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

Q u a n t u m M a r t i n g a l e T h e o r y f o r E n t r o p y P r o d u c t i o n R O S A R I O F A Z I O G O N Z A L O M A N Z A N O 1 , 2 É D G A R R O L D Á N 1 , 2 1 1 Abdus Salam ICTP, Trieste (Italy). 2 Scuola Normale Superiore, Pisa (Italy). Workshop on Martingales in Finance and Physics ICTP, 24 May 2019

* Outline O u t l i n e : ● I n t r o d u c t i o n ● F r o m s t o c h a s t i c t o 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 o s i t i o n a n d c o 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 o n s f o 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 o r i e s ● E n t r o p y p r o d u c t i o n a n d fm u c t u a t i o n t h e o r e ms ● Q u a n t u m Ma r t i n g a l e T h e o r y ● C l a s s i c a l - Q u a n t u m s p l i t o f e n t r o p y p r o d u c t i o n ● S t o 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 o n c l u s i o n s

* From stochastic to quantum thermodynamics Q u a n t u m fm u c t u a t i o n s , T h e r ma l fm u c t u a t i o n s c o h e r e n c e , e n t a n g l e me n t . . . Quantum Thermodynamics Stochastic Thermodynamics 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 o n s : ● H o w h e a t , w o r k a n d e n t r o p y a r e d e fj n e d ? ● H o w t o d e fj n e e fg e c t i v e “ t r a j e c t o 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 o r ? ● I n fm u e n c e o f q u a n t u m me a s u r e me n t s ? ● H o 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)]

* Superposition and coherence S u p e r p o s i t i o n p r i n c i p l e : and are possible states of a quantum system If too Quantum superposition: Classical particles: both “left” and “right” slits either “left” or “right” slits Modern which-path experiment with stochastically arriving phthalocyanine (PcH2) molecules (one at a time) [T. Juffmann et al. Nat. Nano (2012)]

* 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 o s i t i o n s t a t e s : Example: Two-level system (qubit): density operator (matrix) ● Compare the following two states: state of the system is either with probs. with state of the system is i.e. both COHERENCES

* Outline O u t l i n e : ● I n t r o d u c t i o n ● F r o m s t o c h a s t i c t o 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 o s i t i o n a n d c o 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 o n s f o 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 o r i e s ● E n t r o p y p r o d u c t i o n a n d fm u c t u a t i o n t h e o r e ms ● Q u a n t u m Ma r t i n g a l e T h e o r y ● C l a s s i c a l - Q u a n t u m s p l i t o f e n t r o p y p r o d u c t i o n ● S t o 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 o n c l u s i o n s

* Quantum fluctuation theorems F l u c t u a t i o n T h e o r e ms i n q u a n t u m s y s t e ms Thermal fluctuations + Quantum fluctuations ● Thermodynamic quantities are defined through (projective) quantum measurements ● which allow us to define “trajectories” using the measurement outcomes. Useful for work Fluctuation Theorems for isolated driven quantum systems ● Open 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 ? ● Reviews: M. C a mp i s i R e v . Mo d . P h y s . ( 2 0 1 1 ) ; M. E s p o s i t o R e v . Mo d . P h y s . ( 2 0 0 9 ) e t a l . e t a l .

* 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

* Quantum jump trajectories Quantum-jump trajectories: Probability during any dt: Measurements backaction can be recasted as: smooth evolution quantum jump of type k Example : Optical cavity Click! trajectory average Photo-detector

* Quantum jump trajectories Evolution under environmental monitoring Assuming an initial pure state and keeping the record of the outcomes: S t o c h a s t i c S c h r ö d i n g e r e q u a t i o n ( L a n g e v i n - l i k e ) Introducing Poisson increments Smooth evolution (No jump) Jump of type k The average evolution is a Lindblad master equation (Fokker-Planck-like): STEADY STATE: micro-states populations/probabilities of micro-states

* Entropy production and FT’s Trajectories: Initial and final measurements (system) + jumps and times (environment): ● with environmental record environment Entropy production: system ● entropy entropy Local detailed-balance ● e.g. for a thermal bath: For Lindblad operators coming in pairs: ● For any self-adjoint Lindblad operator ● Fluctuation theorems: ● [ G . Ma n z a n o , J . M. H o r o w i t z , a n d J . M. R . P a r r o n d o , P R X ( 2 0 1 8 ) ; ] J . M. H o r o w i t z a n d J . M. R . P a r r o n d o , N J P ( 2 0 1 3 ) ; J . M. H o r o w i t z , P R E ( 2 0 1 2 )

* Outline O u t l i n e : ● I n t r o d u c t i o n ● F r o m s t o c h a s t i c t o 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 o s i t i o n a n d c o 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 o n s f o 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 o r i e s ● E n t r o p y p r o d u c t i o n a n d fm u c t u a t i o n t h e o r e ms ● Q u a n t u m Ma r t i n g a l e T h e o r y ● C l a s s i c a l - Q u a n t u m s p l i t o f e n t r o p y p r o d u c t i o n ● S t o 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 o n c l u s i o n s

* Quantum Martingale Theory Does classical martingale theory for entropy production apply to quantum thermo? ● for average conditioned on trajectory at past times [ I . N e r i , É . R o l d á n , a n d F . J ü l i c h e r , P R X ( 2 0 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 superposition of eigenstates (of the steady state) [EP would depend on an eventual measurement] in a eigenstate (microstate) of the steady state [well defined without measurements] Classical Markov

* Classical-Quantum split Quantum fluctuations spoil the Martingale property! ● for The extra term measures the entropic value of the uncertainty in : ● which fulfills: is the “average probability” when measuring Decomposition of the stochastic EP: ● “classicalization” of EP and ● is an exponential martingale Both terms fulfill fluctuation theorems: ● time

* Stopping-time fluctuations and Extreme-value statistics Stopping-time fluctuation theorem ● either positive or negative stochastic stopping-time Example: 2-level system with orthogonal jumps Minimum between first-passage time with 1 or 2 thresholds and a fixed maximum t Finite-time infimum inequality: ● Modified infimum law: max and min eigenvalues of the steady state