Abstract
Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.
Similar content being viewed by others
References
Alon N., Spencer J.H.: The probabilistic method. Wiley, Newyork (2000)
Lloyd S., Pagels H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186 (1988)
Hayden P., Leung D., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95 (2006)
Hayden P., Leung D.W., Shor P.W., Winter A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371 (2004)
Horodecki M., Oppenheim J., Winter A.: Quantum information can be negative. Nature 436, 673 (2005)
Hastings M.B.: A counterexample to additivity of minimum output entropy. Nature Phys. 5, 255 (2009)
Gross D., Flammia S.T., Eisert J.: Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 102, 190501 (2009)
Bremner M.J., Mora C., Winter A.: Are random pure states useful for quantum computation. Phys. Rev. Lett. 102, 190502 (2009)
Goldstein S., Lebowitz J.L., Tumulka R., Zanghi N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)
Popescu S., Short A.J., Winter A.: Entanglement and the foundations of statistical mechanics. Nature Phys. 2, 754 (2006)
Reimann P.: Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008)
Gogolin C.: Einselection without pointer states. Phys. Rev. E 81, 051127 (2010)
Srednicki M.: Chaos and quantum thermalization. Phys. Rev. E 50, 888 (1994)
Garnerone S., de Oliveira T.R., Zanardi P.: Typicality in random matrix product states. Phys. Rev. A 81, 032336 (2010)
Kollath C., Läuchli A., Altman E.: Quench dynamics and non equilibrium phase diagram of the Bose-Hubbard model. Phys. Rev. B 74, 174508 (2006)
Rigol M., Dunjko V., Yurovsky V., Olshanii M.: Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)
Cramer M., Dawson C.M., Eisert J., Osborne T.J.: Exact relaxation in a class of non-equilibrium quantum lattice systems. Phys. Rev. Lett. 100, 030602 (2008)
Linden, N., Popescu, S., Short, A.J., Winter, A.: On the speed of fluctuations around thermodynamic equilibrium. http://arXiv.org/abs/0907.1267v1 [quant-ph], 2009
Brody D.C., Hook D.W., Hughston L.P.: Quantum phase transitions without thermodynamic limits. Proc. R. Soc. A 463, 2021 (2007)
Bender C.M., Brody D.C., Hook D.W.: Solvable model of quantum microcanonical states. J. Phys. A 38, L607 (2005)
Fresch B., Moro G.J.: Typicality in ensembles of quantum states: Monte Carlo sampling versus analytical approximations. J. Phys. Chem. A 113, 14502 (2009)
Jiang, Z., Chen, Q.: Understanding Statistical Mechanics from a Quantum Point of View. In preparation
Federer H.: Geometric measure theory. Springer-Verlag, Berlin-Heidelberg-New York (1969)
Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs 89, Providence, RI: Amer. Math. Soc., 2001
Cover T.M., Thomas J.M.: Elements of information theory, Second Edition. Wiley, New York (2006)
Gromov, M.: Metric structures for Riemannian and Non-Riemannian spaces. Modern Birkhäuser Classics, Basel-Boston: Birkhäuser, 2007
Zyckowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34(35), 7111 (2001)
Hall M.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123 (1998)
Bhatia R.: Matrix analysis. Springer, Berlin-Heidelberg-New York (1997)
Santaló L.A.: Integral geometry and geometric probability. Addison-Wesley, Reading, MA (1972)
Tasaki H.: Geometry of reflective submanifolds in Riemannian symmetric spaces. J. Math. Soc. Japan 58(1), 275–297 (2006)
Schneider R., Weil W.: Stochastic and integral geometry. Springer, Reading, MA (2008)
Funano K.: Concentration of 1-Lipschitz Maps into an infinite dimensional ℓ p-ball with the ℓ q-distance function. Proc. Amer. Math. Soc. 137, 2407 (2009)
Funano K.: Observable concentration of mm-spaces into nonpositively curved manifolds. Geometriae Dedicata 127, 49 (2007)
Elstrodt J.: Maß–und Integrationstheorie. Springer, Reading, MA (1996)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Mathematics 1200. Reading, MA: Springer, 2001
Blumenson L.E.: A derivation of n-dimensional spherical coordinates.. American Mathematical Monthly 67(1), 63 (1960)
Bengtsson I., Zyczkowski K.: Geometry of quantum states - an introduction to quantum entanglement. Cambridge University Press, Cambridge (2006)
Dempster A.P., Kleyle R.M.: Distributions determined by cutting a simplex with hyperplanes. Ann. Math. Stat. 39(5), 1473 (1968)
Barvinok: Measure concentration in optimization. Springer, Reading, MA (2007)
Furuta T.: Short proof that the arithmetic mean is greater than the harmonic mean and its reverse inequality. Math Ineq and Appl. 8(4), 751 (2005)
Müller M.E.: A note on a method for generating points uniformly on N-dimensional spheres. Comm. Assoc. Comput. Mach. 2, 19 (1959)
Marsaglia G.: Choosing a point from the surface of a sphere. The Annals of Mathematical Statistics 43(2), 645 (1972)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M.B. Ruskai
Rights and permissions
About this article
Cite this article
Müller, M.P., Gross, D. & Eisert, J. Concentration of Measure for Quantum States with a Fixed Expectation Value. Commun. Math. Phys. 303, 785–824 (2011). https://doi.org/10.1007/s00220-011-1205-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1205-1