Bounds in Sequential Unambiguous Discrimination of Multiple Pure Quantum States
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AbstractSequential methods for quantum hypothesis testing offer significant advantages over fixed-length approaches, which rely on a predefined number of state copies. Despite their potential, these methods remain underexplored for unambiguous discrimination. In this work, we derive performance bounds for such methods when applied to the discrimination of a set of pure states. The performance is evaluated based on the expected number of copies required. We establish a lower bound applicable to any sequential method and an upper bound on the optimal sequential method. The upper bound is derived using a novel and simple non-adaptive method. Importantly, the gap between these bounds is minimal, scaling logarithmically with the number of distinct states.► BibTeX data@article{PerezGuijarro2025boundsinsequential, doi = {10.22331/q-2025-11-20-1919}, url = {https://doi.org/10.22331/q-2025-11-20-1919}, title = {Bounds in {S}equential {U}nambiguous {D}iscrimination of {M}ultiple {P}ure {Q}uantum {S}tates}, author = {P{\'{e}}rez-Guijarro, Jordi and Pag{\`{e}}s-Zamora, Alba and Fonollosa, Javier R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1919}, month = nov, year = {2025} }► References [1] Ryuji Takagi, Suguru Endo, Shintaro Minagawa, and Mile Gu. ``Fundamental limits of quantum error mitigation''. npj Quantum Information 8, 114 (2022). https://doi.org/10.1038/s41534-022-00618-z [2] Antonio Acín, Joonwoo Bae, E Bagan, M Baig, Ll Masanes, and Ramon Muñoz-Tapia. ``Secrecy properties of quantum channels''. Physical Review A—Atomic, Molecular, and Optical Physics 73, 012327 (2006). https://doi.org/10.1103/PhysRevA.73.012327 [3] Nicolas Gisin and Rob Thew. ``Quantum communication''. Nature photonics 1, 165–171 (2007). https://doi.org/10.1038/nphoton.2007.22 [4] C. W. Helstrom. ``Quantum detection and estimation''. Academic Press. New York (1976). https://doi.org/10.1007/BF01007479 [5] K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete. ``Discriminating states: The quantum chernoff bound''. Phys. Rev. Lett. 98, 160501 (2007). https://doi.org/10.1103/PhysRevLett.98.160501 [6] M. Nussbaum and A. Szkoła. ``The chernoff lower bound for symmetric quantum hypothesis testing''. The Annals of Statistics 37, 1040–1057 (2009). https://doi.org/10.1214/08-AOS593 [7] Ke Li. ``Discriminating quantum states: The multiple chernoff distance''. Ann. Statist. 44(4): 1661-1679 (2016). https://doi.org/10.1214/16-AOS1436 [8] Ulrike Herzog and János A Bergou. ``Optimum unambiguous discrimination of two mixed quantum states''. Physical Review A—Atomic, Molecular, and Optical Physics 71, 050301 (2005). https://doi.org/10.1103/PhysRevA.71.050301 [9] Gregg Jaeger and Abner Shimony. ``Optimal distinction between two non-orthogonal quantum states''. Physics Letters A 197, 83–87 (1995). https://doi.org/10.1016/0375-9601(94)00919-G [10] Igor D Ivanovic. ``How to differentiate between non-orthogonal states''. Physics Letters A 123, 257–259 (1987). https://doi.org/10.1016/0375-9601(87)90222-2 [11] Dennis Dieks. ``Overlap and distinguishability of quantum states''. Physics Letters A 126, 303–306 (1988). https://doi.org/10.1016/0375-9601(88)90840-7 [12] Asher Peres. ``How to differentiate between non-orthogonal states''. Physics Letters A 128, 19 (1988). https://doi.org/10.1016/0375-9601(88)91034-1 [13] János A Bergou, Ulrike Futschik, and Edgar Feldman. ``Optimal unambiguous discrimination of pure quantum states''. Physical review letters 108, 250502 (2012). https://doi.org/10.1103/PhysRevLett.108.250502 [14] Gael Sentís, Esteban Martínez-Vargas, and Ramon Muñoz-Tapia. ``Online identification of symmetric pure states''. Quantum 6, 658 (2022). https://doi.org/10.22331/q-2022-02-21-658 [15] Sergei Slussarenko, Morgan M Weston, Jun-Gang Li, Nicholas Campbell, Howard M Wiseman, and Geoff J Pryde. ``Quantum state discrimination using the minimum average number of copies''. Physical review letters 118, 030502 (2017). https://doi.org/10.1103/PhysRevLett.118.030502 [16] Esteban Martínez Vargas, Christoph Hirche, Gael Sentís, Michalis Skotiniotis, Marta Carrizo, Ramon Muñoz-Tapia, and John Calsamiglia. ``Quantum sequential hypothesis testing''. Physical review letters 126, 180502 (2021). https://doi.org/10.1103/PhysRevLett.126.180502 [17] Yonglong Li, Vincent YF Tan, and Marco Tomamichel. ``Optimal adaptive strategies for sequential quantum hypothesis testing''. Communications in Mathematical Physics 392, 993–1027 (2022). https://doi.org/10.1007/s00220-022-04362-5 [18] Jordi Pérez-Guijarro, Alba Pagès-Zamora, and Javier Rodríguez Fonollosa. ``Quantum multiple hypothesis testing based on a sequential discarding scheme''. IEEE access 10, 13813–13826 (2022). https://doi.org/10.1109/ACCESS.2022.3143706 [19] Anthony Chefles. ``Unambiguous discrimination between linearly independent quantum states''. Physics Letters A 239, 339–347 (1998). https://doi.org/10.1016/S0375-9601(98)00064-4 [20] J. Calsamiglia, R. Muñoz Tapia, Ll. Masanes, A. Acin, and E. Bagan. ``Quantum chernoff bound as a measure of distinguishability between density matrices: Application to qubit and gaussian states''. Phys. Rev. A 77, 032311 (2008). https://doi.org/10.1103/PhysRevA.77.032311Cited byCould not fetch Crossref cited-by data during last attempt 2025-11-20 16:54:33: Could not fetch cited-by data for 10.22331/q-2025-11-20-1919 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-11-20 16:54:33: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractSequential methods for quantum hypothesis testing offer significant advantages over fixed-length approaches, which rely on a predefined number of state copies. Despite their potential, these methods remain underexplored for unambiguous discrimination. In this work, we derive performance bounds for such methods when applied to the discrimination of a set of pure states. The performance is evaluated based on the expected number of copies required. We establish a lower bound applicable to any sequential method and an upper bound on the optimal sequential method. The upper bound is derived using a novel and simple non-adaptive method. Importantly, the gap between these bounds is minimal, scaling logarithmically with the number of distinct states.► BibTeX data@article{PerezGuijarro2025boundsinsequential, doi = {10.22331/q-2025-11-20-1919}, url = {https://doi.org/10.22331/q-2025-11-20-1919}, title = {Bounds in {S}equential {U}nambiguous {D}iscrimination of {M}ultiple {P}ure {Q}uantum {S}tates}, author = {P{\'{e}}rez-Guijarro, Jordi and Pag{\`{e}}s-Zamora, Alba and Fonollosa, Javier R.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1919}, month = nov, year = {2025} }► References [1] Ryuji Takagi, Suguru Endo, Shintaro Minagawa, and Mile Gu. ``Fundamental limits of quantum error mitigation''. npj Quantum Information 8, 114 (2022). https://doi.org/10.1038/s41534-022-00618-z [2] Antonio Acín, Joonwoo Bae, E Bagan, M Baig, Ll Masanes, and Ramon Muñoz-Tapia. ``Secrecy properties of quantum channels''. Physical Review A—Atomic, Molecular, and Optical Physics 73, 012327 (2006). https://doi.org/10.1103/PhysRevA.73.012327 [3] Nicolas Gisin and Rob Thew. ``Quantum communication''. Nature photonics 1, 165–171 (2007). https://doi.org/10.1038/nphoton.2007.22 [4] C. W. Helstrom. ``Quantum detection and estimation''. Academic Press. New York (1976). https://doi.org/10.1007/BF01007479 [5] K. M. R. Audenaert, J. Calsamiglia, R. Muñoz Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete. ``Discriminating states: The quantum chernoff bound''. Phys. Rev. Lett. 98, 160501 (2007). https://doi.org/10.1103/PhysRevLett.98.160501 [6] M. Nussbaum and A. Szkoła. ``The chernoff lower bound for symmetric quantum hypothesis testing''. The Annals of Statistics 37, 1040–1057 (2009). https://doi.org/10.1214/08-AOS593 [7] Ke Li. ``Discriminating quantum states: The multiple chernoff distance''. Ann. Statist. 44(4): 1661-1679 (2016). https://doi.org/10.1214/16-AOS1436 [8] Ulrike Herzog and János A Bergou. ``Optimum unambiguous discrimination of two mixed quantum states''. Physical Review A—Atomic, Molecular, and Optical Physics 71, 050301 (2005). https://doi.org/10.1103/PhysRevA.71.050301 [9] Gregg Jaeger and Abner Shimony. ``Optimal distinction between two non-orthogonal quantum states''. Physics Letters A 197, 83–87 (1995). https://doi.org/10.1016/0375-9601(94)00919-G [10] Igor D Ivanovic. ``How to differentiate between non-orthogonal states''. Physics Letters A 123, 257–259 (1987). https://doi.org/10.1016/0375-9601(87)90222-2 [11] Dennis Dieks. ``Overlap and distinguishability of quantum states''. Physics Letters A 126, 303–306 (1988). https://doi.org/10.1016/0375-9601(88)90840-7 [12] Asher Peres. ``How to differentiate between non-orthogonal states''. Physics Letters A 128, 19 (1988). https://doi.org/10.1016/0375-9601(88)91034-1 [13] János A Bergou, Ulrike Futschik, and Edgar Feldman. ``Optimal unambiguous discrimination of pure quantum states''. Physical review letters 108, 250502 (2012). https://doi.org/10.1103/PhysRevLett.108.250502 [14] Gael Sentís, Esteban Martínez-Vargas, and Ramon Muñoz-Tapia. ``Online identification of symmetric pure states''. Quantum 6, 658 (2022). https://doi.org/10.22331/q-2022-02-21-658 [15] Sergei Slussarenko, Morgan M Weston, Jun-Gang Li, Nicholas Campbell, Howard M Wiseman, and Geoff J Pryde. ``Quantum state discrimination using the minimum average number of copies''. Physical review letters 118, 030502 (2017). https://doi.org/10.1103/PhysRevLett.118.030502 [16] Esteban Martínez Vargas, Christoph Hirche, Gael Sentís, Michalis Skotiniotis, Marta Carrizo, Ramon Muñoz-Tapia, and John Calsamiglia. ``Quantum sequential hypothesis testing''. Physical review letters 126, 180502 (2021). https://doi.org/10.1103/PhysRevLett.126.180502 [17] Yonglong Li, Vincent YF Tan, and Marco Tomamichel. ``Optimal adaptive strategies for sequential quantum hypothesis testing''. Communications in Mathematical Physics 392, 993–1027 (2022). https://doi.org/10.1007/s00220-022-04362-5 [18] Jordi Pérez-Guijarro, Alba Pagès-Zamora, and Javier Rodríguez Fonollosa. ``Quantum multiple hypothesis testing based on a sequential discarding scheme''. IEEE access 10, 13813–13826 (2022). https://doi.org/10.1109/ACCESS.2022.3143706 [19] Anthony Chefles. ``Unambiguous discrimination between linearly independent quantum states''. Physics Letters A 239, 339–347 (1998). https://doi.org/10.1016/S0375-9601(98)00064-4 [20] J. Calsamiglia, R. Muñoz Tapia, Ll. Masanes, A. Acin, and E. Bagan. ``Quantum chernoff bound as a measure of distinguishability between density matrices: Application to qubit and gaussian states''. Phys. Rev. A 77, 032311 (2008). https://doi.org/10.1103/PhysRevA.77.032311Cited byCould not fetch Crossref cited-by data during last attempt 2025-11-20 16:54:33: Could not fetch cited-by data for 10.22331/q-2025-11-20-1919 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-11-20 16:54:33: No response from ADS or unable to decode the received json data when getting the list of citing works.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.