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