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 vertical bar psi >> that have a fixed expectation value < psi vertical bar H vertical bar psi > = 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.
Author details: | Markus P. Müller, David Gross, Jens Eisert |
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DOI: | https://doi.org/10.1007/s00220-011-1205-1 |
ISSN: | 0010-3616 |
Title of parent work (English): | Communications in mathematical physics |
Publisher: | Springer |
Place of publishing: | New York |
Publication type: | Article |
Language: | English |
Year of first publication: | 2011 |
Publication year: | 2011 |
Release date: | 2017/03/26 |
Volume: | 303 |
Issue: | 3 |
Number of pages: | 40 |
First page: | 785 |
Last Page: | 824 |
Funding institution: | EU; EURYI |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
Peer review: | Referiert |