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.
Similar content being viewed by others
References
Ruskai M.B.: N-representability problem: conditions on geminals. Phys. Rev. 183, 129–141 (1969)
Coleman, A.J., Yukalov, V.I.: Reduced Density Matrices: Coulson’s Challenge. Lecture Notes in Chemistry, vol. 72. Springer, Berlin (2000)
Stillinger, F.H.: Mathematical Challenges from Theoretical/Computational Chemistry. National Academy Press, Atlanta (1995)
Liu, Y.-K.: Consistency of local density matrices is QMA-complete. In: Proceedings of RANDOM, pp. 438–449 (2006)
Liu Y.-K., Christandl M., Verstraete F.: Quantum computational complexity of the N-representability problem: QMA complete. Phys. Rev. Lett. 98, 110503 (2007)
Klyachko, A.: Quantum marginal problem and representations of the symmetric group. arXiv:quant-ph/0409113 (2004)
Daftuar S., Hayden P.: Quantum state transformations and the Schubert calculus. Ann. Phys. 315, 80–122 (2004)
Klyachko A.: Quantum marginal problem and N-representability. J. Phys. Conf. Ser. 36, 72–86 (2006)
Christandl M., Mitchison G.: The spectra of quantum states and the Kronecker coefficients of the symmetric group. Commun. Math. Phys. 261, 789–797 (2006)
Coleman A.J.: Structure of fermion density matrices. Rev. Mod. Phys. 35, 668–686 (1963)
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)
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)
Klyachko A., Altunbulak M.: The Pauli principle revisited. Commun. Math. Phys. 282, 287–322 (2008)
Klyachko, A.: The Pauli exclusion principle and beyond. arXiv:0904.2009 (2009)
Higuchi, A.: On the one-particle reduced density matrix of a pure three-qutrit quantum state. arXiv:quant-ph/0309186v2 (2003)
Higuchi A., Sudbery A., Szulc J.: One-qubit reduced states of a pure many-qubit state: polygon inequalities. Phys. Rev. Lett. 90, 107902 (2003)
Bravyi S.: Requirements for compatibility between local and multipartite quantum states. Quantum Inf. Comput. 4, 012–026 (2004)
Eisert J., Tyc T., Rudolph T., Sanders B.C.: Gaussian quantum marginal problem. Commun. Math. Phys. 280, 263–280 (2008)
Walter M., Doran B., Gross D., Christandl M.: Entanglement polytopes: multiparticle entanglement from single-particle information. Science 340, 1205–1208 (2013)
Huang, K.: Statistical Mechanics. Wiley, New York (1990)
Popescu S., Short A.J., Winter A.: Entanglement and the foundations of statistical mechanics. Nat. Phys. 2, 754–758 (2006)
Lloyd S.: Excuse our ignorance. Nat. Phys. 2, 727–728 (2006)
Goldstein S., Lebowitz J.L., Tumulka R., Zanhi N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)
Lloyd S., Pagels H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186–213 (1988)
Berenstein A., Sjamaar R.: Coadjoint orbits, moment polytopes, and the Hilbert–Mumford criterion. J. Am. Math. Soc. 13, 433–466 (2000)
Ressayre N.: Geometric invariant theory and the generalized eigenvalue problem. Invent. Math. 180, 389–441 (2010)
Lubkin E.: Entropy of an n-system from its correlation with a k-reservoir. J. Math. Phys. 19, 1028–1031 (1978)
Page D.N.: Average entropy of a subsystem. Phys. Rev. Lett. 71, 1291–1294 (1993)
Page, D.N.: Black hole information. In: Proceedings of the 5th Canadian Conference on General Relativity and Relativistic Astrophysics. American Mathematical Society, Providence (1994)
Hayden P., Preskill J.: Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007, 120 (2007)
Heckman G.J.: Projections of orbits and asymptotic behaviour of multiplicities for compact connected Lie groups. Invent. Math. 67, 333–356 (1982)
Guillemin V., Sternberg S.: Geometric quantization and multiplicities of group representations. Invent. Math. 67, 515–538 (1982)
Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, London (1984)
Guillemin V., Lerman E., Sternberg S.: On the Kostant multiplicity formula. J. Geom. Phys. 5, 721–750 (1988)
Guillemin, V., Lerman, E., Sternberg, S.: Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press, London (1996)
Guillemin V., Prato E. Heckman, Kostant, and Steinberg formulas for symplectic manifolds. Adv. Math. 82, 160–179 (1990)
Weyl H.: Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–479 (1912)
Helmke U., Rosenthal J.: Eigenvalue inequalities and Schubert calculus. Math. Nachr. 171, 207–225 (1995)
Klyachko A.: Stable vector bundles and Hermitian operators. Sel. Math. New Ser. 4, 419–445 (1998)
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)
Fulton W.: Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Am. Math. Soc. 37, 209–249 (2000)
Knutson A., Tao T.: Honeycombs and sums of Hermitian matrices. Not. Am. Math. Soc. 38, 175–186 (2001)
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)
Dooley A.H., Repka J., Wildberger N.J.: Sums of adjoint orbits. Linear Multilinear Algebra 36, 79–101 (1993)
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)
Harish-Chandra: Differential operators on a semisimple Lie algebra. Am. J. Math. 79, 87–120 (1957)
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)
Sjamaar R.: Holomorphic slices, symplectic reduction and multiplicities of representations. Ann. Math. 141, 87–129 (1995)
Meinrenken E.: On Riemann–Roch formulas for multiplicities. J. Am. Math. Soc. 9, 373–389 (1996)
Meinrenken E., Sjamaar R.: Singular reduction and quantization. Topology 38, 699–762 (1999)
Vergne M.: Quantization of algebraic cones and Vogan’s conjecture. Pac. J. Math. 182, 113–135 (1998)
Fulton, W.: Young Tableaux. Student Texts. London Mathematical Society (1997)
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)
Mulmuley K., Sohoni M.: Geometric complexity theory II: towards explicit obstructions for embeddings among class varieties. SIAM J. Comput. 38, 1175–1206 (2008)
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)
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)
Christandl M., Harrow A.W., Mitchison G.: On nonzero Kronecker coefficients and their consequences for spectra. Commun. Math. Phys. 270, 575–585 (2007)
Knutson, A.: Schubert calculus and quantum information. In: Quantum Marginals and Density Matrices Workshop, Field Institute, Toronto (2009)
Bürgisser P., Christandl M., Ikenmeyer C.: Nonvanishing of Kronecker coefficients for rectangular shapes. Adv. Math. 227, 2082–2091 (2011)
Bürgisser P., Christandl M., Ikenmeyer C.: Even partitions in plethysms. J. Algebra 328, 322–329 (2011)
Lidskii B.V.: Spectral polyhedron of a sum of two Hermitian matrices. Funct. Anal. Appl. 16, 139–140 (1982)
Knutson A.: The symplectic and algebraic geometry of Horn’s problem. Linear Algebra Appl. 319, 61–81 (2000)
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)
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).
Barvinok A.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19, 769–779 (1994)
Carter, R.W., Segal, G., MacDonald, I.G.: Lectures on Lie groups and Lie algebras. London Mathematical Society (1995)
Kirillov, Jr. A.: An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics. Cambridge University Press, London (2008)
Cannas da Silva, A.: Lectures on Symplectic Geometry, 2nd edn. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2008)
Guillemin V., Sternberg S.: Convexity properties of the moment mapping. Invent. Math. 67, 491–513 (1982)
Kirwan F.: Convexity properties of the moment mapping, III. Invent. Math. 77, 547–552 (1984)
Guillemin, V., Sjamaar, R.: Convexity Properties of Hamiltonian Group Actions. American Mathematical Society, Providence (2005)
Lerman E., Meinrenken E., Tolman S., Woodward C.: Non-abelian convexity by symplectic cuts. Topology 37, 245–259 (1998)
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)
Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (2003)
Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys and Monographs, vol. 14. American Mathematical Society, Providence, revised edition (1977)
Woodhouse, N.M.J.: Geometric Quantization, 2nd edn. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, Oxford/New York (1992)
Venuti, L.C., Zanardi, P.: Probability density of quantum expectation values. arXiv:1202.4810 (2012)
Zyczkowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34, 7111–7125 (2001)
Hayden P., Leung D., Shor P.W., Winter A.: Randomizing quantum states: constructions and applications. Commun. Math. Phys. 250, 371–391 (2004)
Hayden P., Leung D.W., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95–117 (2006)
Aubrun, G., Szarek, S., Ye, D.: Entanglement thresholds for random induced states. arXiv:1106.2264 (2011)
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)
Collins B., Nechita I., Ye D.: The absolute positive partial transpose property for random induced states. Random Matrices Theory Appl. 01, 1250002 (2012)
Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, 453–472 (2004)
Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255–257 (2009)
Aubrun G., Szarek S., Werner E.: Hasting’s additivity counterexample via Dvoretzky’s theorem. Commun. Math. Phys. 305, 85–97 (2011)
Christandl M., Winter A.: “Squashed entanglement”—an additive entanglement measure. J. Math. Phys. 45, 829–840 (2004)
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)
Kirwan, F.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Mathematical Notes. Princeton University Press, New Jersey (1984)
Barvinok A.: Computing the volume, counting integral points, and exponential sums. Discret. Comput. Geom. 10, 123–141 (1993)
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)
Verdoolaege, S., Bruynooghe, M.: Algorithms for weighted counting over parametric polytopes: a survey and a practical comparison. ITSL, pp. 60–66 (2008)
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)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, 2nd edn. Wiley, New York (1971)
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)
Kirillov A.A.: Merits and demerits of the orbit method. Bull. Am. Math. Soc. 36, 433–488 (1999)
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)
Steinberg R.: A general Clebsch–Gordan theorem. Bull. Am. Math. Soc. 67, 406–407 (1961)
Knapp, A.: Lie Groups: Beyond an Introduction, 2nd edition. Progress in Mathematics, vol. 140. Birkhäuser, Boston (2002)
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)
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)
Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Springer, Berlin (2009)
MacDonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs (1995)
Springer, T.A.: Invariant Theory. Lecture Notes in Mathematics, vol. 585. Springer, Berlin (1977)
Kac, V., Cheung, P.: Quantum Calculus. Universitext. Springer, Berlin (2002)
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)
Beardon A.F., Minda D.: On the pointwise limit of complex analytic functions. Am. Math. Mon. 110, 289–297 (2003)
Prodinger H.: On the moments of a distribution defined by the Gaussian polynomials. J. Stat. Plan. Inference 119, 237–239 (2004)
Panny W.: A note on the higher moments of the expected behavior of straight insertion sort. Inf. Process. Lett. 22, 175–177 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. B. Ruskai
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2144-4