C. Mastrodonato. Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
Associated with any density matrix $\rho$ on a Hilbert space, there
is a GAP measure, a natural probability measure $GAP(\rho)$ on the
unit sphere of the Hilbert space, having covariance $\rho$. A system
whose wave function is random with distribution $GAP(\rho)$ is a
system which, according to quantum mechanics, is in the state
$\rho$. For a system in thermal equilibrium with canonical density
matrix $\rho_\exp(-H)$, its wave function should
be regarded as random, with distribution $GAP(\rho_\beta)$ on the
unit sphere of the system's Hilbert space.
The proof of this claim is our main result. Crucial ingredients are:
(i) the general claim that for a system 1 in interaction with a much
larger system 2 and such that the composite is in a pure state
$\psi$ for which the reduced density matrix of system 1 is fixed to
be $\rho_1$, then a typical such $\psi$ yields a wave function
$\psi_1$ for system 1 that is random with distribution
$GAP(\rho_1)$, and (ii) canonical typicality, the fact that when
system 2 is a heat bath, then for a typical pure state $\psi$ of the
composite, the reduced density matrix $\rho_1$ of system 1 is
canonical.
The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R.
Tumulka and N. Zangh\`ı.
%0 Book Section
%1 statphys23_0638
%A Mastrodonato, C.
%B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics
%C Genova, Italy
%D 2007
%E Pietronero, Luciano
%E Loreto, Vittorio
%E Zapperi, Stefano
%K canonical ensemble function gap gaussian group haar measure measures mechanics on quantum statphys23 topic-1 typical unitary wave
%T Typicality of the GAP Measure
%U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=638
%X Associated with any density matrix $\rho$ on a Hilbert space, there
is a GAP measure, a natural probability measure $GAP(\rho)$ on the
unit sphere of the Hilbert space, having covariance $\rho$. A system
whose wave function is random with distribution $GAP(\rho)$ is a
system which, according to quantum mechanics, is in the state
$\rho$. For a system in thermal equilibrium with canonical density
matrix $\rho_\exp(-H)$, its wave function should
be regarded as random, with distribution $GAP(\rho_\beta)$ on the
unit sphere of the system's Hilbert space.
The proof of this claim is our main result. Crucial ingredients are:
(i) the general claim that for a system 1 in interaction with a much
larger system 2 and such that the composite is in a pure state
$\psi$ for which the reduced density matrix of system 1 is fixed to
be $\rho_1$, then a typical such $\psi$ yields a wave function
$\psi_1$ for system 1 that is random with distribution
$GAP(\rho_1)$, and (ii) canonical typicality, the fact that when
system 2 is a heat bath, then for a typical pure state $\psi$ of the
composite, the reduced density matrix $\rho_1$ of system 1 is
canonical.
The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R.
Tumulka and N. Zangh\`ı.
@incollection{statphys23_0638,
abstract = {Associated with any density matrix $\rho$ on a Hilbert space, there
is a GAP measure, a natural probability measure $GAP(\rho)$ on the
unit sphere of the Hilbert space, having covariance $\rho$. A system
whose wave function is random with distribution $GAP(\rho)$ is a
system which, according to quantum mechanics, is in the state
$\rho$. For a system in thermal equilibrium with canonical density
matrix $\rho_\beta \propto \exp(-\beta H)$, its wave function should
be regarded as random, with distribution $GAP(\rho_\beta)$ on the
unit sphere of the system's Hilbert space.
The proof of this claim is our main result. Crucial ingredients are:
(i) the general claim that for a system 1 in interaction with a much
larger system 2 and such that the composite is in a pure state
$\psi$ for which the reduced density matrix of system 1 is fixed to
be $\rho_1$, then a typical such $\psi$ yields a wave function
$\psi_1$ for system 1 that is random with distribution
$GAP(\rho_1)$, and (ii) canonical typicality, the fact that when
system 2 is a heat bath, then for a typical pure state $\psi$ of the
composite, the reduced density matrix $\rho_1$ of system 1 is
canonical.
The talk is based on joint work with S. Goldstein, J.L. Lebowitz, R.
Tumulka and N. Zangh\`\i.},
added-at = {2007-06-20T10:16:09.000+0200},
address = {Genova, Italy},
author = {Mastrodonato, C.},
biburl = {https://www.bibsonomy.org/bibtex/21ccfd7417907399e836fb10f990ff57f/statphys23},
booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano},
interhash = {9c723055af13f75459df7bd21f761387},
intrahash = {1ccfd7417907399e836fb10f990ff57f},
keywords = {canonical ensemble function gap gaussian group haar measure measures mechanics on quantum statphys23 topic-1 typical unitary wave},
month = {9-13 July},
timestamp = {2007-06-20T10:16:25.000+0200},
title = {Typicality of the GAP Measure},
url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=638},
year = 2007
}