HEXAGONbeta

Tsallis Metrics on Density Matrices via the S-Divergence

A mix of human-written and AI-generated textHuman understanding: all partsmath.FA — Functional Analysisquant-ph — Quantum Physicsmath.IT — Information Theory

Contributed by Suvrit Sra ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We study symmetrized divergences induced by centroids of α\alpha-Tsallis relative entropies between density matrices σ\sigma and ρ\rho; up to scaling, the induced divergence equals ∣2−∥ρα+σα∥1/α∣|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|. Earlier work (S. Sra, "Metrics induced by Jensen–Shannon and related divergences on positive definite matrices," Linear Algebra Appl., 2021) studied such divergences and metrics induced by them. That paper allowed centroids to range over the positive definite cone, while we require that the centroid itself be a density matrix. This (more natural) choice makes the metricity in this paper considerably stronger than that of the aforementioned paper. Specifically, we prove that ∣2−∥ρα+σα∥1/α∣1/2|2-\|\rho^\alpha+\sigma^\alpha\|_{1/\alpha}|^{1/2} is a metric on density matrices for 1/2≤α≤21/2\le\alpha\le2, α≠1\alpha\ne1. For α<1\alpha<1, the proof uses a new four-point theorem for the S-divergence that might be of independent interest. For α>1\alpha>1, we first prove the endpoint α=2\alpha=2, and then transfer it to the other orders by a tensoring trick.

Provenance statement

The results in this paper were developed through extensive use of AI assistants: OpenAI's GPT Pro for the main line of argument, and Anthropic's Claude for the endpoint results in Section 3.4 and Appendix A.2, Remark A.2, Proposition 3.6, and the writing, positioning, and literature search. OpenAI assistants also developed the four-point consequences in Appendix B and assisted with proof simplification and reorganization. The arguments emerged through sustained work involving failed attempts, counterexamples, numerical checks, proof audits, and repairs.

Tools used

OpenAI
GPT Pro
Anthropic
Claude

References

  1. Rajendra Bhatia. Matrix Analysis . Graduate Texts in Mathematics 169. Springer, New York, 1997. 10.1007/978-1-4612-0653-8.DOI
  2. Jop Bri ë t and Peter Harremo ë s. Properties of classical and quantum J ensen-- S hannon divergence. Physical Review A , 79 0 (5): 0 052311, 2009. 10.1103/PhysRevA.79.052311.DOI
  3. Anoop Cherian, Suvrit Sra, Arindam Banerjee, and Nikolaos Papanikolopoulos. Jensen-- B regman logdet divergence with application to efficient similarity search for covariance matrices. IEEE Transactions on Pattern Analysis and Machine Intelligence , 35 0 (9): 0 2161--2174, 2013. 10.1109/TPAMI.2012.259.DOI
  4. Marco Dalai. Lower bounds on the probability of error for classical and classical-quantum channels. IEEE Transactions on Information Theory , 59 0 (12): 0 8027--8056, 2013. 10.1109/TIT.2013.2283794.DOI
  5. Michel Marie Deza and Monique Laurent. Geometry of Cuts and Metrics , volume 15 of Algorithms and Combinatorics . Springer, Berlin, 1997. ISBN 3-540-61611-X. 10.1007/978-3-642-04295-9.DOI
  6. D. M. Endres and J. E. Schindelin. A new metric for probability distributions. IEEE Transactions on Information Theory , 49 0 (7): 0 1858--1860, 2003. 10.1109/TIT.2003.813506.DOI
  7. Bent Fuglede and Flemming Topsøe. J ensen-- S hannon divergence and H ilbert space embedding. In Proceedings of the IEEE International Symposium on Information Theory (ISIT) , page 31, 2004. 10.1109/ISIT.2004.1365067.DOI
  8. Loo-Keng Hua. Inequalities involving determinants. Acta Mathematica Sinica , 5: 0 463--470, 1955.
  9. Robert K ö nig and Stephanie Wehner. A strong converse for classical channel coding using entangled inputs. Physical Review Letters , 103: 0 070504, 2009. 10.1103/PhysRevLett.103.070504. Appendix of arXiv:0903.2838.arXivDOI
  10. P. W. Lamberti, A. P. Majtey, A. Borras, M. Casas, and A. Plastino. Metric character of the quantum J ensen-- S hannon divergence. Physical Review A , 77 0 (5): 0 052311, 2008. 10.1103/PhysRevA.77.052311.DOI
Show all 21 referencesShow fewer references
  1. Shunlong Luo and Qiang Zhang. Informational distance on quantum-state space. Physical Review A , 69: 0 032106, 2004. 10.1103/PhysRevA.69.032106.DOI
  2. Andr é F. T. Martins, Noah A. Smith, Eric P. Xing, Pedro M. Q. Aguiar, and M á rio A. T. Figueiredo. Nonextensive information theoretic kernels on measures. Journal of Machine Learning Research , 10: 0 935--975, 2009.
  3. Ferdinand Ö sterreicher and Igor Vajda. A new class of metric divergences on probability spaces and its applicability in statistics. Annals of the Institute of Statistical Mathematics , 55 0 (3): 0 639--653, 2003. 10.1007/BF02517812.DOI
  4. D é nes Petz. Quasi-entropies for finite quantum systems. Reports on Mathematical Physics , 23 0 (1): 0 57--65, 1986. 10.1016/0034-4877(86)90067-4.DOI
  5. I. J. Schoenberg. Remarks to Maurice Fr é chet 's article `` Sur la d é finition axiomatique d'une classe d'espace distanci é s vectoriellement applicable sur l'espace de Hilbert ''. Annals of Mathematics , 36 0 (3): 0 724--732, 1935. 10.2307/1968654.DOI
  6. Naresh Sharma and Naqueeb Ahmad Warsi. Fundamental bound on the reliability of quantum information transmission. Physical Review Letters , 110: 0 080501, 2013. 10.1103/PhysRevLett.110.080501.DOI
  7. Suvrit Sra. Positive definite matrices and the S -divergence. Proceedings of the American Mathematical Society , 144 0 (7): 0 2787--2797, 2016. 10.1090/proc/12953.DOI
  8. Suvrit Sra. Positive definite functions of noncommuting contractions, H ua-- B ellman matrices, and a new distance metric, 2021 a .
  9. Suvrit Sra. Metrics induced by J ensen-- S hannon and related divergences on positive definite matrices. Linear Algebra and its Applications , 616: 0 125--138, 2021 b . 10.1016/j.laa.2020.12.023.DOI
  10. Constantino Tsallis. Possible generalization of B oltzmann-- G ibbs statistics. Journal of Statistical Physics , 52 0 (1--2): 0 479--487, 1988. 10.1007/BF01016429.DOI
  11. D á niel Virosztek. The metric property of the quantum J ensen-- S hannon divergence. Advances in Mathematics , 380: 0 107595, 2021. 10.1016/j.aim.2021.107595.DOI

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 28, 2026