Papers
Topics
Authors
Recent
Detailed Answer
Quick Answer
Concise responses based on abstracts only
Detailed Answer
Well-researched responses based on abstracts and relevant paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses
Gemini 2.5 Flash
Gemini 2.5 Flash 95 tok/s
Gemini 2.5 Pro 46 tok/s Pro
GPT-5 Medium 15 tok/s Pro
GPT-5 High 19 tok/s Pro
GPT-4o 90 tok/s Pro
GPT OSS 120B 449 tok/s Pro
Kimi K2 192 tok/s Pro
2000 character limit reached

Involutive Markov categories and the quantum de Finetti theorem (2312.09666v3)

Published 15 Dec 2023 in math.CT, math.OA, and quant-ph

Abstract: Markov categories have recently emerged as a powerful high-level framework for probability theory and theoretical statistics. Here we study a quantum version of this concept, called involutive Markov categories. These are equivalent to Parzygnat's quantum Markov categories, but we argue that they offer a simpler and more practical approach. Our main examples of involutive Markov categories have pre-C*-algebras, including infinite-dimensional ones, as objects, together with completely positive unital maps as morphisms in the picture of interest. In this context, we prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms, and we develop a categorical description of such quantum de Finetti theorems which amounts to a universal property of state spaces.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (48)
  1. Erik M. Alfsen. Compact convex sets and boundary integrals, volume Band 57 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, New York-Heidelberg, 1971.
  2. Generalized no-broadcasting theorem. Phys. Rev. Lett., 99:240501, 2007. arXiv:0707.0620.
  3. Bruce Blackadar. Shape theory for C*-algebras. Mathematica Scandinavica, 56(2):249–275, 1985.
  4. Variational methods in mathematical physics. A unified approach. Transl. from the German by Gillian M. Hayes. Texts Monogr. Phys. Springer-Verlag, 1992.
  5. C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras and finite-dimensional approximations, volume 88 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2008.
  6. Ulrich Bunke. Lecture course on C*-algebras and C*-categories, 2021. https://bunke.app.uni-regensburg.de/VorlesungCastcats-1.pdf.
  7. Disintegration and Bayesian inversion via string diagrams. Math. Structures Comput. Sci., 29:938–971, 2019. arXiv:1709.00322.
  8. Picturing classical and quantum Bayesian inference. Synthese, 186(3):651–696, 2012. arXiv:1102.2368.
  9. John B. Conway. A course in operator theory, volume 21 of Grad. Stud. Math. Providence, RI: American Mathematical Society, 2000.
  10. Complete family of separability criteria. Phys. Rev. A, 69:022308, 2004. arXiv:quant-ph/0308032.
  11. Morita equivalence of smooth noncommutative tori. Acta Math., 199(1):1–27, 2007. arXiv:math/0311502.
  12. Supplying bells and whistles in symmetric monoidal categories, 2020. arXiv:1908.02633.
  13. Tobias Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370:107239, 2020. arXiv:1908.07021.
  14. From gs-monoidal to oplax cartesian categories: Constructions and functorial completeness. Appl. Categ. Struct., 31(42), 2023. arXiv:2205.06892.
  15. Dilations and information flow axioms in categorical probability. Math. Struct. Comp. Sci., page 1–45, 2023. arXiv:2211.02507.
  16. Absolute continuity, supports and idempotent splitting in categorical probability. arXiv:2308.00651.
  17. Representable Markov categories and comparison of statistical experiments in categorical probability. Theoret. Comput. Sci., 961(113896), 2023. arXiv:2010.07416.
  18. De Finetti’s theorem in categorical probability. J. Stoch. Anal., 2(4), 2021. arXiv:2105.02639.
  19. The d𝑑ditalic_d-separation criterion in categorical probability. J. Mach. Learn. Res., 24(46):1–49, 2023. arXiv:2207.05740.
  20. Infinite products and zero-one laws in categorical probability. Compositionality, 2:3, 2020. arXiv:1912.02769.
  21. 5 Unknown Quantum States and Operations,a Bayesian View, pages 147–187. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004. arXiv:quant-ph/0404156.
  22. Robert W. J. Furber and Bart P. F. Jacobs. From Kleisli categories to commutative C*-algebras: Probabilistic Gelfand duality. Logical Methods in Computer Science, 11, 2015. arXiv:1303.1115.
  23. Alain Guichardet. Produits tensoriels infinis et représentations des relations d’anticommutation. Ann. Sci. École Norm. Sup. (3), 83:1–52, 1966.
  24. Nicholas Gauguin Houghton-Larsen. A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing. PhD thesis, University of Copenhagen, 2021. arXiv:2103.02302.
  25. A. Hulanicki and R.R. Phelps. Some applications of tensor products of partially-ordered linear spaces. J. Funct. Anal., 2(2):177–201, 1968.
  26. Bart Jacobs. Involutive categories and monoids, with a GNS-correspondence. Found. Phys., 42(7):874–895, 2012.
  27. Gregg Jaeger. Quantum information. Springer, New York, 2007. An overview, With a foreword by Tommaso Toffoli.
  28. Fundamentals of the theory of operator algebras. Vol. I, volume 15 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Elementary theory, Reprint of the 1983 original.
  29. Fundamentals of the theory of operator algebras. Vol. II, volume 16 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. Advanced theory, Corrected reprint of the 1986 original.
  30. Quotients, exactness, and nuclearity in the operator system category. Advances in Mathematics, 235:321–360, 2013. arXiv:1008.2811.
  31. Masoud Khalkhali. Basic noncommutative geometry. EMS Ser. Lect. Math. Zürich: European Mathematical Society (EMS), 2nd updated ed. edition, 2013.
  32. C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras generated by operator systems. J. Funct. Anal., 155(2):324–351, 1998.
  33. Huaxin Lin. An introduction to the classification of amenable C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras. Singapore: World Scientific, 2001.
  34. Probability monads with submonads of deterministic states. In Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 1–13, 2022. arXiv:2204.07003.
  35. A category-theoretic proof of the ergodic decomposition theorem. Ergodic Theory Dynam. Systems, pages 1–27, 2023. arXiv:2207.07353.
  36. Gerard J. Murphy. C*{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT-algebras and operator theory. Boston, MA etc.: Academic Press, Inc., 1990.
  37. Quantum computation and quantum information. Cambridge University Press, Cambridge, 2000.
  38. Arthur J. Parzygnat. Inverses, disintegrations, and Bayesian inversion in quantum Markov categories, 2020. arXiv:2001.08375.
  39. Non-commutative disintegrations: Existence and uniqueness in finite dimensions. J. Noncommut. Geom., 2023. arXiv:1907.09689.
  40. Vern Paulsen. Completely bounded maps and operator algebras, volume 78 of Camb. Stud. Adv. Math. Cambridge: Cambridge University Press, 2002.
  41. Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289–355. Berlin: Springer, 2011. arXiv:0908.3347.
  42. Quantum de Finetti theorems as categorical limits, and limits of state spaces of C*-algebras. EPTCS, 394:400–414, 2023. arXiv:2207.05832.
  43. Erling Størmer. Symmetric states of infinite tensor products of C*-algebras. J. Funct. Anal., 3(1):48–68, 1969.
  44. T. Świrszcz. Monadic functors and convexity. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 22:39–42, 1974.
  45. Masamichi Takesaki. On the cross-norm of the direct product of C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras. Tôhoku Math. J. (2), 16:111–122, 1964.
  46. Lev Vaidman. The paradoxes of the interaction-free measurements. Zeitschrift für Naturforschung A, 56(1–2):100–107, 2001. arXiv:quant-ph/0102049.
  47. John von Neumann. Mathematical foundations of quantum mechanics. Princeton University Press, Princeton, NJ, new edition, 2018. Translated from the German and with a preface by Robert T. Beyer, Edited and with a preface by Nicholas A. Wheeler.
  48. Niels Erik Wegge-Olsen. K𝐾Kitalic_K-theory and C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras: a friendly approach. Oxford: Oxford University Press, 1993.
Citations (1)
View on Semantic Scholar
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Ai Generate Text Spark Streamline Icon: https://streamlinehq.com

Paper Prompts

Sign up for free to create and run prompts on this paper using GPT-5.

List Streamline Icon: https://streamlinehq.com Explore 10 Community Prompts
Dice Question Streamline Icon: https://streamlinehq.com

Follow-up Questions

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
Youtube Logo Streamline Icon: https://streamlinehq.com

YouTube