Equating quantum and thermodynamic entropy productions (Information erasure in closed system: Nature may operate twirling) Lajos Di´ osi Wigner Research Centre, Budapest 21 Sept 2016, Sopot Acknowledgements: EU COST Action MP1209 ‘Thermodynamics in Lajos Di´ osiWigner Research Centre, Budapest Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 1 / 8
A new entropy theorem 1 Microscopic reversibility 2 Mechanical friction in ideal Maxwell gas 3 Nature may forget ... 4 ‘Friction’ in abstract quantum gas 5 Summary 6 Lajos Di´ osiWigner 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 : = σ ′ ⊗ σ ⊗ ( N − 1) + σ ⊗ σ ′ ⊗ σ ⊗ ( N − 2) + · · · + σ ⊗ ( N − 1) ⊗ σ ′ � σ ′ ⊗ σ ⊗ ( N − 1) � T N Limit theorem for entropy production: � � S [ T ( σ ′ ⊗ σ ⊗ ( N − 1) )] − S [ σ ′ ⊗ σ ⊗ ( N − 1) ] = S [ σ ′ | σ ] . lim N = ∞ 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´ osiWigner 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 → U ρ U † , S [ U ρ U † ] = S [ ρ ] Experience: entropy production in large closed systems Some irreversible mechanism superseds unitary evolution. ρ → U ρ U † → 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 ∆ S thermo is calculable ⇒ ∆ S micro = S [ M ? ρ ] − S [ ρ ] is strictly given by ∆ S thermo Then you construct M ? ! Lajos Di´ osiWigner 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. Friction force: 2 ν mV v v (Epstein 1910) �� �� �� �� 1 2 �� �� �� �� 2V−v ν collision rate �� �� �� �� 1 �� �� m molecular mass �� �� �� �� �� �� V disc velocity �� �� �� �� �� �� �� �� Thermodynamic entropy- �� �� v 3 -production rate: 2 βν mV 2 kg v V� n − β m − β m − β m v )= � � 2 v 2 � � V − v 1 ) 2 �� � 2 v 2 � ρ ( k exp ⇒ ρ 1 ( v )=exp 2 (2 k � =1 exp k k ∆ S micro / collision = S [ ρ 1 ] − S [ ρ ] = 0 ∆ S thermo / collision = 2 β mV 2 Impose M ? : ρ 1 → M ? ρ 1 such that is dynamically ‘innocent’ and S [ M ? ρ 1 ] − S [ ρ ] = 2 β mV 2 Lajos Di´ osiWigner 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 ρ 1 = ( ρ 1 + ρ 2 + ... + ρ N ) / N where � �� � � − β m − β m V − v n ) 2 2 v 2 ρ n ( v )=exp 2 (2 exp k k � = n Indeed, in thermodynamic limit N → ∞ : ∆ S micro = S [ T ρ 1 ] − S [ ρ 1 ] − → 2 β mV 2 + O ( V 4 ) = ∆ S thermo + O ( V 4 ) Twirling Maxwell gas: dynamically ‘innocent’: T [ H , ρ ] = [ H , T ρ ] erases information ∆ S micro coinciding with ∆ S thermo / k B D. 2002 Lajos Di´ osiWigner 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: � ⊗ N � e − β H ≡ σ ⊗ N ρ = Z ( β ) Collision on outside field/object (cf.: ‘disk’): ρ ⇒ ρ 1 = σ ′ ⊗ σ ⊗ ( N − 1) where σ ′ = U σ U † . Identity for energy change: ∆ E = tr( H σ ′ ) − tr( H σ ) = S [ σ ′ | σ ] /β Suppose ∆ E is dissipated, then ∆ S thermo = S [ σ ′ | σ ] . Twirl T generates exactly this amount: S [ T ρ 1 ] − S [ ρ 1 ] = ∆ S thermo . � � S [ T ( σ ′ ⊗ σ ⊗ ( N − 1) )] − S [ σ ′ ⊗ σ ⊗ ( N − 1) ] = S [ σ ′ | σ ] . lim N = ∞ Conjecture D.-Feldmann-Kosloff 2006. Proof Csisz´ ar-Hiai-Petz 2007. Lajos Di´ osiWigner 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´ osi: Shannon information and rescue in friction, Physics/020638 L.Di´ osi, 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´ osi, AIP Conf.Proc. 1469 (2012) Lajos Di´ osiWigner Research Centre, Budapest Equating quantum and thermodynamic entropy productions (Information erasure in closed system: 21 Sept 2016, Sopot 8 / 8
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