Equating quantum and thermodynamic entropy
productions
(Information erasure in closed system: Nature
may operate twirling)
Lajos Diósi
Wigner Research Centre, Budapest
21 Sept 2016, Sopot
Acknowledgements: EU COST Action MP1209 ‘Thermodynamics in
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
21 (Information
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erasure in 1closed
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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 DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
21 (Information
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erasure in 2closed
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A new entropy theorem
A new entropy theorem
Product state ρ = σ 0 ⊗ σ ⊗ σ ⊗ . . . ⊗ σ entropy:
1
2
3
...
N
S[σ 0 ⊗σ ⊗(N−1) ] = S[σ 0 ] + (N −1)S[σ]
Irreversible operation twirling T :
σ 0 ⊗σ ⊗(N−1) + σ⊗σ 0 ⊗σ ⊗(N−2) + · · · + σ ⊗(N−1) ⊗σ 0
T σ 0 ⊗σ ⊗(N−1) =
N
Limit theorem for entropy production:
0
⊗(N−1)
0
⊗(N−1)
lim S[T (σ ⊗σ
)] − S[σ ⊗σ
] = S[σ 0 |σ].
N=∞
Csiszár-Hiai-Petz: We don’t see how you got the conjecture.
D.-Feldmann-Kosloff: We don’t see how you prove it.
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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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 ∆S thermo is calculable
⇒ ∆S micro = S[M? ρ] − S[ρ] is strictly given by ∆S thermo
Then you construct M? !
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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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: 2νmV
(Epstein 1910)
ν collision rate
m molecular mass
V disc velocity
Thermodynamic entropyv3
-production rate: 2βνmV 2
kg
vn
V
Q
Q
v 2 ⇒ ρ1 (v)=exp − βm
(2V−v1)2 k6=1 exp − βm
v2
ρ(v)= k exp − βm
2 k
2
2 k
v2
11
00
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00
11
00
11
00
11
00
11
00
11
00
11
00
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00
11
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11
00
11
00
11
v1
2V−v1
∆S micro /collision = S[ρ1 ] − S[ρ] = 0
∆S thermo /collision = 2βmV 2
Impose M? : ρ1 → M? ρ1 such that is dynamically ‘innocent’ and
S[M? ρ1 ] − S[ρ] = 2βmV 2
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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Nature may forget ...
Nature may forget...
which one of the N molecules has just collided: M? = T
T ρ1 = (ρ1 + ρ2 + ... + ρN )/N
where
Y βm
βm 2
2
ρn (v)=exp −
(2V−vn)
exp −
vk
2
2
k6=n
Indeed, in thermodynamic limit N → ∞:
∆S micro = S[T ρ1 ] − S[ρ1 ] −→ 2βmV 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 /kB
D. 2002
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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‘Friction’ in abstract quantum gas
‘Friction’ in abstract quantum gas
⊗N
e −βH
ρ=
≡ σ ⊗N
Z (β)
Collision on outside field/object (cf.: ‘disk’):
Initial Gibbs state:
ρ ⇒ ρ1 = σ 0 ⊗ σ ⊗(N−1)
where σ 0 = UσU † . Identity for energy change:
∆E = tr(Hσ 0 ) − tr(Hσ) = S[σ 0 |σ]/β
Suppose ∆E is dissipated, then
∆S thermo = S[σ 0 |σ].
Twirl T generates exactly this amount: S[T ρ1 ] − S[ρ1 ] = ∆S thermo .
lim S[T (σ 0 ⊗σ ⊗(N−1) )] − S[σ 0 ⊗σ ⊗(N−1) ] = S[σ 0 |σ].
N=∞
Conjecture D.-Feldmann-Kosloff 2006. Proof Csiszár-Hiai-Petz 2007.
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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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.Diósi: Shannon information and rescue in friction, Physics/020638
L.Diósi, T.Feldman, R.Kosloff, Int.J.Quant.Info 4, 99 (2006)
I. Csiszár, F.Hiai, D.Petz, J.Math.Phys. 48, 092102 (2007)
L.Diósi, AIP Conf.Proc. 1469 (2012)
Lajos DiósiWigner Research Centre, BudapestEquating quantum and thermodynamic entropy productions
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