Concentration of Measure for Quantum States with a Fixed Expectation Value

Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors that have a fixed expectation value with respect to H . Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 303; no. 3; pp. 785 - 824
Main Authors: Müller, Markus P., Gross, David, Eisert, Jens
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.05.2011
Springer
Subjects:
Classical and Quantum Gravitation
Complex Systems
Exact sciences and technology
Mathematical analysis
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Measure and integration
Operator theory
Other topics in mathematical methods in physics
Physics
Physics and Astronomy
Quantum Physics
Real functions
Relativity Theory
Sciences and techniques of general use
Theoretical
Reduce Density Matrix
Gibbs State
Isoperimetric Inequality
Quantum State
Pure Quantum State
Mathematical manifolds
Gauss method
Continuous function
Spheres
Continuous set
Quantum mechanics
Distribution function
Quantum information
Proof theory
Statistical mechanics
Mathematical physics
Quantum theory
Information theory
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors that have a fixed expectation value 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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-011-1205-1