement''. url: http:/​/​qetlab.com.
http:/​/​qetlab.com

[65] Beata Zjawin, David Schmid, Matty J. Hoban, and Ana Belén Sainz. code: beatazjawin/​Quantifying-EPR.
https:/​/​github.com/​beatazjawin/​Quantifying-EPR

[66] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Rev. Mod. Phys. 81, 865–942 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[67] Cédric Bamps and Stefano Pironio. ``Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing''. Physical Review A 91, 052111 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052111

[68] Eric G Cavalcanti, Qiongyi Y He, Margaret D Reid, and Howard M Wiseman. ``Unified criteria for multipartite quantum nonlocality''. Physical Review A 84, 032115 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.032115

[69] Alexander A Klyachko. ``Quantum marginal problem and n-representability''. Journal of Physics: Conference Series 36, 014 (2006).
https:/​/​doi.org/​10.1088/​1742-6596/​36/​1/​014

[70] Daniel Cavalcanti and Paul Skrzypczyk. ``Quantum steering: A review with focus on semidefinite programming''. Reports on Progress in Physics 80, 024001 (2016).
https:/​/​doi.org/​10.1088/​1361-6633/​80/​2/​024001

[71] Márcio M Taddei, Thais de Lima Silva, Ranieri V. Nery, Gabriel Aguilar, Stephen P Walborn, and Leandro Aolita. ``Exposure of subtle multipartite quantum nonlocality''. npj Quantum Information 7, 1–8 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00402-5

[72] Flavio Baccari, Remigiusz Augusiak, Ivan Šupić, Jordi Tura, and Antonio Acín. ``Scalable Bell inequalities for qubit graph states and robust self-testing''. Physical review letters 124, 020402 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.020402

[73] Sandu Popescu and Daniel Rohrlich. ``Quantum nonlocality as an axiom''. Foundations of Physics 24, 379–385 (1994).
https:/​/​doi.org/​10.1007/​BF02058098

[74] Lucien Hardy. ``Quantum theory from five reasonable axioms'' (2001).
arXiv:quant-ph/0101012

[75] Jonathan Barrett. ``Information processing in generalized probabilistic theories''. Physical Review A 75, 032304 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.032304

[76] Matty J Hoban and Ana Belén Sainz. ``A channel-based framework for steering, non-locality and beyond''. New Journal of Physics 20, 053048 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aabea8

[77] Ivan Šupić and Matty J Hoban. ``Self-testing through EPR-steering''. New Journal of Physics 18, 075006 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​7/​075006

[78] Alexandru Gheorghiu, Petros Wallden, and Elham Kashefi. ``Rigidity of quantum steering and one-sided device-independent verifiable quantum computation''. New Journal of Physics 19, 023043 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa5cff

[79] Shin-Liang Chen, Huan-Yu Ku, Wenbin Zhou, Jordi Tura, and Yueh-Nan Chen. ``Robust self-testing of steerable quantum assemblages and its applications on device-independent quantum certification''. Quantum 5, 552 (2021).
https:/​/​doi.org/​10.22331/​q-2021-09-28-552

Cited by

[1] David Schmid, Thomas C. Fraser, Ravi Kunjwal, Ana Belen Sainz, Elie Wolfe, and Robert W. Spekkens, "Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory", Quantum 7, 1194 (2023).

[2] Beata Zjawin, David Schmid, Matty J. Hoban, and Ana Belén Sainz, "The resource theory of nonclassicality of channel assemblages", Quantum 7, 1134 (2023).

[3] Vinicius P. Rossi, Matty J. Hoban, and Ana Belén Sainz, "On characterising assemblages in Einstein-Podolsky-Rosen scenarios", Journal of Physics A Mathematical General 55 26, 264002 (2022).

[4] Anna Jenčová, "Assemblages and steering in general probabilistic theories", Journal of Physics A Mathematical General 55 43, 434001 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-15 07:38:14) and SAO/NASA ADS (last updated successfully 2024-04-15 07:38:15). The list may be incomplete as not all publishers provide suitable and complete citation data.