HEXAGONbeta

Bures Geometry of Projection States

Primarily human-written textHuman understanding: all partsmath.OA — Operator Algebrasmath-ph — Mathematical Physicsquant-ph — Quantum Physics

Contributed by Zhenyu Sun ↗, Jilin Wang

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

We study the Bures metric on states obtained by conditioning a faithful normal state on nonzero projections of a von Neumann algebra. The metric determines the reference state, projection probabilities, orthogonality, and order. For algebras with no type I2I_2 direct summand, surjective isometries are exactly those induced by reference-state-preserving normal Jordan ∗^*-isomorphisms; the same conclusion holds for M2(C)M_2(\mathbb C). A semifinite tracial version determines the trace up to a single positive factor. In finite tracial algebras, we characterize optimal finite projective measurements and express their fidelity deficit as a sum of squared L2L^2 errors. Spectral measurements with at most n+1n+1 outcomes approximate fidelity with error O(n−2)O(n^{-2}); the interval grid is minimax optimal, and the order is sharp within spectral measurements. We compare the fidelity defect of a common commutative context with approximation by commuting projections, obtain sharp spectral rounding constants, and solve both optimization problems in the algebra generated by two projections. The space of normalized finite-trace projection states is complete; in a finite tracial algebra its rectifiable components are unitary orbits, and within each orbit the intrinsic Bures distance is the root mean square of the principal angles. Finally, a fidelity bound on local unitary orbits characterizes tracial reference states.

Author comments: 38 pages.

Provenance statement

OpenAI Codex was used to revise the English exposition, check selected arguments and references, and prepare and compile the LaTeX submission source.

References

  1. Peter M. Alberti and Armin Uhlmann. On Bures distance and ∗^{*}-algebraic transition probability between inner derived positive linear forms over W∗W^{*} -algebras. Acta Applicandae Mathematicae , 60 0 (1): 0 1--37, 2000. 10.1023/A:1006317508252.DOI
  2. Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar é . Gradient Flows: In Metric Spaces and in the Space of Probability Measures . Lectures in Mathematics ETH Z ü rich. Birkh ä user, Basel, 2 edition, 2008. 10.1007/978-3-7643-8722-8.DOI
  3. Esteban Andruchow. Geodesics of projections in von Neumann algebras. Proceedings of the American Mathematical Society , 149 0 (10): 0 4501--4513, 2021. 10.1090/proc/15568.DOI
  4. Esteban Andruchow and L á zaro Recht. Grassmannians of a finite algebra in the strong operator topology. International Journal of Mathematics , 17 0 (4): 0 477--491, 2006. 10.1142/S0129167X06003552.DOI
  5. Esteban Andruchow, Gabriel Larotonda, and L á zaro Recht. Conjugate points in the Grassmann manifold of a C∗C^* -algebra. Journal of Geometry and Physics , 218: 0 105654, 2025. 10.1016/j.geomphys.2025.105654.DOI
  6. Huzihiro Araki and Guido A. Raggio. A remark on transition probability. Letters in Mathematical Physics , 6: 0 237--240, 1982. 10.1007/BF00403278.DOI
  7. J á nos A. Bergou, Edgar Feldman, and Mark Hillery. Optimal unambiguous discrimination of two subspaces as a case in mixed-state discrimination. Physical Review A , 73: 0 032107, 2006. 10.1103/PhysRevA.73.032107.DOI
  8. Garrett Birkhoff and John von Neumann. The logic of quantum mechanics. Annals of Mathematics , 37 0 (4): 0 823--843, 1936. 10.2307/1968621.DOI
  9. Samuel L. Braunstein and Carlton M. Caves. Statistical distance and the geometry of quantum states. Physical Review Letters , 72 0 (22): 0 3439--3443, 1994. 10.1103/PhysRevLett.72.3439.DOI
  10. Donald Bures. An extension of Kakutani 's theorem on infinite product measures to the tensor product of semifinite W∗W^{*} -algebras. Transactions of the American Mathematical Society , 135: 0 199--212, 1969. 10.1090/S0002-9947-1969-0236719-2.DOI
Show all 26 referencesShow fewer references
  1. Datong Chen and Huangjun Zhu. Optimal quantum measurements with respect to the fidelity, 2025. URL https://arxiv.org/abs/2511.22487. arXiv:2511.22487.arXiv
  2. H. A. Dye. On the geometry of projections in certain operator algebras. Annals of Mathematics , 61 0 (1): 0 73--89, 1955. 10.2307/1969620.DOI
  3. Douglas Farenick, Samuel Jaques, and Mizanur Rahaman. The fidelity of density operators in an operator-algebraic framework. Journal of Mathematical Physics , 57 0 (10): 0 102202, 2016. 10.1063/1.4965876.DOI
  4. Nikolay Filonov and Ilya Kachkovskiy. A Hilbert--Schmidt analog of Huaxin Lin 's theorem, 2010. URL https://arxiv.org/abs/1008.4002. arXiv:1008.4002.arXiv
  5. Christopher A. Fuchs and Carlton M. Caves. Mathematical techniques for quantum communication theory. Open Systems & Information Dynamics , 3 0 (3): 0 345--356, 1995. 10.1007/BF02228997.DOI
  6. Lev Glebsky. Almost commuting matrices with respect to normalized Hilbert--Schmidt norm, 2010. URL https://arxiv.org/abs/1002.3082. arXiv:1002.3082.arXiv
  7. Paul R. Halmos. Two subspaces. Transactions of the American Mathematical Society , 144: 0 381--389, 1969. 10.1090/S0002-9947-1969-0251519-5.DOI
  8. Jin Li, Rajesh Pereira, and Sarah Plosker. Some geometric interpretations of quantum fidelity. Linear Algebra and its Applications , 487: 0 158--171, 2015. 10.1016/j.laa.2015.08.037.DOI
  9. Lajos Moln á r. Bures isometries between density spaces of C∗C^* -algebras. Linear Algebra and its Applications , 557: 0 22--33, 2018. 10.1016/j.laa.2018.07.008.DOI
  10. Lajos Moln á r and Werner Timmermann. Isometries of quantum states. Journal of Physics A: Mathematical and General , 36 0 (1): 0 267--273, 2003. 10.1088/0305-4470/36/1/318.DOI
  11. Michiya Mori. Isometries between projection lattices of von Neumann algebras. Journal of Functional Analysis , 276 0 (11): 0 3511--3528, 2019. 10.1016/j.jfa.2018.10.011.DOI
  12. D é nes Petz and Csaba Sud á r. Geometries of quantum states. Journal of Mathematical Physics , 37 0 (6): 0 2662--2673, 1996. 10.1063/1.531535.DOI
  13. Wenhua Qian, Liguang Wang, Wenming Wu, and Wei Yuan. Wigner -type theorem on transition probability preserving maps in semifinite factors. Journal of Functional Analysis , 276 0 (6): 0 1773--1787, 2019. 10.1016/j.jfa.2018.05.019.DOI
  14. Masamichi Takesaki. Theory of Operator Algebras I , volume 124 of Encyclopaedia of Mathematical Sciences . Springer, Berlin, 2002. 10.1007/978-3-642-61836-9.DOI
  15. Armin Uhlmann. The ``transition probability'' in the state space of a ∗^{*}-algebra. Reports on Mathematical Physics , 9 0 (2): 0 273--279, 1976. 10.1016/0034-4877(76)90060-4.DOI
  16. John von Neumann. Continuous Geometry , volume 25 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1960. With a foreword by Israel Halperin.

Version history

  1. v1 · 38 pagesSubmitted by Zhenyu Sun38 pages.CurrentNov 06, 2026