Quantum Wasserstein distance based on an optimization over separable states

Géza Tóth1,2,3,4,5 and József Pitrik5,6,7

1Theoretical Physics, University of the Basque Country UPV/EHU, ES-48080 Bilbao, Spain
2EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, ES-48940 Leioa, Biscay, Spain
3Donostia International Physics Center (DIPC), ES-20080 San Sebastián, Spain
4IKERBASQUE, Basque Foundation for Science, ES-48011 Bilbao, Spain
5Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, HU-1525 Budapest, Hungary
6Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15., HU-1053 Budapest, Hungary
7Department of Analysis and Operations Research, Institute of Mathematics, Budapest University of Technology and Economics, Müegyetem rkp. 3., HU-1111 Budapest, Hungary

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Abstract

We define the quantum Wasserstein distance such that the optimization of the coupling is carried out over bipartite separable states rather than bipartite quantum states in general, and examine its properties. Surprisingly, we find that the self-distance is related to the quantum Fisher information. We present a transport map corresponding to an optimal bipartite separable state. We discuss how the quantum Wasserstein distance introduced is connected to criteria detecting quantum entanglement. We define variance-like quantities that can be obtained from the quantum Wasserstein distance by replacing the minimization over quantum states by a maximization. We extend our results to a family of generalized quantum Fisher information quantities.

In the everyday life, the distance of two cities tells us how many kilometers we have to drive from one to the other. It is also possible to characterize how easily we can get from one city to the other is to measure the fuel consumption during our journey. The latter is more informative in the sense that it reflects the cost of the travel related to the topography of the road, i. e., it is sensitive to the underlying metric. Next, let us imagine that we need to move a heap of sand from one place to another one and the new heap might have a different shape. In this case, again, we can characterize the effort of moving the sand by the cost of the transport.

Distances play a central role in mathematics, physics and engineering. A fundamental problem in probability and statistics is to come up with useful measures of distance between two probability distributions. Unfortunately, many notions of distance between probability distributions, say p(x) and q(x), are maximal if they do not overlap with each other, i. e., one is always zero when the other is non-zero. This is impractical for many applications. For instance, returning to the sand analogy, two non-overlapping piles of sand seem to be equally far from each other, regardless whether their distance is 10km or 100km. Optimal transport theory is a way to construct an alternative notion of distance between probability distributions, the so-called Wasserstein distance. It can be non-maximal even if the distributions do not overlap with each other, it is sensitive to the underlying metric (i.e., the cost of the transport), and essentially, it expresses the effort we need to move one to the other, as if they were sand piles.

Recently, the quantum Wasserstein distance has been defined generalizing the classical Wasserstein distance. It is based on the minimization of a cost function over the quantum states of a bipartite quantum system. It has the property analogous to the one mentioned above in the quantum world. It can be non-maximal for orthogonal states, which is useful, for instance, when we need to teach quantum data to an algorithm.

As we can expect, quantum Wasserstein distance also has properties that are very different from those of its classical counterpart. For instance, when we measure the distance of a quantum state from itself, it can be nonzero. While this is already puzzling, it has also been found that the self-distance is related to the Wigner-Yanase skew information, introduced in 1963 by the Nobel laureate E. P. Wigner, who has vital contributions to the foundations of quantum physics and M. M. Yanase.

In our paper, we look at this mysterious finding from yet another direction. We restrict the minimization mentioned above to so-called separable states. These are the quantum states that do not contain entanglement. We find that the self-distance becomes the quantum Fisher information, a quantity central in quantum metrology and quantum estimation theory, and appearing for instance in the famous Cramer-Rao bound. By examining the properties of such a Wasserstein distance, our work paves the way to connect the theory of quantum Wasserstein distance to the theory of quantum entanglement.

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► References

[1] G. Monge. ``Mémoire sur la théory des déblais et des remblais''. Mémoires de l'Académie Royale de Sciences de Paris (1781).

[2] L. Kantorovitch. ``On the translocation of masses''. Management Science 5, 1–4 (1958). url: http:/​/​www.jstor.org/​stable/​2626967.
http:/​/​www.jstor.org/​stable/​2626967

[3] Emmanuel Boissard, Thibaut Le Gouic, and Jean-Michel Loubes. ``Distribution's template estimate with wasserstein metrics''. Bernoulli 21, 740–759 (2015).
https:/​/​doi.org/​10.3150/​13-bej585

[4] Oleg Butkovsky. ``Subgeometric rates of convergence of Markov processes in the Wasserstein metric''. Ann. Appl. Probab. 24, 526–552 (2014).
https:/​/​doi.org/​10.1214/​13-AAP922

[5] M. Hairer, J.-C. Mattingly and M. Scheutzow. ``Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations''. Probab. Theory Relat. Fields 149, 223–259 (2011).
https:/​/​doi.org/​10.1007/​s00440-009-0250-6

[6] M. Hairer and J.C. Mattingly. ``Spectral Gaps in Wasserstein Distances and the 2D Stochastic Navier-Stokes Equations''. Ann. Probab. 36, 2050–2091 (2008).
https:/​/​doi.org/​10.1214/​08-AOP392

[7] A. Figalli, F. Maggi and A. Pratelli. ``A mass transportation approach to quantitative isoperimetric inequalities''. Invent. Math. 182, 167–211. (2010).
https:/​/​doi.org/​10.1007/​s00222-010-0261-z

[8] A. Figalli and F. Maggi. ``On the shape of liquid drops and crystals in the small mass regime''. Arch. Ration. Mech. Anal. 201, 143–207 (2011).
https:/​/​doi.org/​10.1007/​s00205-010-0383-x

[9] J. Lott and C. Villani. ``Ricci curvature for metric-measure spaces via optimal transport''. Ann. of Math. 169 (3), 903–991 (2009).
https:/​/​doi.org/​10.48550/​arXiv.math/​0412127

[10] Max-K. von Renesse and Karl-Theodor Sturm. ``Transport inequalities, gradient estimates, entropy, and Ricci curvature''. Comm. Pure Appl. Math. 58, 923–940 (2005).
https:/​/​doi.org/​10.1002/​cpa.20060

[11] Karl-Theodor Sturm. ``On the geometry of metric measure spaces I''. Acta Math. 196, 65–131 (2006).
https:/​/​doi.org/​10.1007/​s11511-006-0002-8

[12] Karl-Theodor Sturm. ``On the geometry of metric measure spaces II''. Acta Math. 196, 133–177 (2006).
https:/​/​doi.org/​10.1007/​s11511-006-0003-7

[13] Benoı̂t Kloeckner. ``A geometric study of Wasserstein spaces: Euclidean spaces''. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Scuola Normale Superiore 2010 IX (2), 297–323 (2010).
https:/​/​doi.org/​10.2422/​2036-2145.2010.2.03

[14] György Pál Gehér, Tamás Titkos, and Dániel Virosztek. ``On isometric embeddings of wasserstein spaces – the discrete case''. J. Math. Anal. Appl. 480, 123435 (2019).
https:/​/​doi.org/​10.1016/​j.jmaa.2019.123435

[15] György Pál Gehér, T. Titkos, Dániel Virosztek. ``Isometric study of Wasserstein spaces – the real line''. Trans. Amer. Math. Soc. 373, 5855–5883 (2020).
https:/​/​doi.org/​10.1090/​tran/​8113

[16] György Pál Gehér, Tamás Titkos, and Dániel Virosztek. ``The isometry group of Wasserstein spaces: the Hilbertian case''. J. Lond. Math. Soc. 106, 3865–3894 (2022).
https:/​/​doi.org/​10.1112/​jlms.12676

[17] György Pál Gehér, Tamás Titkos, and Dániel Virosztek. ``Isometric rigidity of wasserstein tori and spheres''. Mathematika 69, 20–32 (2023).
https:/​/​doi.org/​10.1112/​mtk.12174

[18] Gergely Kiss and Tamás Titkos. ``Isometric rigidity of wasserstein spaces: The graph metric case''. Proc. Am. Math. Soc. 150, 4083–4097 (2022).
https:/​/​doi.org/​10.1090/​proc/​15977

[19] György Pál Gehér, Tamás Titkos, and Dániel Virosztek. ``On the exotic isometry flow of the quadratic wasserstein space over the real line''. Linear Algebra Appl. (2023).
https:/​/​doi.org/​10.1016/​j.laa.2023.02.016

[20] S. Kolouri, S. R. Park and G. K. Rohde. ``The Radon cumulative distribution transform and its application to image classification''. IEEE Trans. Image Process. 25, 920–934 (2016).
https:/​/​doi.org/​10.1109/​TIP.2015.2509419

[21] W. Wang, D. Slepc̆ev, S. Basu, J. A. Ozolek and G. K. Rohde. ``A linear optimal transportation framework for quantifying and visualizing variations in sets of images''. Int. J. Comput. Vis. 101, 254–269 (2013).
https:/​/​doi.org/​10.1007/​s11263-012-0566-z

[22] S. Kolouri, S. Park, M. Thorpe, D. Slepc̆ev, G. K. Rohde. ``Optimal Mass Transport: Signal processing and machine-learning applications''. IEEE Signal Processing Magazine 34, 43–59 (2017).
https:/​/​doi.org/​10.1109/​MSP.2017.2695801

[23] A. Gramfort, G. Peyré and M. Cuturi. ``Fast Optimal Transport Averaging of Neuroimaging Data''. Information Processing in Medical Imaging. IPMI 2015. Lecture Notes in Computer Science 9123, 261–272 (2015).
https:/​/​doi.org/​10.1007/​978-3-319-19992-4_20

[24] Z. Su, W. Zeng, Y. Wang, Z. L. Lu and X. Gu. ``Shape classification using Wasserstein distance for brain morphometry analysis''. Information Processing in Medical Imaging. IPMI 2015. Lecture Notes in Computer Science 24, 411–423 (2015).
https:/​/​doi.org/​10.1007/​978-3-319-19992-4_32

[25] Martin Arjovsky, Soumith Chintala, and Léon Bottou. ``Wasserstein generative adversarial networks''. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning. Volume 70 of Proceedings of Machine Learning Research, pages 214–223. PMLR (2017). arXiv:1701.07875.
arXiv:1701.07875

[26] T. A. El Moselhy and Y. M. Marzouk. ``Bayesian inference with optimal maps''. J. Comput. Phys. 231, 7815–7850 (2012).
https:/​/​doi.org/​10.1016/​j.jcp.2012.07.022

[27] Gabriel Peyré and Marco Cuturi. ``Computational Optimal Transport: With Applications to Data Science''. Found. Trends Machine Learn. 11, 355–602 (2019).
https:/​/​doi.org/​10.1561/​2200000073

[28] Charlie Frogner, Chiyuan Zhang, Hossein Mobahi, Mauricio Araya, and Tomaso A Poggio. ``Learning with a wasserstein loss''. In C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, editors, Advances in Neural Information Processing Systems. Volume 28. Curran Associates, Inc. (2015). arXiv:1506.05439.
arXiv:1506.05439

[29] A. Ramdas, N. G. Trillos and M. Cuturi. ``On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests''. Entropy 19, 47. (2017).
https:/​/​doi.org/​10.3390/​e19020047

[30] S. Srivastava, C. Li and D. B. Dunson. ``Scalable Bayes via Barycenter in Wasserstein Space''. J. Mach. Learn. Res. 19, 1–35 (2018). arXiv:1508.05880.
arXiv:1508.05880

[31] Karol Życzkowski and Wojeciech Slomczynski. ``The Monge distance between quantum states''. J. Phys. A: Math. Gen. 31, 9095–9104 (1998).
https:/​/​doi.org/​10.1088/​0305-4470/​31/​45/​009

[32] Karol Życzkowski and Wojciech Slomczynski. ``The Monge metric on the sphere and geometry of quantum states''. J. Phys. A: Math. Gen. 34, 6689–6722 (2001).
https:/​/​doi.org/​10.1088/​0305-4470/​34/​34/​311

[33] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states: An introduction to quantum entanglement''. Cambridge University Press. (2006).
https:/​/​doi.org/​10.1017/​CBO9780511535048

[34] P. Biane and D. Voiculescu. ``A free probability analogue of the Wasserstein metric on the trace-state space''. GAFA, Geom. Funct. Anal. 11, 1125–1138 (2001).
https:/​/​doi.org/​10.1007/​s00039-001-8226-4

[35] Eric A. Carlen and Jan Maas. ``An Analog of the 2-Wasserstein Metric in Non-Commutative Probability Under Which the Fermionic Fokker-Planck Equation is Gradient Flow for the Entropy''. Commun. Math. Phys. 331, 887–926 (2014).
https:/​/​doi.org/​10.1007/​s00220-014-2124-8

[36] Eric A. Carlen and Jan Maas. ``Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance''. J. Funct. Anal. 273, 1810–1869 (2017).
https:/​/​doi.org/​10.1016/​j.jfa.2017.05.003

[37] Eric A. Carlen and Jan Maas. ``Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems''. J. Stat. Phys. 178, 319–378 (2020).
https:/​/​doi.org/​10.1007/​s10955-019-02434-w

[38] Nilanjana Datta and Cambyse Rouzé. ``Concentration of quantum states from quantum functional and transportation cost inequalities''. J. Math. Phys. 60, 012202 (2019).
https:/​/​doi.org/​10.1063/​1.5023210

[39] Nilanjana Datta and Cambyse Rouzé. ``Relating relative entropy, optimal transport and Fisher information: A quantum HWI inequality''. Ann. Henri Poincaré 21, 2115–2150 (2020).
https:/​/​doi.org/​10.1007/​s00023-020-00891-8

[40] François Golse, Clément Mouhot, and Thierry Paul. ``On the mean field and classical limits of quantum mechanics''. Commun. Math. Phys. 343, 165–205 (2016).
https:/​/​doi.org/​10.1007/​s00220-015-2485-7

[41] François Golse and Thierry Paul. ``The Schrödinger equation in the mean-field and semiclassical regime''. Arch. Ration. Mech. Anal. 223, 57–94 (2017).
https:/​/​doi.org/​10.1007/​s00205-016-1031-x

[42] François Golse and Thierry Paul. ``Wave packets and the quadratic Monge-Kantorovich distance in quantum mechanics''. Comptes Rendus Math. 356, 177–197 (2018).
https:/​/​doi.org/​10.1016/​j.crma.2017.12.007

[43] François Golse. ``The quantum $N$-body problem in the mean-field and semiclassical regime''. Phil. Trans. R. Soc. A 376, 20170229 (2018).
https:/​/​doi.org/​10.1098/​rsta.2017.0229

[44] E. Caglioti, F. Golse, and T. Paul. ``Quantum optimal transport is cheaper''. J. Stat. Phys. 181, 149–162 (2020).
https:/​/​doi.org/​10.1007/​s10955-020-02571-7

[45] Emanuele Caglioti, François Golse, and Thierry Paul. ``Towards optimal transport for quantum densities''. arXiv:2101.03256 (2021).
https:/​/​doi.org/​10.48550/​arXiv.2101.03256
arXiv:2101.03256

[46] Giacomo De Palma and Dario Trevisan. ``Quantum optimal transport with quantum channels''. Ann. Henri Poincaré 22, 3199–3234 (2021).
https:/​/​doi.org/​10.1007/​s00023-021-01042-3

[47] Giacomo De Palma, Milad Marvian, Dario Trevisan, and Seth Lloyd. ``The quantum Wasserstein distance of order 1''. IEEE Trans. Inf. Theory 67, 6627–6643 (2021).
https:/​/​doi.org/​10.1109/​TIT.2021.3076442

[48] Shmuel Friedland, Michał Eckstein, Sam Cole, and Karol Życzkowski. ``Quantum Monge–Kantorovich problem and transport distance between density matrices''. Phys. Rev. Lett. 129, 110402 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.129.110402

[49] Sam Cole, Michał Eckstein, Shmuel Friedland, and Karol Życzkowski. ``Quantum optimal transport''. arXiv:2105.06922 (2021).
https:/​/​doi.org/​10.48550/​arXiv.2105.06922
arXiv:2105.06922

[50] R. Bistroń, M. Eckstein, and K. Życzkowski. ``Monotonicity of a quantum 2-Wasserstein distance''. J. Phys. A: Math. Theor. 56, 095301 (2023).
https:/​/​doi.org/​10.1088/​1751-8121/​acb9c8

[51] György Pál Gehér, József Pitrik, Tamás Titkos, and Dániel Virosztek. ``Quantum Wasserstein isometries on the qubit state space''. J. Math. Anal. Appl. 522, 126955 (2023).
https:/​/​doi.org/​10.1016/​j.jmaa.2022.126955

[52] Lu Li, Kaifeng Bu, Dax Enshan Koh, Arthur Jaffe, and Seth Lloyd. ``Wasserstein complexity of quantum circuits''. arXiv: 2208.06306 (2022).
https:/​/​doi.org/​10.48550/​arXiv.2208.06306

[53] Bobak Toussi Kiani, Giacomo De Palma, Milad Marvian, Zi-Wen Liu, and Seth Lloyd. ``Learning quantum data with the quantum earth mover's distance''. Quantum Sci. Technol. 7, 045002 (2022).
https:/​/​doi.org/​10.1088/​2058-9565/​ac79c9

[54] E. P. Wigner and Mutsuo M. Yanase. ``Information contents of distributions''. Proc. Natl. Acad. Sci. U.S.A. 49, 910–918 (1963).
https:/​/​doi.org/​10.1073/​pnas.49.6.910

[55] 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

[56] Otfried Gühne and Géza Tóth. ``Entanglement detection''. Phys. Rep. 474, 1–75 (2009).
https:/​/​doi.org/​10.1016/​j.physrep.2009.02.004

[57] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber. ``Entanglement certification from theory to experiment''. Nat. Rev. Phys. 1, 72–87 (2019).
https:/​/​doi.org/​10.1038/​s42254-018-0003-5

[58] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum-enhanced measurements: Beating the standard quantum limit''. Science 306, 1330–1336 (2004).
https:/​/​doi.org/​10.1126/​science.1104149

[59] Matteo G. A. Paris. ``Quantum estimation for quantum technology''. Int. J. Quant. Inf. 07, 125–137 (2009).
https:/​/​doi.org/​10.1142/​S0219749909004839

[60] Rafal Demkowicz-Dobrzanski, Marcin Jarzyna, and Jan Kolodynski. ``Chapter four - Quantum limits in optical interferometry''. Prog. Optics 60, 345 – 435 (2015). arXiv:1405.7703.
https:/​/​doi.org/​10.1016/​bs.po.2015.02.003
arXiv:1405.7703

[61] Luca Pezze and Augusto Smerzi. ``Quantum theory of phase estimation''. In G.M. Tino and M.A. Kasevich, editors, Atom Interferometry (Proc. Int. School of Physics 'Enrico Fermi', Course 188, Varenna). Pages 691–741. IOS Press, Amsterdam (2014). arXiv:1411.5164.
arXiv:1411.5164

[62] Géza Tóth and Dénes Petz. ``Extremal properties of the variance and the quantum Fisher information''. Phys. Rev. A 87, 032324 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.032324

[63] Sixia Yu. ``Quantum Fisher Information as the Convex Roof of Variance''. arXiv:1302.5311 (2013).
https:/​/​doi.org/​10.48550/​arXiv.1302.5311
arXiv:1302.5311

[64] Géza Tóth and Florian Fröwis. ``Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices''. Phys. Rev. Research 4, 013075 (2022).
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.013075

[65] Shao-Hen Chiew and Manuel Gessner. ``Improving sum uncertainty relations with the quantum Fisher information''. Phys. Rev. Research 4, 013076 (2022).
https:/​/​doi.org/​10.1103/​PhysRevResearch.4.013076

[66] C. W. Helstrom. ``Quantum detection and estimation theory''. Academic Press, New York. (1976). url: www.elsevier.com/​books/​quantum-detection-and-estimation-theory/​helstrom/​978-0-12-340050-5.
https:/​/​www.elsevier.com/​books/​quantum-detection-and-estimation-theory/​helstrom/​978-0-12-340050-5

[67] A. S. Holevo. ``Probabilistic and statistical aspects of quantum theory''. North-Holland, Amsterdam. (1982).

[68] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Phys. Rev. Lett. 72, 3439–3443 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.72.3439

[69] Samuel L Braunstein, Carlton M Caves, and Gerard J Milburn. ``Generalized uncertainty relations: Theory, examples, and Lorentz invariance''. Ann. Phys. 247, 135–173 (1996).
https:/​/​doi.org/​10.1006/​aphy.1996.0040

[70] Dénes Petz. ``Quantum information theory and quantum statistics''. Springer, Berlin, Heilderberg. (2008).
https:/​/​doi.org/​10.1007/​978-3-540-74636-2

[71] Géza Tóth and Iagoba Apellaniz. ``Quantum metrology from a quantum information science perspective''. J. Phys. A: Math. Theor. 47, 424006 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424006

[72] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018).
https:/​/​doi.org/​10.1103/​RevModPhys.90.035005

[73] Marco Barbieri. ``Optical quantum metrology''. PRX Quantum 3, 010202 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010202

[74] Zoltán Léka and Dénes Petz. ``Some decompositions of matrix variances''. Probab. Math. Statist. 33, 191–199 (2013). arXiv:1408.2707.
arXiv:1408.2707

[75] Dénes Petz and Dániel Virosztek. ``A characterization theorem for matrix variances''. Acta Sci. Math. (Szeged) 80, 681–687 (2014).
https:/​/​doi.org/​10.14232/​actasm-013-789-z

[76] Akio Fujiwara and Hiroshi Imai. ``A fibre bundle over manifolds of quantum channels and its application to quantum statistics''. J. Phys. A: Math. Theor. 41, 255304 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​25/​255304

[77] B. M. Escher, R. L. de Matos Filho, and L. Davidovich. ``General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology''. Nat. Phys. 7, 406–411 (2011).
https:/​/​doi.org/​10.1038/​nphys1958

[78] Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin Guţă. ``The elusive Heisenberg limit in quantum-enhanced metrology''. Nat. Commun. 3, 1063 (2012).
https:/​/​doi.org/​10.1038/​ncomms2067

[79] Iman Marvian. ``Operational interpretation of quantum fisher information in quantum thermodynamics''. Phys. Rev. Lett. 129, 190502 (2022).
https:/​/​doi.org/​10.1103/​PhysRevLett.129.190502

[80] Reinhard F. Werner. ``Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model''. Phys. Rev. A 40, 4277–4281 (1989).
https:/​/​doi.org/​10.1103/​PhysRevA.40.4277

[81] K. Eckert, J. Schliemann, D. Bruss, and M. Lewenstein. ``Quantum correlations in systems of indistinguishable particles''. Ann. Phys. 299, 88–127 (2002).
https:/​/​doi.org/​10.1006/​aphy.2002.6268

[82] Tsubasa Ichikawa, Toshihiko Sasaki, Izumi Tsutsui, and Nobuhiro Yonezawa. ``Exchange symmetry and multipartite entanglement''. Phys. Rev. A 78, 052105 (2008).
https:/​/​doi.org/​10.1103/​PhysRevA.78.052105

[83] Pawel Horodecki. ``Separability criterion and inseparable mixed states with positive partial transposition''. Phys. Lett. A 232, 333–339 (1997).
https:/​/​doi.org/​10.1016/​S0375-9601(97)00416-7

[84] Asher Peres. ``Separability criterion for density matrices''. Phys. Rev. Lett. 77, 1413–1415 (1996).
https:/​/​doi.org/​10.1103/​PhysRevLett.77.1413

[85] Paweł Horodecki, Michał Horodecki, and Ryszard Horodecki. ``Bound entanglement can be activated''. Phys. Rev. Lett. 82, 1056–1059 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.82.1056

[86] Géza Tóth and Tamás Vértesi. ``Quantum states with a positive partial transpose are useful for metrology''. Phys. Rev. Lett. 120, 020506 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.020506

[87] Scott Hill and William K. Wootters. ``Entanglement of a pair of quantum bits''. Phys. Rev. Lett. 78, 5022–5025 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.5022

[88] William K. Wootters. ``Entanglement of formation of an arbitrary state of two qubits''. Phys. Rev. Lett. 80, 2245–2248 (1998).
https:/​/​doi.org/​10.1103/​PhysRevLett.80.2245

[89] David P. DiVincenzo, Christopher A. Fuchs, Hideo Mabuchi, John A. Smolin, Ashish Thapliyal, and Armin Uhlmann. ``Entanglement of assistance''. quant-ph/​9803033 (1998).
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​9803033
arXiv:quant-ph/9803033

[90] John A. Smolin, Frank Verstraete, and Andreas Winter. ``Entanglement of assistance and multipartite state distillation''. Phys. Rev. A 72, 052317 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.052317

[91] Holger F. Hofmann and Shigeki Takeuchi. ``Violation of local uncertainty relations as a signature of entanglement''. Phys. Rev. A 68, 032103 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.68.032103

[92] Otfried Gühne. ``Characterizing entanglement via uncertainty relations''. Phys. Rev. Lett. 92, 117903 (2004).
https:/​/​doi.org/​10.1103/​PhysRevLett.92.117903

[93] Otfried Gühne, Mátyás Mechler, Géza Tóth, and Peter Adam. ``Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion''. Phys. Rev. A 74, 010301 (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.74.010301

[94] Giuseppe Vitagliano, Philipp Hyllus, Iñigo L. Egusquiza, and Géza Tóth. ``Spin squeezing inequalities for arbitrary spin''. Phys. Rev. Lett. 107, 240502 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.240502

[95] A. R. Edmonds. ``Angular momentum in quantum mechanics''. Princeton University Press. (1957).
https:/​/​doi.org/​10.1515/​9781400884186

[96] Géza Tóth. ``Entanglement detection in optical lattices of bosonic atoms with collective measurements''. Phys. Rev. A 69, 052327 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.052327

[97] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. ``Optimal spin squeezing inequalities detect bound entanglement in spin models''. Phys. Rev. Lett. 99, 250405 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.99.250405

[98] Géza Tóth and Morgan W Mitchell. ``Generation of macroscopic singlet states in atomic ensembles''. New J. Phys. 12, 053007 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​5/​053007

[99] Géza Tóth. ``Detection of multipartite entanglement in the vicinity of symmetric Dicke states''. J. Opt. Soc. Am. B 24, 275–282 (2007).
https:/​/​doi.org/​10.1364/​JOSAB.24.000275

[100] Géza Tóth, Tobias Moroder, and Otfried Gühne. ``Evaluating convex roof entanglement measures''. Phys. Rev. Lett. 114, 160501 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.160501

[101] Lieven Vandenberghe and Stephen Boyd. ``Semidefinite programming''. SIAM Review 38, 49–95 (1996).
https:/​/​doi.org/​10.1137/​1038003

[102] Géza Tóth. ``Multipartite entanglement and high-precision metrology''. Phys. Rev. A 85, 022322 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.022322

[103] Philipp Hyllus, Wiesław Laskowski, Roland Krischek, Christian Schwemmer, Witlef Wieczorek, Harald Weinfurter, Luca Pezzé, and Augusto Smerzi. ``Fisher information and multiparticle entanglement''. Phys. Rev. A 85, 022321 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.022321

[104] Géza Tóth, Tamás Vértesi, Paweł Horodecki, and Ryszard Horodecki. ``Activating hidden metrological usefulness''. Phys. Rev. Lett. 125, 020402 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.020402

[105] A. C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. ``Distinguishing separable and entangled states''. Phys. Rev. Lett. 88, 187904 (2002).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.187904

[106] Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. ``Complete family of separability criteria''. Phys. Rev. A 69, 022308 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.022308

[107] Andrew C. Doherty, Pablo A. Parrilo, and Federico M. Spedalieri. ``Detecting multipartite entanglement''. Phys. Rev. A 71, 032333 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.032333

[108] Harold Ollivier and Wojciech H. Zurek. ``Quantum discord: A measure of the quantumness of correlations''. Phys. Rev. Lett. 88, 017901 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.88.017901

[109] L. Henderson and V. Vedral. ``Classical, quantum and total correlations''. J. Phys. A: Math. Gen. 34, 6899 (2001).
https:/​/​doi.org/​10.1088/​0305-4470/​34/​35/​315

[110] Anindita Bera, Tamoghna Das, Debasis Sadhukhan, Sudipto Singha Roy, Aditi Sen(De), and Ujjwal Sen. ``Quantum discord and its allies: a review of recent progress''. Rep. Prog. Phys. 81, 024001 (2017).
https:/​/​doi.org/​10.1088/​1361-6633/​aa872f

[111] Dénes Petz. ``Covariance and Fisher information in quantum mechanics''. J. Phys. A: Math. Gen. 35, 929 (2002).
https:/​/​doi.org/​10.1088/​0305-4470/​35/​4/​305

[112] Paolo Gibilisco, Fumio Hiai, and Dénes Petz. ``Quantum covariance, quantum Fisher information, and the uncertainty relations''. IEEE Trans. Inf. Theory 55, 439–443 (2009).
https:/​/​doi.org/​10.1109/​TIT.2008.2008142

[113] D. Petz and C. Ghinea. ``Introduction to quantum Fisher information''. Volume 27, pages 261–281. World Scientific. (2011).
https:/​/​doi.org/​10.1142/​9789814338745_0015

[114] Frank Hansen. ``Metric adjusted skew information''. Proc. Natl. Acad. Sci. U.S.A. 105, 9909–9916 (2008).
https:/​/​doi.org/​10.1073/​pnas.0803323105

[115] Paolo Gibilisco, Davide Girolami, and Frank Hansen. ``A unified approach to local quantum uncertainty and interferometric power by metric adjusted skew information''. Entropy 23, 263 (2021).
https:/​/​doi.org/​10.3390/​e23030263

[116] MATLAB. ``9.9.0.1524771(r2020b)''. The MathWorks Inc. Natick, Massachusetts (2020).

[117] MOSEK ApS. ``The MOSEK optimization toolbox for MATLAB manual. Version 9.0''. (2019). url: docs.mosek.com/​9.0/​toolbox/​index.html.
https:/​/​docs.mosek.com/​9.0/​toolbox/​index.html

[118] J. Löfberg. ``YALMIP : A Toolbox for Modeling and Optimization in MATLAB''. In Proceedings of the CACSD Conference. Taipei, Taiwan (2004).

[119] Géza Tóth. ``QUBIT4MATLAB V3.0: A program package for quantum information science and quantum optics for MATLAB''. Comput. Phys. Commun. 179, 430–437 (2008).
https:/​/​doi.org/​10.1016/​j.cpc.2008.03.007

[120] The package QUBIT4MATLAB is available at https:/​/​www.mathworks.com/​matlabcentral/​ fileexchange/​8433, and at the personal home page https:/​/​gtoth.eu/​qubit4matlab.html.
https:/​/​www.mathworks.com/​matlabcentral/​fileexchange/​8433

Cited by

[1] Guillem Müller-Rigat, Anubhav Kumar Srivastava, Stanisław Kurdziałek, Grzegorz Rajchel-Mieldzioć, Maciej Lewenstein, and Irénée Frérot, "Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach", Quantum 7, 1152 (2023).

[2] Gergely Bunth, József Pitrik, Tamás Titkos, and Dániel Virosztek, "On the metric property of quantum Wasserstein divergences", arXiv:2402.13150, (2024).

[3] Laurent Lafleche, "Quantum Optimal Transport and Weak Topologies", arXiv:2306.12944, (2023).

[4] Khoi-Nguyen Huynh-Vu, Lin Htoo Zaw, and Valerio Scarani, "Certification of genuine multipartite entanglement in spin ensembles with measurements of total angular momentum", Physical Review A 109 4, 042402 (2024).

[5] Xian Shi, "A coherence quantifier based on the quantum optimal transport cost", arXiv:2311.07852, (2023).

[6] Soumyabrata Paul, S. Ramanan, V. Balakrishnan, and S. Lakshmibala, "Wasserstein distance and entropic divergences between quantum states of light", arXiv:2401.16098, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 09:04:22) and SAO/NASA ADS (last updated successfully 2024-05-21 09:04:23). The list may be incomplete as not all publishers provide suitable and complete citation data.