- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Estimation of multivariate traces of states given partial classical information

Kyrylo Simonov1,*, Rafael Wagner2,3,4,5,†, and Ernesto F. Galvão2,6,‡

  • *Contact author: kyrylo.simonov@univie.ac.at
  • †Contact author: rafael.wagner@uni-ulm.de
  • ‡Contact author: ernestogalvao@id.uff.br
PDF

Phys. Rev. A 112, 062435 – Published 17 December, 2025

DOI: https://doi.org/10.1103/mrpz-xq28

Abstract

Bargmann invariants of order n, defined as multivariate traces of quantum states Tr[ρ1ρ2...ρn], are useful in applications ranging from quantum metrology to certification of nonclassicality. A standard quantum circuit used to estimate Bargmann invariants is the cycle test. In this work, we propose generalizations of the cycle test applicable to a situation where n systems are given and unknown, and classical information on m systems (m≤n) is available, allowing estimation of invariants of order n+m. Our main result is a generalization of results on fourth-order invariants appearing in “double” weak values from Chiribella et al. [Phys. Rev. Res. 6, 043043 (2024)]. The use of classical information on some of the states enables circuits on fewer qubits and with fewer gates, decreasing the experimental requirements for their estimation and enabling multiple applications we briefly review.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (106)

  1. V. Bargmann, Note on Wigner's theorem on symmetry operations, J. Math. Phys. 5, 862 (1964).
  2. R. Simon and N. Mukunda, Bargmann invariant and the geometry of the Güoy effect, Phys. Rev. Lett. 70, 880 (1993).
  3. N. Mukunda, Arvind, S. Chaturvedi, and R. Simon, Bargmann invariants and off-diagonal geometric phases for multilevel quantum systems: A unitary-group approach, Phys. Rev. A 65, 012102 (2001).
  4. N. Mukunda, Arvind, E. Ercolessi, G. Marmo, G. Morandi, and R. Simon, Bargmann invariants, null phase curves, and a theory of the geometric phase, Phys. Rev. A 67, 042114 (2003).
  5. N. Mukunda, P. K. Aravind, and R. Simon, Wigner rotations, Bargmann invariants and geometric phases, J. Phys. A: Math. Gen. 36, 2347 (2003).
  6. K. S. Akhilesh, Arvind, S. Chaturvedi, K. S. Mallesh, and N. Mukunda, Geometric phases for finite-dimensional systems—the roles of Bargmann invariants, null phase curves, and the Schwinger–Majorana SU(2) framework, J. Math. Phys. 61, 072103 (2020).
  7. U. Fano, Description of states in quantum mechanics by density matrix and operator techniques, Rev. Mod. Phys. 29, 74 (1957).
  8. V. S. Shchesnovich, Partial indistinguishability theory for multiphoton experiments in multiport devices, Phys. Rev. A 91, 013844 (2015).
  9. V. S. Shchesnovich and M. E. O. Bezerra, Collective phases of identical particles interfering on linear multiports, Phys. Rev. A 98, 033805 (2018).
  10. K.-W. Bong, N. Tischler, R. B. Patel, S. Wollmann, G. J. Pryde, and M. J. W. Hall, Strong unitary and overlap uncertainty relations: Theory and experiment, Phys. Rev. Lett. 120, 230402 (2018).
  11. A. E. Jones, A. J. Menssen, H. M. Chrzanowski, T. A. W. Wolterink, V. S. Shchesnovich, and I. A. Walmsley, Multiparticle interference of pairwise distinguishable photons, Phys. Rev. Lett. 125, 123603 (2020).
  12. M. Pont, R. Albiero, S. E. Thomas, N. Spagnolo, F. Ceccarelli, G. Corrielli, A. Brieussel, N. Somaschi, H. Huet, A. Harouri, A. Lemaître, I. Sagnes, N. Belabas, F. Sciarrino, R. Osellame, P. Senellart, and A. Crespi, Quantifying n-photon indistinguishability with a cyclic integrated interferometer, Phys. Rev. X 12, 031033 (2022).
  13. A. E. Jones, S. Kumar, S. D'Aurelio, M. Bayerbach, A. J. Menssen, and S. Barz, Distinguishability and mixedness in quantum interference, Phys. Rev. A 108, 053701 (2023).
  14. G. Rodari, C. Fernandes, E. Caruccio, A. Suprano, F. Hoch, T. Giordani, G. Carvacho, R. Albiero, N. D. Giano, G. Corrielli, F. Ceccarelli, R. Osellame, D. J. Brod, L. Novo, N. Spagnolo, E. F. Galvão, and F. Sciarrino, Experimental observation of counter-intuitive features of photonic bunching, arXiv:2410.15883.
  15. G. Rodari, L. Novo, R. Albiero, A. Suprano, C. T. Tavares, E. Caruccio, F. Hoch, T. Giordani, G. Carvacho, M. Gardina, N. D. Giano, S. D. Giorgio, G. Corrielli, F. Ceccarelli, R. Osellame, N. Spagnolo, E. F. Galvão, and F. Sciarrino, Semi-device independent characterization of multiphoton indistinguishability, PRX Quantum 6, 020340 (2025).
  16. M. Oszmaniec, D. J. Brod, and E. F. Galvão, Measuring relational information between quantum states, and applications, New J. Phys. 26, 013053 (2024).
  17. E. F. Galvão and D. J. Brod, Quantum and classical bounds for two-state overlaps, Phys. Rev. A 101, 062110 (2020).
  18. T. Giordani, C. Esposito, F. Hoch, G. Carvacho, D. J. Brod, E. F. Galvão, N. Spagnolo, and F. Sciarrino, Witnesses of coherence and dimension from multiphoton indistinguishability tests, Phys. Rev. Res. 3, 023031 (2021).
  19. R. Wagner, R. S. Barbosa, and E. F. Galvão, Inequalities witnessing coherence, nonlocality, and contextuality, Phys. Rev. A 109, 032220 (2024).
  20. R. Wagner, A. Camillini, and E. F. Galvão, Coherence and contextuality in a Mach-Zehnder interferometer, Quantum 8, 1240 (2024).
  21. R. Wagner, F. C. R. Peres, E. Z. Cruzeiro, and E. F. Galvão, Unitary-invariant method for witnessing nonstabilizerness in quantum processors, J. Phys. A Math. Theor. 58, 285302 (2025).
  22. T. Haug and M. S. Kim, Scalable measures of magic resource for quantum computers, PRX Quantum 4, 010301 (2023).
  23. T. Giordani, R. Wagner, C. Esposito, A. Camillini, F. Hoch, G. Carvacho, C. Pentangelo, F. Ceccarelli, S. Piacentini, A. Crespi, N. Spagnolo, R. Osellame, E. F. Galvão, and F. Sciarrino, Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic processor, Sci. Adv. 9, eadj4249 (2023).
  24. D. J. Brod, E. F. Galvão, N. Viggianiello, F. Flamini, N. Spagnolo, and F. Sciarrino, Witnessing genuine multiphoton indistinguishability, Phys. Rev. Lett. 122, 063602 (2019).
  25. T. Giordani, D. J. Brod, C. Esposito, N. Viggianiello, M. Romano, F. Flamini, G. Carvacho, N. Spagnolo, E. F. Galvão, and F. Sciarrino, Experimental quantification of four-photon indistinguishability, New J. Phys. 22, 043001 (2020).
  26. N. Miklin and M. Oszmaniec, A universal scheme for robust self-testing in the prepare-and-measure scenario, Quantum 5, 424 (2021).
  27. M. Hinsche, M. Ioannou, S. Jerbi, L. Leone, J. Eisert, and J. Carrasco, Efficient distributed inner product estimation via Pauli sampling, PRX Quantum 6, 030354 (2025).
  28. R. Bandyopadhyay, A. H. Rubin, M. Radulaski, and M. M. Wilde, Efficient quantum algorithms for testing symmetries of open quantum systems, Open Syst. Inf. Dyn. 30, 2350017 (2023).
  29. H.-Y. Huang, J. Preskill, and M. Soleimanifar, Certifying almost all quantum states with few single-qubit measurements, Nat. Phys. 21, 1834 (2025).
  30. B. Tamir and E. Cohen, Introduction to weak measurements and weak values, Quanta 2, 7 (2013).
  31. J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Colloquium: Understanding quantum weak values: Basics and applications, Rev. Mod. Phys. 86, 307 (2014).
  32. R. Wagner and E. F. Galvão, Simple proof that anomalous weak values require coherence, Phys. Rev. A 108, L040202 (2023).
  33. M. Lostaglio, A. Belenchia, A. Levy, S. Hernández-Gómez, N. Fabbri, and S. Gherardini, Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables, Quantum 7, 1128 (2023).
  34. R. Wagner, Z. Schwartzman-Nowik, I. L. Paiva, A. Te'eni, A. Ruiz-Molero, R. S. Barbosa, E. Cohen, and E. F. Galvão, Quantum circuits for measuring weak values, Kirkwood–Dirac quasiprobability distributions, and state spectra, Quantum Sci. Technol. 9, 015030 (2024).
  35. S. Gherardini and G. De Chiara, Quasiprobabilities in quantum thermodynamics and many-body systems, PRX Quantum 5, 030201 (2024).
  36. D. R. M. Arvidsson-Shukur, W. F. Braasch Jr, S. De Bièvre, J. Dressel, A. N. Jordan, C. Langrenez, M. Lostaglio, J. S. Lundeen, and N. Y. Halpern, Properties and applications of the Kirkwood–Dirac distribution, New J. Phys. 26, 121201 (2024).
  37. D. Schmid, R. D. Baldijão, Y. Yīng, R. Wagner, and J. H. Selby, Kirkwood–Dirac representations beyond quantum states and their relation to noncontextuality, Phys. Rev. A 110, 052206 (2024).
  38. L. Liu and S. Cheng, The boundary of Kirkwood-Dirac quasiprobability, arXiv:2504.09238.
  39. C. M. Alves, P. Horodecki, D. K. L. Oi, L. C. Kwek, and A. K. Ekert, Direct estimation of functionals of density operators by local operations and classical communication, Phys. Rev. A 68, 032306 (2003).
  40. T. Brun, Measuring polynomial functions of states, Quantum Inf. Comput. 4, 401 (2004).
  41. M. S. Leifer, N. Linden, and A. Winter, Measuring polynomial invariants of multiparty quantum states, Phys. Rev. A 69, 052304 (2004).
  42. S. J. van Enk and C. W. J. Beenakker, Measuring Trρn on single copies of ρ using random measurements, Phys. Rev. Lett. 108, 110503 (2012).
  43. T. Tanaka, Y. Ota, M. Kanazawa, G. Kimura, H. Nakazato, and F. Nori, Determining eigenvalues of a density matrix with minimal information in a single experimental setting, Phys. Rev. A 89, 012117 (2014).
  44. M. Shin, J. Lee, S. Lee, and K. Jeong, Resource-efficient algorithm for estimating the trace of quantum state powers, Quantum 9, 1832 (2025).
  45. S. Johri, D. S. Steiger, and M. Troyer, Entanglement spectroscopy on a quantum computer, Phys. Rev. B 96, 195136 (2017).
  46. X. Turkeshi, M. Schirò, and P. Sierant, Measuring nonstabilizerness via multifractal flatness, Phys. Rev. A 108, 042408 (2023).
  47. E. Tirrito, P. S. Tarabunga, G. Lami, T. Chanda, L. Leone, S. F. E. Oliviero, M. Dalmonte, M. Collura, and A. Hamma, Quantifying nonstabilizerness through entanglement spectrum flatness, Phys. Rev. A 109, L040401 (2024).
  48. G. De Chiara, L. Lepori, M. Lewenstein, and A. Sanpera, Entanglement spectrum, critical exponents, and order parameters in quantum spin chains, Phys. Rev. Lett. 109, 237208 (2012).
  49. C. Chamon, A. Hamma, and E. R. Mucciolo, Emergent irreversibility and entanglement spectrum statistics, Phys. Rev. Lett. 112, 240501 (2014).
  50. B. Koczor, Exponential error suppression for near-term quantum devices, Phys. Rev. X 11, 031057 (2021).
  51. W. J. Huggins, S. McArdle, T. E. O'Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Phys. Rev. X 11, 041036 (2021).
  52. Y. Wang, G. Li, and X. Wang, Variational quantum Gibbs state preparation with a truncated Taylor series, Phys. Rev. Appl. 16, 054035 (2021).
  53. J. Cotler, S. Choi, A. Lukin, H. Gharibyan, T. Grover, M. E. Tai, M. Rispoli, R. Schittko, P. M. Preiss, A. M. Kaufman, M. Greiner, H. Pichler, and P. Hayden, Quantum virtual cooling, Phys. Rev. X 9, 031013 (2019).
  54. N. Gao, D. Li, A. Mishra, J. Yan, K. Simonov, and G. Chiribella, Measuring incompatibility and clustering quantum observables with a quantum switch, Phys. Rev. Lett. 130, 170201 (2023).
  55. N. Yunger Halpern, B. Swingle, and J. Dressel, Quasiprobability behind the out-of-time-ordered correlator, Phys. Rev. A 97, 042105 (2018).
  56. J. R. G. Alonso, N. Yunger Halpern, and J. Dressel, Out-of-time-ordered-correlator quasiprobabilities robustly witness scrambling, Phys. Rev. Lett. 122, 040404 (2019).
  57. J. R. G. Alonso, N. Shammah, S. Ahmed, F. Nori, and J. Dressel, Diagnosing quantum chaos with out-of-time-ordered-correlator quasiprobability in the kicked-top model, arXiv:2201.08175.
  58. G. Mitchison, R. Jozsa, and S. Popescu, Sequential weak measurement, Phys. Rev. A 76, 062105 (2007).
  59. S. Pancharatnam, Generalized theory of interference, and its applications: Part I. Coherent pencils, Proc. Indian Acad. Sci. 44, 247 (1956).
  60. Arvind, K. S. Mallesh, and N. Mukunda, A generalized Pancharatnam geometric phase formula for three-level quantum systems, J. Phys. A: Math. Gen. 30, 2417 (1997).
  61. F. Wilczek and A. Shapere, Geometric Phases in Physics, Advanced Series in Mathematical Physics (World Scientific, Singapore, 1989).
  62. D. Chruściński and A. Jamiołkowski, Geometric Phases in Classical and Quantum Mechanics (Birkhäuser Boston, Boston, 2004).
  63. C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Unitary-invariant witnesses of quantum imaginarity, Phys. Rev. Lett. 133, 190201 (2024).
  64. M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392, 45 (1984).
  65. A. Avdoshkin and F. K. Popov, Extrinsic geometry of quantum states, Phys. Rev. B 107, 245136 (2023).
  66. E. M. Rabei, Arvind, N. Mukunda, and R. Simon, Bargmann invariants and geometric phases: A generalized connection, Phys. Rev. A 60, 3397 (1999).
  67. Y. Quek, E. Kaur, and M. M. Wilde, Multivariate trace estimation in constant quantum depth, Quantum 8, 1220 (2024).
  68. G. Chiribella, K. Simonov, and X. Zhao, Dimension-independent weak value estimation via controlled SWAP operations, Phys. Rev. Res. 6, 043043 (2024).
  69. J. C. Garcia-Escartin and P. Chamorro-Posada, Swap test and Hong-Ou-Mandel effect are equivalent, Phys. Rev. A 87, 052330 (2013).
  70. A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa, and C. Macchiavello, Stabilization of quantum computations by symmetrization, SIAM J. Comput. 26, 1541 (1997).
  71. H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, Quantum fingerprinting, Phys. Rev. Lett. 87, 167902 (2001).
  72. D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, Algorithmica 55, 395 (2008).
  73. P. Schiansky, T. Strömberg, D. Trillo, V. Saggio, B. Dive, M. Navascués, and P. Walther, Demonstration of universal time-reversal for qubit processes, Optica 10, 200 (2023).
  74. M. Larocca, F. Sauvage, F. M. Sbahi, G. Verdon, P. J. Coles, and M. Cerezo, Group-invariant quantum machine learning, PRX Quantum 3, 030341 (2022).
  75. K. Fujii, Exchange gate on the qudit space and Fock space, J. Opt. B: Quantum Semiclass. Opt. 5, S613 (2003).
  76. S. Foulds, O. Prove, and V. Kendon, Generalizing multipartite concentratable entanglement for practical applications: Mixed, qudit and optical states, Philos. Trans. R. Soc. A 382, 20240411 (2024).
  77. S. Foulds, V. Kendon, and T. Spiller, The controlled SWAP test for determining quantum entanglement, Quantum Sci. Technol. 6, 035002 (2021).
  78. E. Cohen and E. Pollak, Determination of weak values of quantum operators using only strong measurements, Phys. Rev. A 98, 042112 (2018).
  79. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, 2010).
  80. A. W. Harrow, The church of the symmetric subspace, arXiv:1308.6595.
  81. L. I. Reascos, B. Murta, E. F. Galvão, and J. Fernández-Rossier, Quantum circuits to measure scalar spin chirality, Phys. Rev. Res. 5, 043087 (2023).
  82. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  83. L. Diósi, Structural features of sequential weak measurements, Phys. Rev. A 94, 010103(R) (2016).
  84. D. Georgiev and E. Cohen, Probing finite coarse-grained virtual Feynman histories with sequential weak values, Phys. Rev. A 97, 052102 (2018).
  85. P. K. Faehrmann, J. Eisert, and R. Kueng, In the shadow of the Hadamard test: Using the garbage state for good and further modifications, Phys. Rev. Lett. 135, 150603 (2025).
  86. G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, Beyond quantum computers, , arXiv:0912.0195 (2009).
  87. G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013).
  88. G. Chiribella, Perfect discrimination of no-signalling channels via quantum superposition of causal structures, Phys. Rev. A 86, 040301(R) (2012).
  89. M. Araújo, F. Costa, and Č. Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014).
  90. D. Felce and V. Vedral, Quantum refrigeration with indefinite causal order, Phys. Rev. Lett. 125, 070603 (2020).
  91. X. Zhao, Y. Yang, and G. Chiribella, Quantum metrology with indefinite causal order, Phys. Rev. Lett. 124, 190503 (2020).
  92. G. Chiribella, M. Banik, S. S. Bhattacharya, T. Guha, M. Alimuddin, A. Roy, S. Saha, S. Agrawal, and G. Kar, Indefinite causal order enables perfect quantum communication with zero capacity channels, New J. Phys. 23, 033039 (2021).
  93. J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, Quantum circuits with classical versus quantum control of causal order, PRX Quantum 2, 030335 (2021).
  94. G. Zhu, Y. Chen, Y. Hasegawa, and P. Xue, Charging quantum batteries via indefinite causal order: Theory and experiment, Phys. Rev. Lett. 131, 240401 (2023).
  95. J. Bavaresco, M. Murao, and M. T. Quintino, Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination, Phys. Rev. Lett. 127, 200504 (2021).
  96. K. Simonov, G. Francica, G. Guarnieri, and M. Paternostro, Work extraction from coherently activated maps via quantum switch, Phys. Rev. A 105, 032217 (2022).
  97. K. Simonov, S. Roy, T. Guha, Z. Zimborás, and G. Chiribella, Activation of thermal states by coherently controlled thermalization processes, New J. Phys. 27, 074502 (2025).
  98. S. Koudia, A. S. Cacciapuoti, K. Simonov, and M. Caleffi, How deep the theory of quantum communications goes: Superadditivity, superactivation and causal activation, IEEE Commun. Surv. Tutorials 24, 1926 (2022).
  99. Q. Liu, Z. Hu, H. Yuan, and Y. Yang, Optimal strategies of quantum metrology with a strict hierarchy, Phys. Rev. Lett. 130, 070803 (2023).
  100. M. Caleffi, K. Simonov, and A. S. Cacciapuoti, Beyond Shannon limits: Quantum communications through quantum paths, IEEE J. Sel. Areas Commun. 41, 2707 (2023).
  101. H. Spencer-Wood, Indefinite causal key distribution, J. Phys. A: Math. Theor. 58, 495303 (2025).
  102. K. Simonov, M. Caleffi, J. Illiano, J. Romero, and A. S. Cacciapuoti, Universal quantum computation via superposed orders of single-qubit gates, arXiv:2311.13654.
  103. W.-Q. Liu, Z. Meng, B.-W. Song, J. Li, Q.-Y. Wu, X.-X. Chen, J.-Y. Hong, A.-N. Zhang, and Z.-Q. Yin, Experimentally demonstrating indefinite causal order algorithms to solve the generalized Deutsch's problem, Adv. Quantum Technol. 7, 2400181 (2024).
  104. L. Rozema, T. Strömberg, H. Cao, Y. Guo, B.-H. Liu, and P. Walther, Experimental aspects of indefinite causal order in quantum mechanics, Nat. Rev. Phys. 6, 483 (2024).
  105. B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302(R) (2016).
  106. P. C. Azado, R. Wagner, R. S. Barbosa, and E. F. Galvão, Measuring unitary invariants with the quantum switch, arXiv:2508.02345.

Outline

Information

Phys. Rev. A 112, 062435– Published 17 December, 2025
  • Received 4 June 2025
  • Accepted 13 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation