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Equating quantum and thermodynamic entropy productions (Information erasure in closed system: Nature may operate twirling)

Lajos Di´

• si

Wigner Research Centre, Budapest 21 Sept 2016, Sopot Acknowledgements: EU COST Action MP1209 ‘Thermodynamics in

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 1 / 8

1

A new entropy theorem

2

Microscopic reversibility

3

Mechanical friction in ideal Maxwell gas

4

Nature may forget ...

5

‘Friction’ in abstract quantum gas

6

Summary

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 2 / 8

A new entropy theorem

A new entropy theorem

Product state ρ = σ′ ⊗ σ ⊗ σ ⊗ . . . ⊗ σ entropy:

1 2 3 ... N

S[σ′⊗σ⊗(N−1)] = S[σ′] + (N−1)S[σ] Irreversible operation twirling T : T

• σ′⊗σ⊗(N−1)

= σ′⊗σ⊗(N−1) + σ⊗σ′⊗σ⊗(N−2) + · · · + σ⊗(N−1)⊗σ′ N Limit theorem for entropy production: lim

N=∞

• S[T (σ′⊗σ⊗(N−1))] − S[σ′⊗σ⊗(N−1)]
• = S[σ′|σ].

Csisz´ ar-Hiai-Petz: We don’t see how you got the conjecture. D.-Feldmann-Kosloff: We don’t see how you prove it.

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 3 / 8

Microscopic reversibility

Microscopic reversibility

Theory: reversibility in closed systems ρ → UρU†, S[UρU†] = S[ρ] Experience: entropy production in large closed systems Some irreversible mechanism superseds unitary evolution. ρ → UρU† → M?ρ, S[M?ρ] > S[ρ] What can M? be? Find a system such that: microscopic dynamics U is tractable macroscopic friction force is calculabe from U ⇒ thermodynamic entropy production ∆Sthermo is calculable ⇒ ∆Smicro = S[M?ρ] − S[ρ] is strictly given by ∆Sthermo Then you construct M?!

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 4 / 8

Mechanical friction in ideal Maxwell gas

Mechanical friction in Maxwell gas

Constant force is dragging a disk at velocity V through the gas.

• v

v v

n 2 3

v

1

2V−v V kg

1

Friction force: 2νmV (Epstein 1910) ν collision rate m molecular mass V disc velocity Thermodynamic entropy-

• production rate: 2βνmV 2

ρ( v)=

kexp

• − βm

2 v 2 k

• ⇒ ρ1(

v)=exp

• − βm

2 (2

V−v1)2

k=1exp

• − βm

2 v 2 k

• ∆Smicro/collision = S[ρ1] − S[ρ] = 0

∆Sthermo/collision = 2βmV 2 Impose M? : ρ1 → M?ρ1 such that is dynamically ‘innocent’ and S[M?ρ1] − S[ρ] = 2βmV 2

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 5 / 8

Nature may forget ...

Nature may forget...

which one of the N molecules has just collided: M? = T T ρ1 = (ρ1 + ρ2 + ... + ρN)/N where ρn( v)=exp

• −βm

2 (2 V−vn)2

• k=n

exp

• −βm

2 v 2

k

• Indeed, in thermodynamic limit N → ∞:

∆Smicro = S[T ρ1]−S[ρ1] − → 2βmV 2 +O(V 4) = ∆Sthermo +O(V 4) Twirling Maxwell gas: dynamically ‘innocent’: T [H, ρ] = [H, T ρ] erases information ∆Smicro coinciding with ∆Sthermo/kB

• D. 2002

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 6 / 8

‘Friction’ in abstract quantum gas

‘Friction’ in abstract quantum gas

Initial Gibbs state: ρ = e−βH Z(β) ⊗N ≡ σ⊗N Collision on outside field/object (cf.: ‘disk’): ρ ⇒ ρ1 = σ′ ⊗ σ⊗(N−1) where σ′ = UσU†. Identity for energy change: ∆E = tr(Hσ′) − tr(Hσ) = S[σ′|σ]/β Suppose ∆E is dissipated, then ∆Sthermo = S[σ′|σ]. Twirl T generates exactly this amount: S[T ρ1] − S[ρ1] = ∆Sthermo. lim

N=∞

• S[T (σ′⊗σ⊗(N−1))] − S[σ′⊗σ⊗(N−1)]
• = S[σ′|σ].

Conjecture D.-Feldmann-Kosloff 2006. Proof Csisz´ ar-Hiai-Petz 2007.

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 7 / 8

Summary

Summary

Notorious tension: reversible micro vs. irrev. macro Case study: mechanical friction in Maxwell gas Quantitative entropic constraint on microscopic mechanism Nature may use twirl to erase information Bye-product: new quantum informatic theorem Reality: twirling local perturbation of Gibbs state (D. 2012)

L.Di´

• si: Shannon information and rescue in friction, Physics/020638

L.Di´

• si, T.Feldman, R.Kosloff, Int.J.Quant.Info 4, 99 (2006)
• I. Csisz´

ar, F.Hiai, D.Petz, J.Math.Phys. 48, 092102 (2007) L.Di´

• si, AIP Conf.Proc. 1469 (2012)

Lajos Di´

• siWigner Research Centre, Budapest

Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 8 / 8