X - B a t t e r i e s Majorization and Fluctuations Phys. - - PowerPoint PPT Presentation

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X - B a t t e r i e s Majorization and Fluctuations Phys. - - PowerPoint PPT Presentation

Nov 08, 2023 322 likes •660 views

X - B a t t e r i e s Majorization and Fluctuations Phys. Rev. X 6, 041017 (2016) Fluctuation Theorems for Entanglement arXiv:1709.06139 Alvaro Alhambra, Lluis Masanes, Jonathan Oppenheim, Chris Perry X - B a t t e r i e s s

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X

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a t t e r i e s

Majorization and Fluctuations

  • Phys. Rev. X 6, 041017 (2016)

Fluctuation Theorems for Entanglement arXiv:1709.06139

Alvaro Alhambra, Lluis Masanes, Jonathan Oppenheim, Chris Perry

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SLIDE 2

Carlo Sparaciari et. al. arXiv:1806.04937

1

s t

l a w

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q u a n t u m r e s

  • u

r c e t h e

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i e s

X

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a t t e r i e s

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SLIDE 3

M a x w e l l S z i l a r d L a n d a u e r B e n n e t t

4

W= k T l

  • g

2

I n f

  • r

m a t i

  • n

i s p h y s i c a l R L

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SLIDE 4

What do we mean by W=kTlog2?

(consider the limit of perfect erasure) A)We can achieve W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)Using slightly more work on average than kTlog2, enables you to sometimes gain work by erasing the bit. D)None of these statements are true. E)This quiz is undecidable.

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SLIDE 5

What do we mean by W=kTlog2?

(consider the limit of perfect erasure) A)We can achieve W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)Using slightly more work on average than kTlog2, enables you to sometimes gain work by erasing the bit D)None of these statements are true. E)This quiz is undecidable.

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SLIDE 6

What do we mean by W=kTlog2?

(consider the limit of perfect erasure) A)We can achieve W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)Using slightly more work on average than kTlog2, enables you to sometimes gain work by erasing the bit. D)None of these statements are true. E)This quiz is undecidable.

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SLIDE 7

Corrections to second law

Standard 2nd law

W ≤Δ F

∑

k=1 N β k

k ! ⟨(f s'−f s+w)

k⟩≤0

Jarzynski, Crooks

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SLIDE 8
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SLIDE 9

Fluctuating work in erasure

∑

s ,w

P(s' ,w∣s)e

β( Es'−Es+w )=1

∑

s' , w

P(s' ,w∣s)=1

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SLIDE 10

The 4 questions

1) What does W=kTlog2 mean? Average? 2) Entanglement dilution vs concentration? 3) Is a heat bath a bank? 4) What do we do about embezzlement?

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SLIDE 11

Pure state entanglement theory

entanglement concentration n m m fluctuates around

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SLIDE 12

Pure state entanglement theory

entanglement dilution n m We require Alice compresses and teleports it to Bob using m ebits.

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SLIDE 13

Pure state entanglement theory

Entanglement cycle n m Why does this dilution protocol require: But concentration has: a) Optimal dilution protocol? b) Reversibility? c) Can we characterize the fluctuations? d) Do we require many copies?

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SLIDE 14

The 4 questions

1) What does W=kTlog2 mean? Average? 2) Entanglement dilution vs concentration? 3) Is a heat bath a bank? 4) What do we do about embezzlement?

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SLIDE 15

Banks and interconversion

Can we interconvert between resources? Bank

£ $ £ $

  • Cannot get dollars/pounds for free
  • The bank fjxes an exchange rate
  • Afuer the exchange, the rate does not change

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SLIDE 16

Interconversion and banks: Landauer’s erasure

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Thermal bath

  • Energy (work) is added to the thermal bath
  • Purity (neg-entropy) is taken from the thermal bath
  • Exchange rate depends on temperature
  • The thermal bath is lefu (almost) unchanged
  • The thermal state converts work into purity with temperature as an exchange rate
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SLIDE 17

Interconversion and banks: Maxwell’s demon

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Thermal bath

  • Energy (work) is taken from the thermal bath
  • Purity (neg-entropy) is injected into the thermal bath
  • Exchange rate depends on temperature
  • The thermal bath is lefu (almost) unchanged
  • The thermal state converts purity into work with temperature as an exchange rate
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SLIDE 18

The First Law

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First Law :

  • Main system :
  • Bank : allows for exchange of resources
  • Batuery : exchange fjrst resource
  • Batuery : exchange second resource

For thermodynamics, we get:

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SLIDE 19

The 4 questions

1) What does W=kTlog2 mean? Average? 2) Entanglement dilution vs concentration? 3) Is a heat bath a bank? 4) What do we do about embezzlement?

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SLIDE 20

Asymptotic limit

Pure state entanglement theory

iff

Single copy transformations Single copy transformations

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SLIDE 21

Embezzling state

van Dam, Hayden (2002)

iff

Single copy transformations Single copy transformations

Always

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Pure state entanglement theory with an entanglement battery

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SLIDE 23

Pure state entanglement theory with an entanglement battery

iff

Single copy transformations Single copy transformations Reversible on single copy level

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SLIDE 24

Pure state entanglement theory

entanglement concentration w/ entanglement battery n m

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SLIDE 25

Pure state entanglement theory with an entanglement battery

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no work to fmuctuating work

doubly stochastic maps majorisation Gibbs-stochastic maps thermo-majorisation fmuctuating work linear program

∑

s

P(s'∣s)=1

∑

s'

P(s’∣s)=1

∑

s

P(s'∣s)e

β(E s'−E s)=1

∑

s'

P(s’∣s)=1

∑

s ,w

P(s' ,w∣s)e

β( Es'−Es+w )=1

∑

s'

P(s’∣s)=1

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SLIDE 27

Why I like resource theories

Entanglement theory

  • Majorisation criteria
  • Entanglement fmuctuations

Thermodynamics

  • Many second laws
  • Work fmuctuations

Majorisation and Fluctuations

  • Phys. Rev. X 6, 041017 (2016)
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SLIDE 28

Why I like resource theories

Entanglement theory Something new about entanglement distillation Thermodynamics Something new about Landauer erasure

Majorisation and Fluctuations

  • Phys. Rev. X 6, 041017 (2016)
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SLIDE 29

What do we mean by W=kTlog2?

(consider the limit of perfect erasure) A)We can achieve W=kTlog2 on average, but there will be fluctuations around this value. B)We can achieve perfect erasure. C)By using slightly more work on average than kTlog2, you can sometimes gain work when you erase. D)None of these statements are true. E)This quiz is undecidable.

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SLIDE 30

Fluctuating work in erasure

∑

s ,w

P(s' ,w∣s)e

β( Es'−Es+w )=1

∑

s' , w

P(s' ,w∣s)=1

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SLIDE 31

Summary

  • Majorisation fluctuation relations
  • Pure state entanglement criteria 2nd laws of thermo
  • Entanglement fluctuation theorem Work fluctuation theorem
  • Two kinds of batteries: 1st law of resource theories
  • I want an entanglement battery!
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Outlook and open questions

  • Other theories with fluctuation relations?
  • e.g. Coherence (Morris & Adesso; 1802.059191802.05919)
  • More connections between resource theories:
  • Relative entropy distance as unique measure

(Horodecki, JO; quant-ph/0207177)

  • More 1st law examples? (Sparaciari et. al.)
  • Destruction of the resource (Groisman et. al. 2005)
  • Many second laws for black holes (Alice Bernamonti et. al. 2018)