Hierarchy of correlation quantifiers comparable to negativity

Ray Ganardi; Marek Miller; Tomasz Paterek; Marek Żukowski

doi:10.22331/q-2022-02-16-654

Hierarchy of correlation quantifiers comparable to negativity

Ray Ganardi^1,2, Marek Miller³, Tomasz Paterek¹, and Marek Żukowski²

¹Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland
²International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, 80-308 Gdańsk, Poland
³Centre of New Technologies, University of Warsaw, 02-097 Warsaw, Poland

Published:	2022-02-16, volume 6, page 654
Eprint:	arXiv:2111.11887v2
Doi:	https://doi.org/10.22331/q-2022-02-16-654
Citation:	Quantum 6, 654 (2022).

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

Abstract

Quantum systems generally exhibit different kinds of correlations. In order to compare them on equal footing, one uses the so-called distance-based approach where different types of correlations are captured by the distance to different sets of states. However, these quantifiers are usually hard to compute as their definition involves optimization aiming to find the closest states within the set. On the other hand, negativity is one of the few computable entanglement monotones, but its comparison with other correlations required further justification. Here we place negativity as part of a family of correlation measures that has a distance-based construction. We introduce a suitable distance, discuss the emerging measures and their applications, and compare them to relative entropy-based correlation quantifiers. This work is a step towards correlation measures that are simultaneously comparable and computable.

Featured image: Geometrical illustration. Negativity of state $\rho$ is the trace distance from its partial transposition to the set of states. The closest state is known and given as $\sigma_N$ in the figure. The main claim of the paper is negativity can be computed by taking the partial transpose distance from $\rho$ to the set of PPT states (in the shadowed overlap of states and partially transposed states). This is proven when partial transposition of $\sigma_N$ is positive semi-definite, i.e. a state. In general, the closest PPT state in the partial transpose distance is unknown as depicted with the question mark. We emphasise that this is just an illustration as partial transposition of $\sigma_N$ can produce matrices with negative eigenvalues, i.e. outside the set of states.

Popular summary

Classical bits (bits in our present computers) can be correlated in only one way because there is only one question one can ask a bit: "Are you zero or one?" The correlations are present if the value of one bit tells something new about a value of another bit and these are therefore called "classical correlations." Quite differently, quantum bits can be asked infinitely many more questions and can be prepared in states which give a deterministic, non-random, yes answer to just ONE of them. This leads to many different types of correlations between quantum objects, including entanglement or total correlations in addition to the classical correlations. Currently, they are measured in many ways that are not necessarily comparable. This is because the definition of the measure is often tied to a specific task. For example, negativity shows that in case of some quantum states, a change of sign of one of the parameters that mathematically describe them leads to a prediction of nonsensical negative probabilities. The special thing about negativity is that, unlike many other entanglement measures, we can compute it for any quantum state. This makes negativity a popular choice in practical situations. In this work, we develop a way to measure different types of correlations that are comparable to negativity. This is done as a step towards developing a framework in which different types of correlations can be meaningfully compared and efficiently computed.

► BibTeX data

@article{Ganardi2022hierarchyof,
  doi = {10.22331/q-2022-02-16-654},
  url = {https://doi.org/10.22331/q-2022-02-16-654},
  title = {Hierarchy of correlation quantifiers comparable to negativity},
  author = {Ganardi, Ray and Miller, Marek and Paterek, Tomasz and {\.{Z}}ukowski, Marek},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {654},
  month = feb,
  year = {2022}
}

► References

[1] G. Adesso, T. R. Bromley, and M. Cianciaruso, J. Phys. A: Math. Theor. 49, 473001 (2016).
https://doi.org/10.1088/1751-8113/49/47/473001

[2] K. Jiráková, A. Černoch, K. Lemr, K. Bartkiewicz, and A. Miranowicz, Phys. Rev. A 104, 062436 (2021).
https://doi.org/10.1103/PhysRevA.104.062436

[3] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[4] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. 84, 1655 (2012).
https://doi.org/10.1103/RevModPhys.84.1655

[5] S. Virmani and M. Plenio, Phys. Lett. A 268, 31 (2000).
https://doi.org/10.1016/s0375-9601(00)00157-2

[6] A. Miranowicz and A. Grudka, J. Opt. B: Quantum Semiclass. Opt. 6, 542 (2004).
https://doi.org/10.1088/1464-4266/6/12/009

[7] K. Modi, T. Paterek, W. Son, V. Vedral, and M. Williamson, Phys. Rev. Lett. 104, 080501 (2010).
https://doi.org/10.1103/PhysRevLett.104.080501

[8] V. Vedral, M. Plenio, M. Rippin, and P. Knight, Phys. Rev. Lett. 78, 2275 (1997a).
https://doi.org/10.1103/PhysRevLett.78.2275

[9] E. Chitambar and G. Gour, Rev. Mod. Phys. 91, 025001 (2019).
https://doi.org/10.1103/RevModPhys.91.025001

[10] V. Vedral, M. Plenio, K. Jacobs, and P. Knight, Phys. Rev. A 56, 4452 (1997b).
https://doi.org/10.1103/PhysRevA.56.4452

[11] A. Miranowicz and S. Ishizaka, Phys. Rev. A 78, 032310 (2008).
https://doi.org/10.1103/PhysRevA.78.032310

[12] S. Friedland and G. Gour, J. Math. Phys. 52, 052201 (2011).
https://doi.org/10.1063/1.3591132

[13] K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58, 883 (1998).
https://doi.org/10.1103/PhysRevA.58.883

[14] K. Życzkowski, Phys. Rev. A 60, 3496 (1999).
https://doi.org/10.1103/PhysRevA.60.3496

[15] G. Vidal and R. Werner, Phys. Rev. A 65, 032314 (2002).
https://doi.org/10.1103/PhysRevA.65.032314

[16] M. Plenio, Phys. Rev. Lett. 95, 090503 (2005).
https://doi.org/10.1103/PhysRevLett.95.090503

[17] J. Lee, M. Kim, Y. Park, and S. Lee, J. Mod. Optic. 47, 2151 (2000).
https://doi.org/10.1080/09500340008235138

[18] K. Audenaert, M. Plenio, and J. Eisert, Phys. Rev. Lett. 90, 027901 (2003).
https://doi.org/10.1103/PhysRevLett.90.027901

[19] S. Ishizaka, Phys. Rev. A 69, 020301 (2004).
https://doi.org/10.1103/PhysRevA.69.020301

[20] C. Eltschka and J. Siewert, Phys. Rev. Lett. 111, 100503 (2013).
https://doi.org/10.1103/PhysRevLett.111.100503

[21] T. R. Bromley, M. Cianciaruso, R. L. Franco, and G. Adesso, J. Phys. A: Math. Theor. 47, 405302 (2014).
https://doi.org/10.1088/1751-8113/47/40/405302

[22] W. Roga, D. Spehner, and F. Illuminati, J. Phys. A: Math. Theor. 49, 235301 (2016).
https://doi.org/10.1088/1751-8113/49/23/235301

[23] F. Paula, J. Montealegre, A. Saguia, T. R. de Oliveira, and M. Sarandy, Europhys. Lett. 103, 50008 (2013).
https://doi.org/10.1209/0295-5075/103/50008

[24] E. Rains, IEEE Trans. Inf. Theory 47, 2921 (2001).
https://doi.org/10.1109/18.959270

[25] E. Rains, Phys. Rev. A 60, 179 (1999a).
https://doi.org/10.1103/PhysRevA.60.179

[26] H. Fawzi, J. Saunderson, and P. A. Parrilo, Found. Comp. Math. 19, 259 (2018).
https://doi.org/10.1007/s10208-018-9385-0

[27] H. Fawzi and O. Fawzi, J. Phys. A: Math. Theor. 51, 154003 (2018).
https://doi.org/10.1088/1751-8121/aab285

[28] B. Regula, J. Phys. A: Math. Theor. 51, 045303 (2017).
https://doi.org/10.1088/1751-8121/aa9100

[29] G. Vidal and R. Tarrach, Phys. Rev. A 59, 141 (1999).
https://doi.org/10.1103/PhysRevA.59.141

[30] X. Wang and M. M. Wilde, Phys. Rev. Lett. 125, 040502 (2020a).
https://doi.org/10.1103/PhysRevLett.125.040502

[31] X. Wang and M. M. Wilde, Phys. Rev. A 102, 032416 (2020b).
https://doi.org/10.1103/PhysRevA.102.032416

[32] S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 (1997).
https://doi.org/10.1103/physrevlett.78.5022

[33] D. Mundarain and J. Stephany, arXiv:0712.1015 [quant-ph] (2007).
arXiv:0712.1015

[34] D. Girolami and G. Adesso, Phys. Rev. A 84, 052110 (2011).
https://doi.org/10.1103/PhysRevA.84.052110

[35] T. Debarba, T. O. Maciel, and R. O. Vianna, Phys. Rev. A 86, 024302 (2012).
https://doi.org/10.1103/PhysRevA.86.024302

[36] M. Ozawa, Phys. Lett. A 268, 158 (2000).
https://doi.org/10.1016/s0375-9601(00)00171-7

[37] M. Piani, Phys. Rev. A 86, 034101 (2012).
https://doi.org/10.1103/PhysRevA.86.034101

[38] C. Helstrom, Quantum detection and estimation theory (Academic Press, New York, 1976).

[39] B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, Phys. Rev. Lett. 86, 5807 (2001).
https://doi.org/10.1103/PhysRevLett.86.5807

[40] D. DiVincenzo, D. Leung, and B. Terhal, IEEE Trans. Inf. Theory 48, 580 (2002).
https://doi.org/10.1109/18.985948

[41] W. Matthews, S. Wehner, and A. Winter, Commun. Math. Phys. 291, 813 (2009).
https://doi.org/10.1007/s00220-009-0890-5

[42] E. Rains, Phys. Rev. A 60, 173 (1999b).
https://doi.org/10.1103/PhysRevA.60.173

[43] T. Eggeling and R. Werner, Phys. Rev. Lett. 89, 097905 (2002).
https://doi.org/10.1103/PhysRevLett.89.097905

[44] M. B. Ruskai, Rev. Math. Phys. 06, 1147 (1994).
https://doi.org/10.1142/S0129055X94000407

[45] M. Khasin, R. Kosloff, and D. Steinitz, Phys. Rev. A 75, 052325 (2007).
https://doi.org/10.1103/PhysRevA.75.052325

[46] K. Audenaert, B. De Moor, K. Vollbrecht, and R. Werner, Phys. Rev. A 66, 032310 (2002).
https://doi.org/10.1103/PhysRevA.66.032310

[47] L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).
https://doi.org/10.1137/1038003

[48] J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018).
https://doi.org/10.1017/9781316848142

[49] K. Zyczkowski and H.-J. Sommers, J. Phys. A: Math. Gen. 34, 7111 (2001).
https://doi.org/10.1088/0305-4470/34/35/335

[50] S. L. Braunstein, Phys. Lett. A 219, 169 (1996).
https://doi.org/10.1016/0375-9601(96)00365-9

[51] M. J. Hall, Phys. Lett. A 242, 123 (1998).
https://doi.org/10.1016/s0375-9601(98)00190-x

[52] M. W. Girard, G. Gour, and S. Friedland, J. Phys. A: Math. Theor. 47, 505302 (2014).
https://doi.org/10.1088/1751-8113/47/50/505302

[53] V. Vedral and M. Plenio, Phys. Rev. A 57, 1619 (1998).
https://doi.org/10.1103/PhysRevA.57.1619

[54] B. Aaronson, R. L. Franco, G. Compagno, and G. Adesso, New J. Phys. 15, 093022 (2013).
https://doi.org/10.1088/1367-2630/15/9/093022

[55] T. Krisnanda, R. Ganardi, S.-Y. Lee, J. Kim, and T. Paterek, Phys. Rev. A 98, 052321 (2018).
https://doi.org/10.1103/PhysRevA.98.052321

[56] Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014).
https://doi.org/10.1088/0034-4885/77/9/094001

[57] H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).
https://doi.org/10.1103/PhysRevLett.103.210401

[58] B. Bylicka, M. Johansson, and A. Acín, Phys. Rev. Lett. 118, 120501 (2017).
https://doi.org/10.1103/PhysRevLett.118.120501

[59] J. Kołodyński, S. Rana, and A. Streltsov, Phys. Rev. A 101, 020303 (2020).
https://doi.org/10.1103/PhysRevA.101.020303

Cited by

[1] Mirko Consiglio, Tony J. G. Apollaro, and Marcin Wieśniak, "Variational approach to the quantum separability problem", Physical Review A 106 6, 062413 (2022).

[2] Mikhail A. Yurischev and Saeed Haddadi, "Local quantum Fisher information and local quantum uncertainty for general X states", Physics Letters A 476, 128868 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2023-05-29 20:58:49) and SAO/NASA ADS (last updated successfully 2023-05-29 20:58:50). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.