entanglement and the foundations of statistical mechanics sandu popescu bristol university...
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Entanglement and the Foundations of Statistical Mechanics
Sandu Popescu
Bristol University Hewlett-Packard
Tony Short Bristol University
Andreas Winter Bristol University
Yakir Aharonov Tel Aviv University
Noah Linden Bristol University
Large Number of Particles
Lack of Knowledge
Use of Averages
Great Paradox
Subjective lack of knowledge hasobjective physical implications!
Macro state (determined by macroscopic parameters P,V,E)
Number of (micro) states compatible with the macrostate, N
S=lnN Entropy
dS/dE =1/T
Actual (micro) state
ENSEMBLE AVERAGE
Postulate of equal a-priory probabilities
Gibbs ensemble
TIME AVERAGE
Ergodicity
Quantum Mechanics: Objective Lack of KnowledgeYakir Aharonov
(0) (t)
S((t) ) = 0
system
environment environment
EntanglementSsystem(t) ≠ 0
Universe (big isolated system)
system environment
HR
R restriction (macroscopic parameters)
Postulate of equal a-priori probabilities: All states in HR are equally probable.
U = IR/dim R S = TrE U = TrE IR /dim R
canonical state
HU =HSHE
Universe (big isolated system)
system environment
HR
R restriction (macroscopic parameters)
The universe is in a well-defined pure state
U =U U S = TrE U = TrE U U
HU =HSHE
S≈ S for almost all U
Universe (big isolated system)
system environment
HR
R restriction (macroscopic parameters)
Postulate of equal a-priori probabilities
U = IR/dim R
S = TrE U = TrE IR /dim R
HU =HSHE
U = U U
S= TrE U = TrE U U
S≈ S for almost all U
Principle of apparent equal a-priori probabilities
Main Result 1.
|| S- S ||1 ≤ √ (dS / dEeff)
Main Result 2.
V( || S- S ||1 ≥ + √ (dS / dEeff)) ≤ V0 exp(-C dR 2)
HR
Main Tool: Levy’s Lemma
PSd
f: Sd R
V( f(P) - <f> ≥ ) ≤ V0 exp (- C (d+1) 2/2)
2=max f
f() = TrE UU- S1
M1=Tr √(MM+)
Dynamics
HR
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