Sang Wook Kim (Pusan N. Univ) Collaborators Simone de Liberato - PowerPoint PPT Presentation

Sang Wook Kim (Pusan N. Univ)

Collaborators Simone de Liberato (Univ. of Southampton) Hee Jun Jeon (PNU) Takahiro Sagawa (Univ. of Tokyo) Jung Jun Park (Singapore Natl. U.) Masahito Ueda Kang-Hwan Kim (Univ. of Tokyo) (KAIST)

Maxwell’s Demon Now let us suppose … that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the 2 nd law of thermodynamics. - J. C. Maxwell (1871)

Szilard’s engine (1929)   W pdV T

Flow of entropy ln 2 0 S 0 S 0 ln 2 ln 2 increase decrease ln 2 ln 2 S 0 - ln 2 ln 2 0 S 0

Information heat engine     ( : ) W F kTI S M M S Toyabe, Sagawa, Ueda, Muneyuki & Sano, Sagawa & Ueda, PRL (2008) Nature Phys. (2010)

Quantum dynamical demon?

Thermodynamic work in Q-world     E dP dU P dE n n n n n n      TdS dU dW H E n n n    ln S k P P T n n n   q-thermodynamic heat U E n P n   dQ E n dP n   n E e n  n in equilibrium P n Z q-thermodynamic work partition function    dW P n dE  n   ( ) dU E dP P dE n n n n n Kieu, PRL (2004) n

Inserting a wall    dW P n dE n n Inserting a wall is considered as an isothermal process.

Adiabatic process for inserting a wall     0 0 dP dQ E n dP T should n n   be changed n   0 dQ E n dP 0 dP n n quantum adiabatic n    T  E ? e T n T  0 P n Z The final state should be in non-equilibrium, so that the irreversible process inevitably occurs in isothermal expansion.

Q-work in an isothermal process   E n e  X   2 W dE n Z X 1 n   ln Z E  X  2 n kT dE   E X X 1 n n     ln ( ) ln ( ) kT Z X Z X 2 1 Helmholtz free energy difference (Note) Due to isothermal process, we don’t have to consider a full density matrix.

Thermodynamic process Expansion T Removal Insertion T T Measurement

Q-work of q-Szilard engine    W W W W exp tot ins rem   N f      m ln kT f   m *   f  0 m m * f f m m m ( ) Z l  m eq * f m m ( ) Z l eq N   m m ( ) ( ) Z l Z l eq n eq  0 n *  N  1 f m  1 m SWK, Sagawa, De Liberato & Ueda, PRL (2011)

Single particle q-Szilard engine W tot k T B  p 1 p W      tot p ln p ( 1 p ) ln( 1 p ) k T B The 3 rd law of thermodynamics   0 as 0 S T K.-H. Kim & SWK, J. Korean Phys. Soc. (2012)

Two particle q-Szilard engine I   N f      m ln W kT f   tot m *   f  0 m m L l  2 0  * 1 f f 0  * f f f 1 1 1 2  f  * 1 f f 2 0

Two particle q-Szilard engine II   2 0 ln W tot kTf f 0 Bosons Fermions (spinless)     2 0 for 0 W T     ln 3 for 0 tot W kT T tot      3 ln 2 for W kT T tot    ln 2 for W kT T tot Both and 1   as 0 f T are prohibited due to Pauli 0 3 exclusion principle in the low T. 1    (cf) classical work as f T 0 4 1   W classical  2 ln 2 kT 2

Two particle q-Szilard engine Bosons ln 2 Classical W tot kT Fermions SWK, Sagawa, De Liberato & Ueda, PRL (2011)

Irreversible process I Time-forward Time-backward x 1 x 1 x 0 L 0 1 L x 1 x 1 Inherently (cf) Murashita, Funo & Ueda, PRE (2014) irreversible! Ashida, Funo, Murashita & Ueda, PRE (2014)

Irreversible process II T     N N        *   ln ln W kT f f f f tot m m m m       0 0 m m

(Option 1) Make the protocol reversible Horowitz & Parrond, NJP (2011)

(Option 2) Optimize work via math   N f      ( , { }) m W l x ln W kT f   tot m * tot m   f  0 m m Optimal condition:   ( , { }) 0 W l x , l x tot m m   1 1 Z Z     m ( ) ( ) F x F x   m m p m p Z x Z x m m m  0 W tot

Physical meaning of optimal condition Time-backward * ( 1 ) f 0 x Time-forward x 1 * ( 1 ) f 1 x x 1 x * 0 0 ( 1 ) L f 2 x L 1 x 1 * ( 1 ) f 3 x    left right (cf) ( ) 0 F x F F x 1 N   *  ( ) ( ) F x f F x ( 1 ) F 1 x 1 1 p p m p  0 p

Numerical check I N  3 Boson k T E  / 1 0 B W F 1 1   F p p

Numerical check II kT  E 0  ( 1 ) 0 F x p p T  kT  5 E 0

Casimir force N    * ( ) ( ) 0 F x f F x 1 1 p p m p  0 p Why? Time-backward x 1 x 1 0 L x 1 x 1

The optimal condition of the q-SZE with intrinsic irreversibility is achieved once the time-forward force is equivalent to the time back-ward force:    ( ) ( ) F x F x m m p m p

Remark and a new question   N f      m ln W kT f   tot m *   f  0 m m In fact, this equation can be derived from fully classical consideration. The point is that the above expression is mainly ascribed to multi-particle nature of SZE. Is work extractable from a heat engine by using purely quantum mechanical information? If yes, what is its mathematical formula? K.-H. Kim & SWK, PRE (2011)

Quantum information demon?

- Previous works Oppenheim, Horodeki, Horodeki & Horodeki, PRL (2002) Zurek, PRA (2003) Rio, Aberg, Renner, Dahlsten & Vedral, Nature (2011) Funo, Watanabe & Ueda, PRA (2013) Our Set-up         ( ) ( ) i i exp exp H H      ( ) ( ) i i S R AB ( ) ( ) i i Z Z S R Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

B A

Thermodynamic process Stage 1 (unitary evolution)     ( 1 ) ( 1 ) ( ) ( 1 ) i   ( 1 ) ( 1 ) U U U I U AB SR This can also describe isothermal process. Stage 2 (POVM)            ( 2 ) ( 2 ) ( 1 ) ( 2 ) ( 2 ) k k k U U p k k A A k BSR A k                ( 2 ) ( 1 ) ( 2 ) ( 2 ) ( 2 ) ( 1 ) ( 2 ) k k k k k tr tr p U U U U A k A A BSR A A Stage 3 (feedback control)     ( ) ( 3 ) ( 2 ) ( 3 ) f U U

Entropy consideration        von Neumann entropy ( ) tr ln S      (1) concavity    ( 2 ) ( ) k f p S S k SR SR k     i    (2) POVM ( ) ( 2 ) S S        S  (3) Klein inequality ( ) ( ) can ( ) f f f tr ln SR SR SR        W ΔF kT ΔS ΔS kTΔI S A B        ( 2 ) ( 2 ) ( ) ( ) i i : : I I A B I A B        mutual information ( : ) ( ) ( ) ( ) ( : ) I A B S S S I B A A B AB Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

Mutual information and Discord classical mutual information      ( : ) ( ) ( ) ( , ) ( ) ( | ) J B A H B H A H B A H B H B A  Shanon information   ( ) ( ) ln ( ) H A p A p A A  conditional entropy   H ( B | A ) p ( A , B ) ln p ( B | A ) A , B quantum analogue ~   ( : ) ( ) ( | ) ??? J B A S B S B A     ~   )    i ( : ) ( J   B A S S  i B B A A   ~   quantum discord ( | ) min I(A : B) - ( : ) B A J   B A  i A Ollivier & Zurek, PRL (2002)

Final formula                ( ) ( ) i i W F kT S S kT J kT B A S A B Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

Szilard engine containing a heteronuclear diatomic molecule semi-permeable wall   Von Neumann, A. Peres, V. Vedral, L. B. Levitin