Q u a n t u m M a r t i n g a l e T h e o r y - - PowerPoint PPT Presentation

SLIDE 1

G O N Z A L O M A N Z A N O

1 , 2

Q u a n t u m M a r t i n g a l e T h e

• r

y f

• r

E n t r

• p

y P r

• d

u c t i

• n

1 Abdus Salam ICTP, Trieste (Italy).

Workshop on Martingales in Finance and Physics ICTP, 24 May 2019

R O S A R I O F A Z I O

1 , 2

É D G A R R O L D Á N

1

2 Scuola Normale Superiore, Pisa (Italy).

SLIDE 2

*

Outline

• I

n t r

• d

u c t i

• n
• F

r

• m

s t

• c

h a s t i c t

• q

u a n t u m t h e r mo d y n a mi c s

• S

u p e r p

• s

i t i

• n

a n d c

• h

e r e n c e

• T

h e r mo d y n a mi c s a n d fm u c t u a t i

• n

s f

• r

q u a n t u m s y s t e ms

• Q

u a n t u m t r a j e c t

• r

i e s

• E

n t r

• p

y p r

• d

u c t i

• n

a n d fm u c t u a t i

• n

t h e

• r

e ms

• Q

u a n t u m Ma r t i n g a l e T h e

• r

y

• C

l a s s i c a l

• Q

u a n t u m s p l i t

• f

e n t r

• p

y p r

• d

u c t i

• n
• S

t

• p

p i n g

• t

i me s a n d fj n i t e

• t

i me i n fj mu m

• Ma

i n c

• n

c l u s i

• n

s

O u t l i n e

:

SLIDE 3

*

From stochastic to quantum thermodynamics

T h e r ma l fm u c t u a t i

• n

s

Stochastic Thermodynamics Quantum Thermodynamics

Q u a n t u m fm u c t u a t i

• n

s , c

• h

e r e n c e , e n t a n g l e me n t . . .

F u n d a me n t a l a n d p r a c t i c a l q u e s t i

• n

s :

• H
• w

h e a t , w

• r

k a n d e n t r

• p

y a r e d e fj n e d ?

• H
• w

t

• d

e fj n e e fg e c t i v e “ t r a j e c t

• r

i e s ” ?

• C

a n q u a n t u m e fg e c t s mo d i f y t h e r mo d y n a mi c b e h a v i

• r

?

• I

n fm u e n c e

• f

q u a n t u m me a s u r e me n t s ?

• H
• w

s ma l l c a n t h e r ma l ma c h i n e s b e ?

[J. Roßnagel, et al. Science (2016)]

SLIDE 4

*

Superposition and coherence

If and are possible states of a quantum system too

Quantum superposition: Classical particles:

Modern which-path experiment with stochastically arriving phthalocyanine (PcH2) molecules (one at a time) [T. Juffmann et al. Nat. Nano (2012)]

both “left” and “right” slits either “left” or “right” slits

S u p e r p

• s

i t i

• n

p r i n c i p l e :

SLIDE 5

*

Superposition and coherence

C l a s s i c a l mi x t u r e s v s . s u p e r p

• s

i t i

• n

s t a t e s :

Example: Two-level system (qubit):

with

density operator (matrix)

• Compare the following two states:

state of the system is either with probs. state of the system is COHERENCES i.e. both

SLIDE 6

*

Outline

• I

n t r

• d

u c t i

• n
• F

r

• m

s t

• c

h a s t i c t

• q

u a n t u m t h e r mo d y n a mi c s

• S

u p e r p

• s

i t i

• n

a n d c

• h

e r e n c e

• T

h e r mo d y n a mi c s a n d fm u c t u a t i

• n

s f

• r

q u a n t u m s y s t e ms

• Q

u a n t u m t r a j e c t

• r

i e s

• E

n t r

• p

y p r

• d

u c t i

• n

a n d fm u c t u a t i

• n

t h e

• r

e ms

• Q

u a n t u m Ma r t i n g a l e T h e

• r

y

• C

l a s s i c a l

• Q

u a n t u m s p l i t

• f

e n t r

• p

y p r

• d

u c t i

• n
• S

t

• p

p i n g

• t

i me s a n d fj n i t e

• t

i me i n fj mu m

• Ma

i n c

• n

c l u s i

• n

s

O u t l i n e

:

SLIDE 7

*

Quantum fluctuation theorems

F l u c t u a t i

• n

T h e

• r

e ms i n q u a n t u m s y s t e ms

• Thermodynamic quantities are defined through (projective) quantum measurements

which allow us to define “trajectories” using the measurement outcomes.

Reviews: M. C a mp i s i e t a l . R e v . Mo d . P h y s . ( 2 1 1 ) ; M. E s p

• s

i t

• e

t a l . R e v . Mo d . P h y s . ( 2 9 )

• Thermal fluctuations + Quantum fluctuations
• Useful for work Fluctuation Theorems for isolated driven quantum systems
• Scheme needs to be extended to the environment → Environmental monitoring
• Usually the environment is assumed to be a thermal reservoir
• More general environments such as finite-size and/or engineered quantum reservoirs ?

Open quantum systems?

SLIDE 8

*

Collision-like open system evolution

System interacts “sequentially” with the environment:

• Trajectories now comprise all the measurements in system and environmental ancillas:
• The continuous limit can be obtained if the following limit exist:

= finite

SLIDE 9

*

Quantum jump trajectories

Quantum-jump trajectories:

quantum jump of type k smooth evolution

Probability during any dt: Measurements backaction can be recasted as:

Example: Optical cavity

Click! trajectory average Photo-detector

SLIDE 10

*

Quantum jump trajectories

Smooth evolution (No jump) Jump of type k

Evolution under environmental monitoring S t

• c

h a s t i c S c h r ö d i n g e r e q u a t i

• n

( L a n g e v i n

• l

i k e )

Assuming an initial pure state and keeping the record of the outcomes:

The average evolution is a Lindblad master equation (Fokker-Planck-like):

Introducing Poisson increments STEADY STATE: micro-states populations/probabilities of micro-states

SLIDE 11

*

Entropy production and FT’s

• Trajectories: Initial and final measurements (system) + jumps and times (environment):
• Local detailed-balance
• For Lindblad operators coming in pairs:

with environmental record

• For any self-adjoint Lindblad operator
• Entropy production:

[ G . Ma n z a n

• ,

J . M. H

• r
• w

i t z , a n d J . M. R . P a r r

• n

d

• ,

P R X ( 2 1 8 ) ; J . M. H

• r
• w

i t z a n d J . M. R . P a r r

• n

d

• ,

N J P ( 2 1 3 ) ; J . M. H

• r
• w

i t z , P R E ( 2 1 2 ) ]

• Fluctuation theorems:

e.g. for a thermal bath: system entropy environment entropy

SLIDE 12

*

Outline

• I

n t r

• d

u c t i

• n
• F

r

• m

s t

• c

h a s t i c t

• q

u a n t u m t h e r mo d y n a mi c s

• S

u p e r p

• s

i t i

• n

a n d c

• h

e r e n c e

• T

h e r mo d y n a mi c s a n d fm u c t u a t i

• n

s f

• r

q u a n t u m s y s t e ms

• Q

u a n t u m t r a j e c t

• r

i e s

• E

n t r

• p

y p r

• d

u c t i

• n

a n d fm u c t u a t i

• n

t h e

• r

e ms

• Q

u a n t u m Ma r t i n g a l e T h e

• r

y

• C

l a s s i c a l

• Q

u a n t u m s p l i t

• f

e n t r

• p

y p r

• d

u c t i

• n
• S

t

• p

p i n g

• t

i me s a n d fj n i t e

• t

i me i n fj mu m

• Ma

i n c

• n

c l u s i

• n

s

O u t l i n e

:

SLIDE 13

*

Quantum Martingale Theory

• Does classical martingale theory for entropy production apply to quantum thermo?

average conditioned on trajectory at past times

for

[ I . N e r i , É . R

• l

d á n , a n d F . J ü l i c h e r , P R X ( 2 1 7 ) ]

• Quantum generalization becomes problematic !
• Entropy production needs

measurements on the system.

• Sometimes it is not well defined at

intermediate times

• How to make meaningful conditions on

past times ?

in a eigenstate (microstate) of the steady state [well defined without measurements] in a superposition of eigenstates (of the steady state) [EP would depend on an eventual measurement] Classical Markov

SLIDE 14

*

Classical-Quantum split

• Decomposition of the stochastic EP:

for

• “classicalization” of EP and

is an exponential martingale

• Both terms fulfill fluctuation theorems:
• Quantum fluctuations spoil the Martingale property!
• The extra term measures the entropic value of the uncertainty in :

which fulfills:

is the “average probability” when measuring

time

SLIDE 15

*

Stopping-time fluctuations and Extreme-value statistics

• Stopping-time fluctuation theorem

stochastic stopping-time max and min eigenvalues of the steady state Example: 2-level system with orthogonal jumps Minimum between first-passage time with 1 or 2 thresholds and a fixed maximum t

Modified infimum law:

either positive

• r negative
• Finite-time infimum inequality:

SLIDE 16

*

Outline

• I

n t r

• d

u c t i

• n
• F

r

• m

s t

• c

h a s t i c t

• q

u a n t u m t h e r mo d y n a mi c s

• S

u p e r p

• s

i t i

• n

a n d c

• h

e r e n c e

• T

h e r mo d y n a mi c s a n d fm u c t u a t i

• n

s f

• r

q u a n t u m s y s t e ms

• Q

u a n t u m t r a j e c t

• r

i e s

• E

n t r

• p

y p r

• d

u c t i

• n

a n d fm u c t u a t i

• n

t h e

• r

e ms

• Q

u a n t u m Ma r t i n g a l e T h e

• r

y

• C

l a s s i c a l

• Q

u a n t u m s p l i t

• f

e n t r

• p

y p r

• d

u c t i

• n
• S

t

• p

p i n g

• t

i me s a n d fj n i t e

• t

i me i n fj mu m

• Ma

i n c

• n

c l u s i

• n

s

O u t l i n e

:

SLIDE 17

*

Main conclusions

Ma i n c

• n

c l u s i

• n

s

• Stochastic thermodynamics can be extended to the quantum realm by properly

defining “trajectories”, trough quantum measurements.

• The quantum jump trajectory formalism can be employed to asses the

thermodynamics of open quantum systems beyond thermal reservoirs.

• For nonequilibrium steady states, the entropy production is not always an

exponential Martingale due to quantum fluctuations.

• A quantum martingale theory can be however developed by performing

a quantum-classical split of the entropy production.

• We obtain quantum corrections in several results for stopping times and finite-time

infimum, whose consequences are still to be fully understood.

SLIDE 18

*

Main conclusions

THANK YOU

for your attention

FOR MORE INFORMATION:

G . Ma n z a n

• .

, R . F a z i

• ,

a n d É . R

• l

d á n , a r X i v : 1 9 3 . 2 9 2 5 ( 2 1 9 ) ; [ a c c e p t e d i n P R L ]