Entanglement cost of discriminating quantum states under locality constraints (2402.18446v2)
Abstract: The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. First, we introduce an efficiently computable quantity called the spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show that it can be a lower bound for the entanglement cost of optimal discrimination by PPT POVMs. Second, we establish an upper bound on the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging this result, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
- C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, “Quantum nonlocality without entanglement,” Physical Review A, vol. 59, no. 2, pp. 1070–1091, feb 1999. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.59.1070
- D. Leung, A. Winter, and N. Yu, “LOCC protocols with bounded width per round optimize convex functions,” Reviews in Mathematical Physics, vol. 33, no. 05, p. 2150013, jan 2021. [Online]. Available: https://doi.org/10.1142%2Fs0129055x21500136
- S. Bandyopadhyay, S. Halder, and M. Nathanson, “Entanglement as a resource for local state discrimination in multipartite systems,” Physical Review A, vol. 94, no. 2, Aug. 2016. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.94.022311
- A. M. Childs, D. Leung, L. Mančinska, and M. Ozols, “A framework for bounding nonlocality of state discrimination,” Communications in Mathematical Physics, vol. 323, no. 3, pp. 1121–1153, 2013.
- S. Bandyopadhyay, “More nonlocality with less purity,” Physical Review Letters, vol. 106, no. 21, p. 210402, 2011.
- J. Calsamiglia, J. I. De Vicente, R. Muñoz-Tapia, and E. Bagan, “Local discrimination of mixed states,” Physical Review Letters, vol. 105, no. 8, pp. 1–4, 2010.
- S. Halder, M. Banik, S. Agrawal, and S. Bandyopadhyay, “Strong Quantum Nonlocality without Entanglement,” Physical Review Letters, vol. 122, no. 4, p. 40403, 2019. [Online]. Available: https://doi.org/10.1103/PhysRevLett.122.040403
- J. Walgate, A. J. Short, L. Hardy, and V. Vedral, “Local distinguishability of multipartite orthogonal quantum states,” Physical Review Letters, vol. 85, no. 23, p. 4972, 2000.
- E. Chitambar, R. Duan, and M.-H. Hsieh, “When do local operations and classical communication suffice for two-qubit state discrimination?” IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1549–1561, 2013.
- E. Chitambar and M.-H. Hsieh, “Revisiting the optimal detection of quantum information,” Physical Review A, vol. 88, no. 2, p. 020302, 2013.
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Reviews of Modern Physics, vol. 74, no. 1, pp. 145–195, mar 2002. [Online]. Available: https://doi.org/10.1103%2Frevmodphys.74.145
- R. Cleve, D. Gottesman, and H.-K. Lo, “How to share a quantum secret,” Physical Review Letters, vol. 83, no. 3, pp. 648–651, jul 1999. [Online]. Available: https://doi.org/10.1103%2Fphysrevlett.83.648
- A. Leverrier and P. Grangier, “Unconditional security proof of long-distance continuous-variable quantum key distribution with discrete modulation,” Physical Review Letters, vol. 102, no. 18, may 2009. [Online]. Available: https://doi.org/10.1103%2Fphysrevlett.102.180504
- N. Brunner, M. Navascué s, and T. Vértesi, “Dimension witnesses and quantum state discrimination,” Physical Review Letters, vol. 110, no. 15, apr 2013. [Online]. Available: https://doi.org/10.1103%2Fphysrevlett.110.150501
- M. Hendrych, R. Gallego, M. Mičuda, N. Brunner, A. Ac\́mathbf{i}n, and J. P. Torres, “Experimental estimation of the dimension of classical and quantum systems,” Nature Physics, vol. 8, no. 8, pp. 588–591, jun 2012. [Online]. Available: https://doi.org/10.1038%2Fnphys2334
- B. M. Terhal, D. P. DiVincenzo, and D. W. Leung, “Hiding bits in bell states,” Phys. Rev. Lett., vol. 86, pp. 5807–5810, Jun 2001. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.86.5807
- T. Eggeling and R. F. Werner, “Hiding classical data in multipartite quantum states,” Phys. Rev. Lett., vol. 89, p. 097905, Aug 2002. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.89.097905
- W. Matthews, S. Wehner, and A. Winter, “Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding,” Communications in Mathematical Physics, vol. 291, no. 3, pp. 813–843, aug 2009. [Online]. Available: https://doi.org/10.1007%2Fs00220-009-0890-5
- S. Bandyopadhyay, A. Cosentino, N. Johnston, V. Russo, J. Watrous, and N. Yu, “Limitations on separable measurements by convex optimization,” IEEE Transactions on Information Theory, vol. 61, no. 6, pp. 3593–3604, 2015.
- E. Chitambar and R. Duan, “Nonlocal entanglement transformations achievable by separable operations,” Physical review letters, vol. 103, no. 11, p. 110502, 2009.
- N. Yu, R. Duan, and M. Ying, “Distinguishability of quantum states by positive operator-valued measures with positive partial transpose,” IEEE Transactions on Information Theory, vol. 60, no. 4, pp. 2069–2079, 2014.
- Y. Li, X. Wang, and R. Duan, “Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario,” Physical Review A, vol. 95, no. 5, p. 052346, 2017.
- H.-C. Cheng, A. Winter, and N. Yu, “Discrimination of quantum states under locality constraints in the many-copy setting,” Communications in Mathematical Physics, pp. 1–33, 2023.
- R. Takagi and B. Regula, “General resource theories in quantum mechanics and beyond: Operational characterization via discrimination tasks,” Physical Review X, vol. 9, no. 3, sep 2019. [Online]. Available: https://doi.org/10.1103%2Fphysrevx.9.031053
- L. Lami, C. Palazuelos, and A. Winter, “Ultimate data hiding in quantum mechanics and beyond,” Communications in Mathematical Physics, vol. 361, no. 2, pp. 661–708, jun 2018. [Online]. Available: https://doi.org/10.1007%2Fs00220-018-3154-4
- S. M. Cohen, “Understanding entanglement as resource: Locally distinguishing unextendible product bases,” Phys. Rev. A, vol. 77, p. 012304, Jan 2008. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.77.012304
- S. Ghosh, G. Kar, A. Roy, A. Sen(De), and U. Sen, “Distinguishability of bell states,” Phys. Rev. Lett., vol. 87, p. 277902, Dec 2001.
- S. Bandyopadhyay and V. Russo, “Entanglement cost of discriminating noisy bell states by local operations and classical communication,” Physical Review A, vol. 104, no. 3, Sep. 2021. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.104.032429
- C. W. Helstrom, “Quantum detection and estimation theory,” Journal of Statistical Physics, vol. 1, pp. 231–252, 1969.
- E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter, “Everything you always wanted to know about locc (but were afraid to ask),” Communications in Mathematical Physics, vol. 328, no. 1, p. 303–326, Mar. 2014. [Online]. Available: http://dx.doi.org/10.1007/s00220-014-1953-9
- W. Matthews, S. Wehner, and A. Winter, “Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding,” Communications in Mathematical Physics, vol. 291, no. 3, pp. 813–843, 2009.
- Ö. Güngör and S. Turgut, “Entanglement-assisted state discrimination and entanglement preservation,” Physical Review A, vol. 94, no. 3, Sep. 2016. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.94.032330
- S. Bandyopadhyay, S. Halder, and M. Nathanson, “Optimal resource states for local state discrimination,” Physical Review A, vol. 97, no. 2, Feb. 2018. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.97.022314
- D. DiVincenzo, D. Leung, and B. Terhal, “Quantum data hiding,” IEEE Transactions on Information Theory, vol. 48, no. 3, pp. 580–598, mar 2002. [Online]. Available: https://doi.org/10.1109%2F18.985948
- M. Oszmaniec and T. Biswas, “Operational relevance of resource theories of quantum measurements,” Quantum, vol. 3, p. 133, apr 2019. [Online]. Available: https://doi.org/10.22331%2Fq-2019-04-26-133
- C. Zhu, Z. Liu, C. Zhu, and X. Wang, “Limitations of classically-simulable measurements for quantum state discrimination,” 2023.
- G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, “The resource theory of informational nonequilibrium in thermodynamics,” Physics Reports, vol. 583, p. 1–58, Jul. 2015. [Online]. Available: http://dx.doi.org/10.1016/j.physrep.2015.04.003
- G. Chiribella, F. Meng, R. Renner, and M.-H. Yung, “The nonequilibrium cost of accurate information processing,” Nature Communications, vol. 13, no. 1, p. 7155, 2022.
- K.-D. Wu, T. V. Kondra, S. Rana, C. M. Scandolo, G.-Y. Xiang, C.-F. Li, G.-C. Guo, and A. Streltsov, “Resource theory of imaginarity: Quantification and state conversion,” Physical Review A, vol. 103, no. 3, Mar. 2021. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.103.032401
- S. Rana, “Negative eigenvalues of partial transposition of arbitrary bipartite states,” Physical Review A, vol. 87, no. 5, may 2013. [Online]. Available: https://doi.org/10.1103%2Fphysreva.87.054301
- M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.
- ——, “Graph implementations for nonsmooth convex programs,” in Recent Advances in Learning and Control, ser. Lecture Notes in Control and Information Sciences, V. Blondel, S. Boyd, and H. Kimura, Eds. Springer-Verlag Limited, 2008, pp. 95–110, http://stanford.edu/~boyd/graph_dcp.html.
- T. M. Inc., “Matlab version: 9.13.0 (r2022b),” Natick, Massachusetts, United States, 2022. [Online]. Available: https://www.mathworks.com
- J. Bavaresco, M. Murao, and M. T. Quintino, “Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination,” Physical Review Letters, vol. 127, no. 20, Nov. 2021. [Online]. Available: http://dx.doi.org/10.1103/PhysRevLett.127.200504