Purification complexity without purifications

Abstract

We generalize the Fubini-Study method for pure-state complexity to generic quantum states by taking Bures metric or quantum Fisher information metric (QFIM) on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purification. We also find the purification complexity is non-increasing under any trace-preserving quantum operations. We also study the mixed Gaussian states as an example to explicitly illustrate our conclusions for purification complexity.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, vol. 931, Springer (2017) [DOI] [arXiv:1609.01287] [INSPIRE].

  3. [3]

    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    D. Harlow, TASI Lectures on the Emergence of Bulk Physics in AdS/CFT, PoS TASI2017 (2018) 002 [arXiv:1802.01040] [INSPIRE].

  9. [9]

    L. Susskind, Three Lectures on Complexity and Black Holes, arXiv:1810.11563 [INSPIRE].

  10. [10]

    R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [Addendum ibid. 64 (2016) 44] [arXiv:1403.5695] [INSPIRE].

  12. [12]

    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, Quant. Inf. Comput. 8 (2008) 861.

  15. [15]

    M.A. Nielsen, A geometric approach to quantum circuit lower bounds, Quant. Inf. Comput. 6 (2006) 213.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602 [arXiv:1707.08582] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  17. [17]

    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. [18]

    B. Czech, Einstein Equations from Varying Complexity, Phys. Rev. Lett. 120 (2018) 031601 [arXiv:1706.00965] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  19. [19]

    M. Guo, J. Hernandez, R.C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [arXiv:1807.07677] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. [20]

    L. Hackl and R.C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [arXiv:1803.10638] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. [21]

    R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory, Phys. Rev. D 98 (2018) 126001 [arXiv:1801.07620] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. [22]

    A. Bhattacharyya, A. Shekar and A. Sinha, Circuit complexity in interacting QFTs and RG flows, JHEP 10 (2018) 140 [arXiv:1808.03105] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. [23]

    S. Chapman et al., Complexity and entanglement for thermofield double states, SciPost Phys. 6 (2019) 034 [arXiv:1810.05151] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. [24]

    T. Ali, A. Bhattacharyya, S. Shajidul Haque, E.H. Kim and N. Moynihan, Time Evolution of Complexity: A Critique of Three Methods, JHEP 04 (2019) 087 [arXiv:1810.02734] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  25. [25]

    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP 11 (2017) 097 [arXiv:1706.07056] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  26. [26]

    A. Bhattacharyya, P. Caputa, S.R. Das, N. Kundu, M. Miyaji and T. Takayanagi, Path-Integral Complexity for Perturbed CFTs, JHEP 07 (2018) 086 [arXiv:1804.01999] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  27. [27]

    H.A. Camargo, P. Caputa, D. Das, M.P. Heller and R. Jefferson, Complexity as a novel probe of quantum quenches: universal scalings and purifications, Phys. Rev. Lett. 122 (2019) 081601 [arXiv:1807.07075] [INSPIRE].

    ADS  Article  Google Scholar 

  28. [28]

    P. Caputa and J.M. Magan, Quantum Computation as Gravity, Phys. Rev. Lett. 122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    H.A. Camargo, M.P. Heller, R. Jefferson and J. Knaute, Path integral optimization as circuit complexity, Phys. Rev. Lett. 123 (2019) 011601 [arXiv:1904.02713] [INSPIRE].

    ADS  Article  Google Scholar 

  30. [30]

    S. Chapman and H.Z. Chen, Complexity for Charged Thermofield Double States, arXiv:1910.07508 [INSPIRE].

  31. [31]

    M. Doroudiani, A. Naseh and R. Pirmoradian, Complexity for Charged Thermofield Double States, JHEP 01 (2020) 120 [arXiv:1910.08806] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  32. [32]

    P. Caputa, M. Miyaji, T. Takayanagi and K. Umemoto, Holographic Entanglement of Purification from Conformal Field Theories, Phys. Rev. Lett. 122 (2019) 111601 [arXiv:1812.05268] [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    J. Erdmenger, M. Gerbershagen and A.-L. Weigel, Complexity measures from geometric actions on Virasoro and Kac-Moody orbits, JHEP 11 (2020) 003 [arXiv:2004.03619] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    M. Guo, Z.-Y. Fan, J. Jiang, X. Liu and B. Chen, Circuit complexity for generalized coherent states in thermal field dynamics, Phys. Rev. D 101 (2020) 126007 [arXiv:2004.00344] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. [35]

    M. Flory and M.P. Heller, Complexity and Conformal Field Theory, Phys. Rev. Res. 2 (2020) 043438 [arXiv:2005.02415] [INSPIRE].

    Article  Google Scholar 

  36. [36]

    C.A. Agón, M. Headrick and B. Swingle, Subsystem Complexity and Holography, JHEP 02 (2019) 145 [arXiv:1804.01561] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  37. [37]

    E. Caceres, S. Chapman, J.D. Couch, J.P. Hernández, R.C. Myers and S.-M. Ruan, Complexity of Mixed States in QFT and Holography, JHEP 03 (2020) 012 [arXiv:1909.10557] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  38. [38]

    I. Bengtsson and K. Życzkowski, Geometry of quantum states: an introduction to quantum entanglement, Cambridge university press (2017).

  39. [39]

    D. Chruscinski and A. Jamiolkowski, Geometric phases in classical and quantum mechanics, vol. 36, Springer Science & Business Media (2012).

  40. [40]

    M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000).

    Google Scholar 

  41. [41]

    A. Uhlmann, The “transition probability” in the state space of a ∗-algebra, Rept. Math. Phys. 9 (1976) 273 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  42. [42]

    S.-J. GU, Fidelity approach to quantum phase transitions, Int. J. Mod. Phys. B 24 (2010) 4371 [arXiv:0811.3127].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. [43]

    J. Watrous, The theory of quantum information, Cambridge University Press (2018).

  44. [44]

    M.M. Wilde, Quantum information theory, Cambridge University Press (2013).

  45. [45]

    R. Jozsa, Fidelity for mixed quantum states, J. Mod. Opt. 41 (1994) 2315.

    ADS  MathSciNet  MATH  Article  Google Scholar 

  46. [46]

    H. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa and B. Schumacher, Noncommuting mixed states cannot be broadcast, Phys. Rev. Lett. 76 (1996) 2818 [quant-ph/9511010] [INSPIRE].

    ADS  Article  Google Scholar 

  47. [47]

    M.A. Nielsen, The Entanglement fidelity and quantum error correction, quant-ph/9606012.

  48. [48]

    D. Bures, An extension of kakutani’s theorem on infinite product measures to the tensor product of semifinite w -algebras, Trans. Am. Math. Soc. 135 (1969) 199.

    MathSciNet  MATH  Google Scholar 

  49. [49]

    J. Liu, H. Yuan, X.-M. Lu and X. Wang, Quantum Fisher information matrix and multiparameter estimation, J. Phys. A 53 (2020) 023001 [arXiv:1907.08037] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  50. [50]

    A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, S.-M. Ruan and J. Simón, First Law of Holographic Complexity, Phys. Rev. Lett. 123 (2019) 081601 [arXiv:1903.04511] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  51. [51]

    A. Bernamonti, F. Galli, J. Hernandez, R.C. Myers, S.-M. Ruan and J. Simón, Aspects of The First Law of Complexity, arXiv:2002.05779 [INSPIRE].

  52. [52]

    J. Twamley, Bures and statistical distance for squeezed thermal states, J. Phys. A 29 (1996) 3723 [quant-ph/9603019] [INSPIRE].

  53. [53]

    V. Link and W.T. Strunz, Geometry of gaussian quantum states, J. Phys. A 48 (2015) 275301.

    ADS  MathSciNet  MATH  Article  Google Scholar 

  54. [54]

    G. Di Giulio and E. Tonni, Complexity of mixed Gaussian states from Fisher information geometry, JHEP 12 (2020) 101 [arXiv:2006.00921] [INSPIRE].

    ADS  Article  Google Scholar 

  55. [55]

    M.G. Paris, Quantum estimation for quantum technology, Int. J. Quant. Inf. 7 (2009) 125.

    MATH  Article  Google Scholar 

  56. [56]

    M. Hübner, Explicit computation of the bures distance for density matrices, Phys. Lett. A 163 (1992) 239.

    ADS  MathSciNet  Article  Google Scholar 

  57. [57]

    A.R. Brown and L. Susskind, Complexity geometry of a single qubit, Phys. Rev. D 100 (2019) 046020 [arXiv:1903.12621] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  58. [58]

    L. Banchi, P. Giorda and P. Zanardi, Quantum information-geometry of dissipative quantum phase transitions, Phys. Rev. E 89 (2014) 022102.

    ADS  Article  Google Scholar 

  59. [59]

    L. Banchi, S.L. Braunstein and S. Pirandola, Quantum fidelity for arbitrary gaussian states, Phys. Rev. Lett. 115 (2015) 260501.

    ADS  Article  Google Scholar 

  60. [60]

    A. Carollo, B. Spagnolo and D. Valenti, Uhlmann curvature in dissipative phase transitions, Sci. Rep. 8 (2018) 9852.

    ADS  Article  Google Scholar 

  61. [61]

    H. Nha and H.J. Carmichael, Distinguishing two single-mode gaussian states by homodyne detection: An information-theoretic approach, Phys. Rev. A 71 (2005) 032336.

    ADS  Article  Google Scholar 

  62. [62]

    J. Kirklin, The Holographic Dual of the Entanglement Wedge Symplectic Form, JHEP 01 (2020) 071 [arXiv:1910.00457] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  63. [63]

    N. Lashkari and M. Van Raamsdonk, Canonical Energy is Quantum Fisher Information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  64. [64]

    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].

    ADS  Article  Google Scholar 

  65. [65]

    A. Trivella, Holographic Computations of the Quantum Information Metric, Class. Quant. Grav. 34 (2017) 105003 [arXiv:1607.06519] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  66. [66]

    D. Bak and A. Trivella, Quantum Information Metric on× Sd−1, JHEP 09 (2017) 086 [arXiv:1707.05366] [INSPIRE].

    ADS  Article  Google Scholar 

  67. [67]

    B. Czech, L. Lamprou, S. Mccandlish and J. Sully, Modular Berry Connection for Entangled Subregions in AdS/CFT, Phys. Rev. Lett. 120 (2018) 091601 [arXiv:1712.07123] [INSPIRE].

    ADS  Article  Google Scholar 

  68. [68]

    M. Alishahiha and A. Faraji Astaneh, Holographic Fidelity Susceptibility, Phys. Rev. D 96 (2017) 086004 [arXiv:1705.01834] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  69. [69]

    S. Banerjee, J. Erdmenger and D. Sarkar, Connecting Fisher information to bulk entanglement in holography, JHEP 08 (2018) 001 [arXiv:1701.02319] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  70. [70]

    M. Moosa and I. Shehzad, Is volume the holographic dual of fidelity susceptibility?, arXiv:1809.10169 [INSPIRE].

  71. [71]

    B. Czech, L. Lamprou and L. Susskind, Entanglement Holonomies, arXiv:1807.04276 [INSPIRE].

  72. [72]

    A. Belin, A. Lewkowycz and G. Sárosi, The boundary dual of the bulk symplectic form, Phys. Lett. B 789 (2019) 71 [arXiv:1806.10144] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  73. [73]

    Y. Suzuki, T. Takayanagi and K. Umemoto, Entanglement Wedges from the Information Metric in Conformal Field Theories, Phys. Rev. Lett. 123 (2019) 221601 [arXiv:1908.09939] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  74. [74]

    J. Erdmenger, K.T. Grosvenor and R. Jefferson, Information geometry in quantum field theory: lessons from simple examples, SciPost Phys. 8 (2020) 073 [arXiv:2001.02683] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  75. [75]

    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  76. [76]

    M. Botta-Cantcheff, P. Martínez and G.A. Silva, On excited states in real-time AdS/CFT, JHEP 02 (2016) 171 [arXiv:1512.07850] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  77. [77]

    D. Marolf, O. Parrikar, C. Rabideau, A. Izadi Rad and M. Van Raamsdonk, From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories, JHEP 06 (2018) 077 [arXiv:1709.10101] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  78. [78]

    R. Arias, M. Botta-Cantcheff, P.J. Martinez and J.F. Zarate, Modular Hamiltonian for holographic excited states, Phys. Rev. D 102 (2020) 026021 [arXiv:2002.04637] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  79. [79]

    M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  80. [80]

    D. Carmi, R.C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [arXiv:1612.00433] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  81. [81]

    O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  82. [82]

    E. Cáceres, J. Couch, S. Eccles and W. Fischler, Holographic Purification Complexity, Phys. Rev. D 99 (2019) 086016 [arXiv:1811.10650] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  83. [83]

    M. Alishahiha, K. Babaei Velni and M.R. Mohammadi Mozaffar, Black hole subregion action and complexity, Phys. Rev. D 99 (2019) 126016 [arXiv:1809.06031] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  84. [84]

    P. Braccia, A.L. Cotrone and E. Tonni, Complexity in the presence of a boundary, JHEP 02 (2020) 051 [arXiv:1910.03489] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  85. [85]

    R. Abt, J. Erdmenger, M. Gerbershagen, C.M. Melby-Thompson and C. Northe, Holographic Subregion Complexity from Kinematic Space, JHEP 01 (2019) 012 [arXiv:1805.10298] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  86. [86]

    A. Bhattacharya, K.T. Grosvenor and S. Roy, Entanglement Entropy and Subregion Complexity in Thermal Perturbations around Pure-AdS Spacetime, Phys. Rev. D 100 (2019) 126004 [arXiv:1905.02220] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  87. [87]

    T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].

    ADS  Article  Google Scholar 

  88. [88]

    H. Hirai, K. Tamaoka and T. Yokoya, Towards Entanglement of Purification for Conformal Field Theories, PTEP 2018 (2018) 063B03 [arXiv:1803.10539] [INSPIRE].

  89. [89]

    S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, arXiv:1905.00577 [INSPIRE].

  90. [90]

    C. Weedbrook et al., Gaussian quantum information, Rev. Mod. Phys. 84 (2012) 621.

    ADS  Article  Google Scholar 

  91. [91]

    A. Ferraro, S. Olivares and M.G. Paris, Gaussian states in continuous variable quantum information, quant-ph/0503237.

  92. [92]

    A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, CRC Press (2017).

  93. [93]

    H. Scutaru, Fidelity for displaced squeezed states and the oscillator semigroup, J. Phys. A 31 (1998) 3659 [quant-ph/9708013] [INSPIRE].

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Ruan, SM. Purification complexity without purifications. J. High Energ. Phys. 2021, 92 (2021). https://doi.org/10.1007/JHEP01(2021)092

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Keywords

  • AdS-CFT Correspondence
  • Gauge-gravity correspondence