Average entropy of a subsystem over a global unitary orbit of a mixed bipartite state

  • Lin ZhangEmail author
  • Hua Xiang


We investigate the average entropy of a subsystem within a global unitary orbit of a given mixed bipartite state in the finite-dimensional space. Without working out the closed-form expression of such average entropy for the mixed state case, we provide an analytical lower bound for this average entropy. In deriving this analytical lower bound, we get some useful by-products of independent interest. We also apply these results to estimate average correlation along a global unitary orbit of a given mixed bipartite state. When the notion of von Neumann entropy is replaced by linear entropy, the similar problem can be considered also, and moreover the exact average linear entropy formula is derived for a subsystem over a global unitary orbit of a mixed bipartite state.


Quantum state Unitary orbit Average entropy Average correlation Page’s formula 



L. Zhang is supported by Natural Science Foundation of Zhejiang Province of China (LY17A010027) and also by National Natural Science Foundation of China (Nos.11301124 and 61673145). H. Xiang is supported by the National Natural Science Foundation of China (Nos.11571265 and 11471253). Michael Walter is also acknowledged for his comments on this manuscript.


  1. 1.
    Bravyi, S.: Compatibility between local and multipartite states. Quantum Inf. Comput. 4(1), 012–026 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975). doi: 10.1016/0024-3795(75)90075-0 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Christandl, M., Doran, B., Kousidis, S., Walter, M.: Eigenvalue distributions of reduced density matrices. Commun. Math. Phys. 332, 1–52 (2014). doi: 10.1007/s00220-014-2144-4 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006). doi: 10.1007/s00220-006-1554-3 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dyer, J.P.: Divergence of Lubkin’s series for a quantum subsystem’s mean entropy, arXiv:1406.5776
  7. 7.
    Foong, S.K., Kano, S.: Proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. Lett. 72, 1148 (1994). doi: 10.1103/PhysRevLett.72.1148 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gessner, M., Breuer, H.-P.: Generic features of the dynamics of complex open quantum systems: statistical approach based on averages over the unitary group. Phys. Rev. E 87, 042128 (2013). doi: 10.1103/PhysRevE.87.042128 ADSCrossRefGoogle Scholar
  9. 9.
    Giorda, P., Allegra, M.: Two-qubit correlations revisited: average mutual information, relevant (and useful) observables and an application to remote state preparation, arXiv: 1606.02197
  10. 10.
    Hiai, F., Petz, D.: Introduction to Matrix Analysis and Applications. Hindustan Book Agency, Springer, Switzerland (2014)Google Scholar
  11. 11.
    Jevtic, S., Jennings, D., Rudolph, T.: Maximally and minimally correlated states attainable within a closed evolving system. Phys. Rev. Lett. 108, 110403 (2012). doi: 10.1103/PhysRevLett.108.110403 ADSCrossRefGoogle Scholar
  12. 12.
    Jevtic, S., Jennings, D., Rudolph, T.: Quantum mutual information along unitary orbits. Phys. Rev. A 85, 052121 (2012). doi: 10.1103/PhysRevA.85.052121 ADSCrossRefGoogle Scholar
  13. 13.
    Klyachko, A.: Quantum marginal problem and \(N\)-representability. J. Phys. Conf. Ser. 36, 72–86 (2006)CrossRefGoogle Scholar
  14. 14.
    Lachal, A.: Probabilistic approach to Page’s formula for the entropy of a quantum system. Stoch. Int. J. Probab. Stoch. Process. 78, 157–178 (2006)CrossRefzbMATHGoogle Scholar
  15. 15.
    Lubkin, E.: Entropy of an \(n\)-system from its correlation with a \(k\)-reservoir. J. Math. Phys. 19(5), 1028 (1978). doi: 10.1063/1.523763 ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Modi, K., Gu, M.: Coherent and incoherent contents of correlations. Int. J. Mod. Phys. B 27, 1345027 (2012). doi: 10.1142/S0217979213450276 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oszmaniec, M., Kuś, M.: Fraction of isospectral states exhibiting quantum correlations. Phys. Rev. A 90, 010302 (2014). doi: 10.1103/PhysRevA.90.010302 ADSCrossRefGoogle Scholar
  18. 18.
    Page, D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291 (1993). doi: 10.1103/PhysRevLett.71.1291 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sen, S.: Average entropy of a quantum subsystem. Phys. Rev. Lett. 77, 1 (1996). doi: 10.1103/PhysRevLett.77.1 ADSCrossRefGoogle Scholar
  20. 20.
    Sánchez-Ruiz, J.: Simple proof of Page’s conjecture on the average entropy of a subsystem. Phys. Rev. E 52, 5653 (1995). doi: 10.1103/PhysRevE.52.5653 ADSCrossRefGoogle Scholar
  21. 21.
    Walter, M., Doran, B., Gross, D., Christandl, M.: Entanglement polytopes: multiparticle entanglement from single-particle information. Science 340, 6137 (2013). doi: 10.1126/science.1232957 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang, L.: Matrix integrals over unitary groups: an application of Schur–Weyl duality, arXiv:1408.3782v3
  23. 23.
    Zhang, L.: Average coherence and its typicality for random mixed quantum states. J. Phys. A: Math. Theor. doi: 10.1088/1751-8121/aa6179
  24. 24.
    Zhang, L., Fei, S.-M.: Quantum fidelity and relative entropy between unitary orbits. J. Phys. A Math. Theor. 47, 055301 (2014). doi: 10.1088/1751-8113/47/5/055301 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, L., Chen, L., Bu, K.: Fidelity between one bipartite quantum state and another undergoing local unitary dynamics. Quantum Inf. Process. 14, 4715 (2015). doi: 10.1007/s11128-015-1117-7 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, L., Singh, U., Pati, A.K.: Average subentropy, coherence and entanglement of random mixed quantum states. Ann. Phys. 377, 125–146 (2017). doi: 10.1016/j.aop.2016.12.024 ADSMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHangzhou Dianzi UniversityHangzhouChina
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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