Quantum marginal problem and incompatibility

Erkka Haapasalo1,2, Tristan Kraft3, Nikolai Miklin4, and Roope Uola5

1Centre for Quantum Technologies, National University of Singapore, Science Drive 2, Block S15-03-18, Singapore 117543
2Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China
3Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany
4Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdańsk, Poland
5Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

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


One of the basic distinctions between classical and quantum mechanics is the existence of fundamentally incompatible quantities. Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.

► BibTeX data

► References

[1] Teiko Heinosaari, Takayuki Miyadera, and Mário Ziman. An invitation to quantum incompatibility. J. Phys. A, 49:123001, 2016.

[2] Marco Túlio Quintino, Tamás Vértesi, and Nicolas Brunner. Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality. Phys. Rev. Lett., 113:160402, 2014.

[3] Roope Uola, Tobias Moroder, and Otfried Gühne. Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett., 113:160403, 2014.

[4] Roope Uola, Costantino Budroni, Otfried Gühne, and Juha-Pekka Pellonpää. One-to-one mapping between steering and joint measurability problems. Phys. Rev. Lett., 115:230402, 2015.

[5] Jukka Kiukas, Constantino Budroni, Roope Uola, and Juha-Pekka Pellonpää. Continuous-variable steering and incompatibility via state-channel dualism. Phys. Rev. A, 96:042331, 2017.

[6] Zhen-Peng Xu and Adán Cabello. Necessary and sufficient condition for contextuality from incompatibility. Phys. Rev. A, 99:020103, 2019.

[7] Armin Tavakoli and Roope Uola. Measurement incompatibility and steering are necessary and sufficient for operational contextuality. Phys. Rev. Research, 2:013011, 2020.

[8] Lucas Clemente and Johannes Kofler. Necessary and sufficient conditions for macroscopic realism from quantum mechanics. Phys. Rev. A, 91:062103, 2015.

[9] Roope Uola, Giuseppe Vitagliano, and Costantino Budroni. Leggett-Garg macrorealism and the quantum nondisturbance conditions. Phys. Rev. A, 100:042117, 2019.

[10] Claudio Carmeli, Teiko Heinosaari, and Alessandro Toigo. Quantum incompatibility witnesses. Phys. Rev. Lett., 122:130402, 2019.

[11] Paul Skrzypczyk, Ivan Šupić, and Daniel Cavalcanti. All sets of incompatible measurements give an advantage in quantum state discrimination. Phys. Rev. Lett., 122:130403, 2019.

[12] Roope Uola, Tristan Kraft, Jiangwei Shang, Xiao-Dong Yu, and Otfried Gühne. Quantifying quantum resources with conic programming. Phys. Rev. Lett., 122:130404, 2019.

[13] Michał Oszmaniec and Tanmoy Biswas. Operational relevance of resource theories of quantum measurements. Quantum, 3:133, 2019.

[14] Leonardo Guerini, Marco Túlio Quintino, and Leandro Aolita. Distributed sampling, quantum communication witnesses, and measurement incompatibility. Phys. Rev. A, 100:042308, 2019.

[15] Claudio Carmeli, Teiko Heinosaari, Takayuki Miyadera, and Alessandro Toigo. Witnessing incompatibility of quantum channels. J. Math. Phys., 60:122202, 2019.

[16] Junki Mori. Operational characterization of incompatibility of quantum channels with quantum state discrimination. Phys. Rev. A, 101:032331, 2020.

[17] Roope Uola, Tristan Kraft, and Alastair A. Abbott. Quantification of quantum dynamics with input-output games. Phys. Rev. A, 101:052306, 2020.

[18] Francesco Buscemi, Eric Chitambar, and Wenbin Zhou. Complete resource theory of quantum incompatibility as quantum programmability. Phys. Rev. Lett., 124:120401, 2020.

[19] Arthur Fine. Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett., 48:291, 1982.

[20] Michael M. Wolf, David Perez-Garcia, and Carlos Fernandez. Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. Phys. Rev. Lett., 103:230402, 2009.

[21] Mary Beth Ruskai. $N$-representability problem: Conditions on geminals. Phys. Rev., 183:129, 1969.

[22] National Research Council. Mathematical Challenges from Theoretical/​Computational Chemistry. The National Academies Press, Washington, DC, 1995.

[23] Yi-Kai Liu. Consistency of local density matrices is QMA-complete. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Lecture Notes in Computer Science, vol 4110. Springer, Berlin, Heidelberg, 2006.

[24] Valerie Coffman, Joydip Kundu, and William K. Wootters. Distributed entanglement. Phys. Rev. A, 61:052306, 2000.

[25] A. J. Coleman. Structure of fermion density matrices. Rev. Mod. Phys., 35:668, 1963.

[26] Alexander Klyachko. Quantum marginal problem and representations of the symmetric group. arXiv:quant-ph/​0409113, 2004.

[27] Alexander A Klyachko. Quantum marginal problem and $N$-representability. J. Phys. Conf. Ser., 36:72, 2006.

[28] Christian Schilling, Carlos L. Benavides-Riveros, and Péter Vrana. Reconstructing quantum states from single-party information. Phys. Rev. A, 96:052312, 2017.

[29] Giulio Chiribella. On quantum estimation, quantum cloning and finite quantum de Finetti theorems. In Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science, vol 6519. Springer, Berlin, Heidelberg, 2011.

[30] Joonwoo Bae and Antonio Acín. Asymptotic quantum cloning is state estimation. Phys. Rev. Lett., 97:030402, 2006.

[31] Erling Størmer. Symmetric states of infinite tensor products of $C^*$-algebras. J. Funct. Anal., 3:48, 1969.

[32] G. A. Raggio and Reinhard F. Werner. Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta, 62:980, 1989.

[33] Teiko Heinosaari and Takayuki Miyadera. Incompatibility of quantum channels. J. Phys. A, 50:135302, 2017.

[34] Lukasz Pankowski, Fernando G.S.L. Brandao, Michal Horodecki, Graeme Smith. Entanglement distillation by extendible maps. Q. Inf. Comp., 13:751, 2013.

[35] Mario Berta, Francesco Borderi, Omar Fawzi, Volkher Scholz. Semidefinite programming hierarchies for quantum error correction. arXiv:1810.12197, 2018.

[36] Eneet Kaur, Siddhartha Das, Mark M. Wilde, and Andreas Winter. Extendibility limits the performance of quantum processors. Phys. Rev. Lett., 123:070502, 2019.

[37] Erkka Haapasalo. The Choi-Jamiolkowski isomorphism and covariant quantum channels. arXiv:1906.11442, 2019.

[38] Xiao Yuan, Yunchao Liu, Qi Zhao, Bartosz Regula, Jayne Thompson, and Mile Gu. Universal and operational benchmarking of quantum memories. arXiv:1907.02521, 2019.

[39] Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Distinguishing separable and entangled states. Phys. Rev. Lett., 88:187904, 2002.

[40] Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. Complete family of separability criteria. Phys. Rev. A, 69:022308, 2004.

[41] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A, 223:1, 1996.

[42] Kai Sun, Xiang-Jun Ye, Ya Xiao, Xiao-Ye Xu, Yu-Chun Wu, Jin-Shi Xu, Jing-Ling Chen, Chuan-Feng Li, and Guang-Can Guo. Demonstration of Einstein–Podolsky–Rosen steering with enhanced subchannel discrimination. npj Quantum Information, 4:12, 2018.

[43] Wenqiang Zheng, Zhihao Ma, Hengyan Wang, Shao-Ming Fei, and Xinhua Peng. Experimental demonstration of observability and operability of robustness of coherence. Phys. Rev. Lett., 120:230504, 2018.

[44] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge University Press, Cambridge, 2004.

[45] Bernd Gärtner and Jiří Matoušek. Approximation Algorithms and Semidefinite Programming. Springer Verlag, Berlin, Heidelberg, 2012.

[46] Eric A Carlen, Joel L Lebowitz, and Elliott H Lieb. On an extension problem for density matrices. J. Math. Phys., 54:062103, 2013.

[47] Elliott Lieb and Mary Beth Ruskai. Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys., 14:1938, 1973.

[48] Huzihiro Araki and Elliott H. Lieb. Entropy inequalities. In Inequalities. Springer, Berlin, Heidelberg, 2002.

[49] J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O'Brien, M. A. Nielsen, and A. G. White. Ancilla-assisted quantum process tomography. Phys. Rev. Lett., 90:193601, 2003.

[50] Erkka Haapasalo. Compatibility of covariant quantum channels with emphasis on Weyl symmetry. Ann. Henri Poincaré, 20:3163, 2019.

[51] Jianxin Chen, Zhengfeng Ji, David Kribs, Norbert Lütkenhaus, and Bei Zeng. Symmetric extension of two-qubit states. Phys. Rev. A, 90:032318, 2014.

[52] Connor Paddock and Jianxin Chen. A characterization of antidegradable qubit channels. arXiv:1712.03399, 2017.

[53] Ludovico Lami, Sumeet Khatri, Gerardo Adesso, and Mark M. Wilde. Extendibility of bosonic gaussian states. Phys. Rev. Lett., 123:050501, 2019.

[54] Chung-Yun Hsieh, Matteo Lostaglio, Antonio Acín. Quantum channel marginal problem. arXiv:2102.10926, 2021.

[55] Ryuji Takagi, Bartosz Regula, Kaifeng Bu, Zi-Wen Liu, and Gerardo Adesso. Operational advantage of quantum resources in subchannel discrimination. Phys. Rev. Lett., 122:140402, 2019.

[56] Ryuji Takagi and Bartosz Regula. General resource theories in quantum mechanics and beyond: operational characterization via discrimination tasks. Phys. Rev. X, 9:031053, 2019.

[57] Tobias Fritz and Rafael Chaves. Entropic inequalities and marginal problems. IEEE Trans. Inf. Theory, 59:803, 2013.

[58] C. D. Cushen and R. L. Hudson. A quantum-mechanical central limit theorem. J. Appl. Probab., 8:454, 1971.

[59] Alexander S. Holevo and Reinhardt F. Werner. Evaluating capacities of bosonic gaussian channels. Phys. Rev. A, 63:032312, 2001.

[60] C.W. Gardiner and P. Zoller. Quantum Noise, A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, 3rd Edition, Springer Verlag, Berlin Heidelberg, 2004.

Cited by

[1] Martin Plávala, "General probabilistic theories: An introduction", arXiv:2103.07469.

[2] Mark Girard, Martin Plávala, and Jamie Sikora, "Jordan products of quantum channels and their compatibility", Nature Communications 12, 2129 (2021).

The above citations are from SAO/NASA ADS (last updated successfully 2021-08-04 07:28:37). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2021-08-04 07:28:35).