Skip to main content
SpringerLink
Log in

Eigenvalue Distributions of Reduced Density Matrices

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices. As a corollary, by taking the distribution’s support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem. We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically. This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ruskai M.B.: N-representability problem: conditions on geminals. Phys. Rev. 183, 129–141 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  2. Coleman, A.J., Yukalov, V.I.: Reduced Density Matrices: Coulson’s Challenge. Lecture Notes in Chemistry, vol. 72. Springer, Berlin (2000)

  3. Stillinger, F.H.: Mathematical Challenges from Theoretical/Computational Chemistry. National Academy Press, Atlanta (1995)

  4. Liu, Y.-K.: Consistency of local density matrices is QMA-complete. In: Proceedings of RANDOM, pp. 438–449 (2006)

  5. Liu Y.-K., Christandl M., Verstraete F.: Quantum computational complexity of the N-representability problem: QMA complete. Phys. Rev. Lett. 98, 110503 (2007)

    Article  ADS  Google Scholar 

  6. Klyachko, A.: Quantum marginal problem and representations of the symmetric group. arXiv:quant-ph/0409113 (2004)

  7. Daftuar S., Hayden P.: Quantum state transformations and the Schubert calculus. Ann. Phys. 315, 80–122 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  8. Klyachko A.: Quantum marginal problem and N-representability. J. Phys. Conf. Ser. 36, 72–86 (2006)

    Article  ADS  Google Scholar 

  9. Christandl M., Mitchison G.: The spectra of quantum states and the Kronecker coefficients of the symmetric group. Commun. Math. Phys. 261, 789–797 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Coleman A.J.: Structure of fermion density matrices. Rev. Mod. Phys. 35, 668–686 (1963)

    Article  ADS  Google Scholar 

  11. Borland R.E., Dennis K.: The conditions on the one-matrix for three-body fermion wavefunctions with one-rank equal to six. J. Phys. B 5, 7–15 (1972)

    Article  ADS  Google Scholar 

  12. Ruskai M.B.: Connecting N-representability to Weyl’s problem: the one-particle density matrix for n = 3 and r = 6. J. Phys. A 40, F961–F967 (2007)

  13. Klyachko A., Altunbulak M.: The Pauli principle revisited. Commun. Math. Phys. 282, 287–322 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Klyachko, A.: The Pauli exclusion principle and beyond. arXiv:0904.2009 (2009)

  15. Higuchi, A.: On the one-particle reduced density matrix of a pure three-qutrit quantum state. arXiv:quant-ph/0309186v2 (2003)

  16. Higuchi A., Sudbery A., Szulc J.: One-qubit reduced states of a pure many-qubit state: polygon inequalities. Phys. Rev. Lett. 90, 107902 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. Bravyi S.: Requirements for compatibility between local and multipartite quantum states. Quantum Inf. Comput. 4, 012–026 (2004)

    MathSciNet  Google Scholar 

  18. Eisert J., Tyc T., Rudolph T., Sanders B.C.: Gaussian quantum marginal problem. Commun. Math. Phys. 280, 263–280 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Walter M., Doran B., Gross D., Christandl M.: Entanglement polytopes: multiparticle entanglement from single-particle information. Science 340, 1205–1208 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  20. Huang, K.: Statistical Mechanics. Wiley, New York (1990)

  21. Popescu S., Short A.J., Winter A.: Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754–758 (2006)

    Article  Google Scholar 

  22. Lloyd S.: Excuse our ignorance. Nat. Phys. 2, 727–728 (2006)

    Article  Google Scholar 

  23. Goldstein S., Lebowitz J.L., Tumulka R., Zanhi N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  24. Lloyd S., Pagels H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186–213 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  25. Berenstein A., Sjamaar R.: Coadjoint orbits, moment polytopes, and the Hilbert–Mumford criterion. J. Am. Math. Soc. 13, 433–466 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Ressayre N.: Geometric invariant theory and the generalized eigenvalue problem. Invent. Math. 180, 389–441 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Lubkin E.: Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028–1031 (1978)

    Article  ADS  MATH  Google Scholar 

  28. Page D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291–1294 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  29. Page, D.N.: Black hole information. In: Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics. American Mathematical Society, Providence (1994)

  30. Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007, 120 (2007)

    Article  MathSciNet  Google Scholar 

  31. Heckman G.J.: Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups. Invent. Math. 67, 333–356 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  32. Guillemin V., Sternberg S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515–538 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, London (1984)

  34. Guillemin V., Lerman E., Sternberg S.: On the Kostant multiplicity formula. J. Geom. Phys. 5, 721–750 (1988)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  35. Guillemin, V., Lerman, E., Sternberg, S.: Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press, London (1996)

  36. Guillemin V., Prato E. Heckman, Kostant, and Steinberg formulas for symplectic manifolds. Adv. Math. 82, 160–179 (1990)

  37. Weyl H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912)

    Article  MATH  MathSciNet  Google Scholar 

  38. Helmke U., Rosenthal J.: Eigenvalue inequalities and Schubert calculus. Math. Nachr. 171, 207–225 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  39. Klyachko A.: Stable vector bundles and Hermitian operators. Sel. Math. New Ser. 4, 419–445 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  40. Knutson A., Tao T.: The honeycomb model of \({{\mathrm{GL}}_n({\mathbb{C}})}\) tensor products I: proof of the saturation conjecture. J. Am. Math. Soc. 12, 1055–1090 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  41. Fulton W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc. 37, 209–249 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Knutson A., Tao T.: Honeycombs and sums of Hermitian matrices. Not. Am. Math. Soc. 38, 175–186 (2001)

    MathSciNet  Google Scholar 

  43. Knutson A., Tao T., Woodward C.: The honeycomb model of \({{\mathrm{GL}}_n({\mathbb{C}})}\) tensor products II: puzzles determine facets of the Littlewood–Richardson cone. J. Am. Math. Soc. 17, 19–48 (2003)

    Article  MathSciNet  Google Scholar 

  44. Dooley A.H., Repka J., Wildberger N.J.: Sums of adjoint orbits. Linear Multilinear Algebra 36, 79–101 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  45. Frumkin A., Goldberger A.: On the distribution of the spectrum of the sum of two hermitian or real symmetric matrices. Adv. Appl. Math. 37, 268–286 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  47. Boysal A., Vergne M.: Paradan’s wall crossing formula for partition functions and Khovanskii–Pukhlikov differential operators. Ann. l’Inst. Fourier 59, 1715–1752 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  48. Sjamaar R.: Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. 141, 87–129 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  49. Meinrenken E.: On Riemann–Roch formulas for multiplicities. J. Am. Math. Soc. 9, 373–389 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  50. Meinrenken E., Sjamaar R.: Singular reduction and quantization. Topology 38, 699–762 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  51. Vergne M.: Quantization of algebraic cones and Vogan’s conjecture. Pac. J. Math. 182, 113–135 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  52. Fulton, W.: Young Tableaux. Student Texts. London Mathematical Society (1997)

  53. Mulmuley K., Sohoni M.: Geometric complexity theory I: an approach to the P vs. NP and related problems. SIAM J. Comput. 31, 496–526 (2001)

  54. Mulmuley K., Sohoni M.: Geometric complexity theory II: towards explicit obstructions for embeddings among class varieties. SIAM J. Comput. 38, 1175–1206 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  55. Mulmuley, K.: Geometric complexity theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry. Technical report, Computer Science Department, The University of Chicago (2007)

  56. Bürgisser P., Landsberg J.M., Manivel L., Weyman J.: An overview of mathematical issues arising in the geometric complexity theory approach to VP ≠  VNP. SIAM J. Comput. 40, 1179–1209 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  57. Christandl M., Harrow A.W., Mitchison G.: On nonzero Kronecker coefficients and their consequences for spectra. Commun. Math. Phys. 270, 575–585 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  58. Knutson, A.: Schubert calculus and quantum information. In: Quantum Marginals and Density Matrices Workshop, Field Institute, Toronto (2009)

  59. Bürgisser P., Christandl M., Ikenmeyer C.: Nonvanishing of Kronecker coefficients for rectangular shapes. Adv. Math. 227, 2082–2091 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  60. Bürgisser P., Christandl M., Ikenmeyer C.: Even partitions in plethysms. J. Algebra 328, 322–329 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  61. Lidskii B.V.: Spectral polyhedron of a sum of two Hermitian matrices. Funct. Anal. Appl. 16, 139–140 (1982)

    Article  MathSciNet  Google Scholar 

  62. Knutson A.: The symplectic and algebraic geometry of Horn’s problem. Linear Algebra Appl. 319, 61–81 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  63. Christandl, M.: A quantum information-theoretic proof of the relation between Horn’s Problem and the Littlewood–Richardson coefficients. In: Proceedings of Computability in Europe: CiE 2008. Lecture Notes in Computer Science, vol. 5028, pp. 120–128. Springer, Berlin (2008)

  64. Okounkov, A.: Why would multiplicities be log-concave? The orbit method in geometry and physics (Marseilk, 2000). Progress in Mathematics, vol. 213, Birkhauser, Boston, pp. 329–347 (2003).

  65. Barvinok A.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19, 769–779 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  66. Carter, R.W., Segal, G., MacDonald, I.G.: Lectures on Lie groups and Lie algebras. London Mathematical Society (1995)

  67. Kirillov, Jr. A.: An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics. Cambridge University Press, London (2008)

  68. Cannas da Silva, A.: Lectures on Symplectic Geometry, 2nd edn. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2008)

  69. Guillemin V., Sternberg S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  70. Kirwan F.: Convexity properties of the moment mapping, III. Invent. Math. 77, 547–552 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  71. Guillemin, V., Sjamaar, R.: Convexity Properties of Hamiltonian Group Actions. American Mathematical Society, Providence (2005)

  72. Lerman E., Meinrenken E., Tolman S., Woodward C.: Non-abelian convexity by symplectic cuts. Topology 37, 245–259 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  73. Duistermaat J.J., Heckman G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69, 259–268 (1982)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  74. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (2003)

  75. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys and Monographs, vol. 14. American Mathematical Society, Providence, revised edition (1977)

  76. Woodhouse, N.M.J.: Geometric Quantization, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford/New York (1992)

  77. Venuti, L.C., Zanardi, P.: Probability density of quantum expectation values. arXiv:1202.4810 (2012)

  78. Zyczkowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34, 7111–7125 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  79. Hayden P., Leung D., Shor P.W., Winter A.: Randomizing quantum states: constructions and applications. Commun. Math. Phys. 250, 371–391 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  80. Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  81. Aubrun, G., Szarek, S., Ye, D.: Entanglement thresholds for random induced states. arXiv:1106.2264 (2011)

  82. Aubrun G., Szarek S., Ye D.: Phase transitions for random states and a semi-circle law for the partial transpose. Phys. Rev. A (Rapid Communications) 85, 030302 (2012)

    Article  ADS  Google Scholar 

  83. Collins B., Nechita I., Ye D.: The absolute positive partial transpose property for random induced states. Random Matrices Theory Appl. 01, 1250002 (2012)

    Article  MathSciNet  Google Scholar 

  84. Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, 453–472 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  85. Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255–257 (2009)

    Article  Google Scholar 

  86. Aubrun G., Szarek S., Werner E.: Hasting’s additivity counterexample via Dvoretzky’s theorem. Commun. Math. Phys. 305, 85–97 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  87. Christandl M., Winter A.: “Squashed entanglement”—an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  88. Woodward C.T.: Localization for the norm-square of the moment map and the two-dimensional Yang–Mills integral. J. Symplectic Geom. 3, 17–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  89. Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes. Princeton University Press, New Jersey (1984)

  90. Barvinok A.: Computing the volume, counting integral points, and exponential sums. Discret. Comput. Geom. 10, 123–141 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  91. Verdoolaege S., Seghir R., Beyls K., Loechner V., Bruynooghe M.: Counting integer points in parametric polytopes using Barvinok’s rational functions. Algorithmica 48, 37–66 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  92. Verdoolaege, S., Bruynooghe, M.: Algorithms for weighted counting over parametric polytopes: a survey and a practical comparison. ITSL, pp. 60–66 (2008)

  93. Greenberger, D.M., Horne, M.A., Zeilinger A.: Going beyond Bell’s Theorem. In Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, pp. 69–72. Kluwer, Dordrecht (1989)

  94. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)

  95. Müller M.P., Dahlsten O.C.O., Vedral V.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories. Commun. Math. Phys. 316(2), 441–487 (2012)

    Article  ADS  MATH  Google Scholar 

  96. Kirillov A.A.: Merits and demerits of the orbit method. Bull. Am. Math. Soc. 36, 433–488 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  97. Brion, M.: Sur l’image de l’application moment. In Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol. 1296, pp. 177–192. Springer, Berlin (1987)

  98. Steinberg R.: A general Clebsch–Gordan theorem. Bull. Am. Math. Soc. 67, 406–407 (1961)

    Article  MATH  Google Scholar 

  99. Knapp, A.: Lie Groups: Beyond an Introduction, 2nd edition. Progress in Mathematics, vol. 140. Birkhäuser, Boston (2002)

  100. Barvinok, A., Pommersheim J.E.: An Algorithmic Theory of Lattice Points in Polyhedra. New Perspectives in Algebraic Combinatorics, vol. 38. MSRI Publications. Cambridge University Press, London (1999)

  101. Christandl, M., Doran, B., Walter, M.: Computing multiplicities of Lie group representations. In: Proceedings of 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pp. 639–648. IEEE Computer Society (2012)

  102. Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer, Berlin (2009)

  103. MacDonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs (1995)

  104. Springer, T.A.: Invariant Theory. Lecture Notes in Mathematics, vol. 585. Springer, Berlin (1977)

  105. Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, Berlin (2002)

  106. Osgood W.F.: Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, second Series. Ann. Math. 3, 25–34 (1901)

    Article  MATH  MathSciNet  Google Scholar 

  107. Beardon A.F., Minda D.: On the pointwise limit of complex analytic functions. Am. Math. Mon. 110, 289–297 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  108. Prodinger H.: On the moments of a distribution defined by the Gaussian polynomials. J. Stat. Plan. Inference 119, 237–239 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  109. Panny W.: A note on the higher moments of the expected behavior of straight insertion sort. Inf. Process. Lett. 22, 175–177 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093, Zurich, Switzerland

    Matthias Christandl, Stavros Kousidis & Michael Walter

  2. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Matthias Christandl

  3. Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zurich, Switzerland

    Brent Doran

  4. Institute of Physics, University of Freiburg, Rheinstrasse 10, 79104, Freiburg, Germany

    Stavros Kousidis

  5. Rosenthalstr. 17, 53859, Niederkassel, Germany

    Stavros Kousidis

Authors
  1. Matthias Christandl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brent Doran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stavros Kousidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michael Walter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Christandl.

Additional information

Communicated by M. B. Ruskai

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Christandl, M., Doran, B., Kousidis, S. et al. Eigenvalue Distributions of Reduced Density Matrices. Commun. Math. Phys. 332, 1–52 (2014). https://doi.org/10.1007/s00220-014-2144-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2144-4

Keywords

Search

Navigation