1. Entanglement Creation in Open Quantum Systems Fabio Benatti Department of Theoretical Physics University of Trieste Milano, December 1, 2006 In collaboration with R. Floreanini

2. Outline Open quantum dynamics: dissipation and decoherence Open quantum dynamics: dissipation and decoherence Entanglement in open quantum systems: creation and its persistence Entanglement in open quantum systems: creation and its persistence

3. Open Quantum Dynamics qubits (S) in interaction with an environment (E): heat bath, external classical noise qubits (S) in interaction with an environment (E): heat bath, external classical noise weak coupling limit: Lindblad equation weak coupling limit: Lindblad equation H S + E = H 0 S ­ 1 E + 1 S ­ H E + ¸P® V ® ­ B ® L am b -s h i f t e d H0S NOISE FRICTION @ t ½(t)= L[½(t)] = ¡ i [ H S ; ½ ( t )] + D [ ½ ( t )] D [ ½ ( t )] = X ij D ij £ F i ½ ( t ) F y j ¡ 1 2 © F y i F j ; ½ ( t ) ª¤

4. Phenomenological Coefficients Kossakowski matrix Kossakowski matrix dissipative generator after ergodic mean dissipative generator after ergodic mean D : = [ D ij ] ¸ 0 V ® ( ! ) = X E a ¡ E b = ! P a V ® P b H 0 S =Xa E a P a h ® ¯ ( ! ) = d t ei! t G ® ¯ ( t ) R V ® ( ! ) = X i ( T r ( F y i V ® ( ! )) F i D [ ½ ] = X ® ; ¯ X ! h ® ¯ ( ! )( V ® ( ! ) ½ V y ¯ ( ! ) ¡ 1 2 f V y ¯ ( ! ) V ® ( ! ) ; ½ g) T r ( Fyi F j ) = ± ij G ® ¯ ( t ) = ! E ( B ® ( t ) B ¯ )

5. Physical Consistency Complete Positivity necessary for the physical consistency of necessary for the physical consistency of equivalent to complete positivity of equivalent to complete positivity of for all entangled states of for all entangled states of any n-level ancilla any n-level ancilla D : = [ D ® ¯ ] ¸ 0 ° t = e t L ; t ¸ 0 ; ½ 7! ½ ( t ) = ° t [ ½ ] ° t ° t ­ i d [ ½ en t ] ¸ 0 S + S n Sn

6. One open 2-level atom S(ystem)+E(nvironment): S(ystem)+E(nvironment): Lindblad equation for 1 qubit density matrices: Lindblad equation for 1 qubit density matrices: D = D 11 D 12 D 13 D 21 D 22 D 13 D 31 D 32 D 33 ¸ 0 + 3 X i ; j = 1 D ij [ ¾ i ½ ( t ) ¾ j ¡ 1 2 f ¾ j ¾ i ; ½ ( t )g] + ¸3X i = 1 ¾ i ­ B i H S + E = ( ! 0 2 3 X i = 1 n i ¾ i ) |{z} H 0 S ­ 1 E + 1 S ­ H E @ t ½ ( t ) = ¡ i [ H S ; ½ ( t )]

7. Two open 2-level atoms S(1)+S(2)+E: no direct S(1)--S(2) interaction S(1)+S(2)+E: no direct S(1)--S(2) interaction ( ! 0 2 3 X i = 1 n i ¾ i ) |{z} H 0 1 H S + E = + 1 S 1 ­­ 1 E ­HE ( ! 0 2 3 X i = 1 n i ¾ i |{z} H 0 2 ) ­ ( 1 S 2 ­ 1 E ) + 1 S 1 ­ 1 S 2 + ¸ 3 X i = 1 ¡ ¾ i ­ 1 S 2 ) ­ B i + ( 1 S 1 ­ ¾ i ) ­ B i + 1

8. Lindblad for two open atoms + 3 X i ; j = 1 B ij [( ¾ i ­ 1 2 ) ½ ( t )( 1 1 ­ ¾ j ) ¡ 1 2 f ¾ i ­ ¾ j ; ½ ( t )g] + 3 X i ; j = 1 B ¤ ji [( 1 1 ­ ¾ i ) ½ ( t )( ¾ j ­ 1 2 ) ¡ 1 2 f ¾ j ­ ¾ i ; ½ ( t )g] + 3 X i ; j = 1 A ij [( ¾ i ­ 1 2 ) ½ ( t )( ¾ j ­ 1 2 ) ¡ 1 2 f ¾ j ¾ i ­ 1 2 ; ½ ( t )g] + 3 X i ; j = 1 C ij [( 1 1 ­ ¾ i ) ½ ( t )( 1 1 ­ ¾ j ) ¡ 1 2 f 1 1 ­ ¾ j ¾ i ; ½ ( t )g] + D [ ½ ( t )] = ¡ i [ H 1 ­ 1 2 + 1 1 ­ H 2 + H 12 ; ½ ( t )] D [ ½ ( t )] : L H [ ½ ( t )] @ t ½ ( t ) = L [ ½ ( t )] = By= [ B ¤ ji ] B = [ B ij ] A = [ A ij ] C =[C ij] Environment induced interaction

9. ¾ ( ® ) = ¾ ® ­ 1 2 ® = 1 ; 2 ; 3 ¾ ( ® ) = 1 1 ­ ¾ ® ¡ 3 ® = 4 ; 5 ; 6 D = AB B y C ¸ 0 Can without direct interaction generate entanglement ? H12 ¡ i [ H 1 ­ 1 2 + 1 1 ­ H 2 ; ½ ( t )] + D [ ½ ( t )] @ t ½ ( t ) = D [ ½ ( t )] =6X ® ; ¯ = 1 D ® ¯ [ ¾ ( ® ) ½ ( t ) ¾ ( ¯ ) ¡ 1 2 f ¾ ( ¯ ) ¾ ( ® ) ; ½ ( t )g]

10. Sufficient Condition ( F.B., R. Floreanini, M. Piani, PRL 2003) an initial 2 qubit separable state an initial 2 qubit separable state gets entangled as soon as if gets entangled as soon as if j Á 1 ih Á 1 j ­ j  1 ih  1 j t > 0 ( R e ( B )) ij = 1 2 ( B ij + B ¤ ij ) j u i =hÁ 1j¾ 1jÁ 2ih Á 1 j ¾ 2 j Á 2 i h Á 1 j ¾ 3 j Á 2 i ; Á 1 ?Á 2 j v i =h 2j¾ 1j 1ih  2 j ¾ 2 j  1 i h  2 j ¾ 3 j  1 i ;  1 ?  2 h u j A j u ih v j C T j v i < jh u j R e ( B )j v ij 2

11. Idea for proof use partial transposition to check whether use partial transposition to check whether as well the the maps the maps as well the the maps the maps form a semigroup with generator form a semigroup with generator i d ­ T ( i d ± T ) ± ° t [j Á 1 ih Á 1 j ­ j  1 ih  1 j] ¸ 0 ° t ; t ¸0; + R [ ½ ( t )] ¡ i [ e H ; ½ ( t )] R [ ½ ] = 6 X ® ; ¯ = 1 Q ® ¯ [ ¾ ( ® ) ½¾ ( ¯ ) ¡ 1 2 f ¾ ( ¯ ) ¾ ( ® ) ; ½ g] G [ ½ ] = g t : = ( i d ­ T ) ± ° t ± ( i d ­ T ) = e tG

12. need not be positive need not be positive need not preserve the positivity of ( i d ­ T )[j Á 1 ih Á 1 j ­ j  1 ih  1 j] need not be positive ( i d ­ T ) ± ° t [j Á 1 ih Á 1 j ­ j  1 ih  1 j] Q = AR e(B)R e ( B T ) C T g t = ( i d ­ T ) ± ° t ± ( i d ­ T )

13. E à ; Á 1 ;  1 ( t ) : = h à j g t [j Á 1 ih Á 1 j ­ j  ¤ 1 ih  ¤ 1 j]j à i T [j  1 ih  1 j] = j  ¤ 1 ih  ¤ 1 j <0 @ t E à ; Á 1 ;  1 ( 0 ) = 6 X ® ; ¯ = 1 Q ® ¯ h à j ¾ ( ® ) (j Á 1 ih Á 1 j ­ j  ¤ 1 ih  ¤ 1 j) ¾ ( ¯ ) j à i E à ; Á 1 ;  1 ( 0 ) = jh à j Á 1 ­  ¤ 1 ij 2 = 0 and get entangled j Á 1 i ­ j  1 i E à ; Á ;  1 ( t ) < 0 t ! 0 +

14. Particular Case: choose choose sufficient condition becomes sufficient condition becomes A = B = C ¸ 0 Á 1 = Â 2 =)jui=jvi h u j A j u ih u j ATj u i < jh u j R e ( A )j u ij 2 (h u j I m ( A )j u i) 2 > 0 A = A y = R e ( A ) + I m ( A ) ; < ( A ) = A + AT2 I m ( A ) : = 1 2 ( A ¡ A T )

15. A = a 1 i b 0 ¡ i b a 2 0 00 a 3 ; a 1 ; 2 ; 3 ¸ 0 ; a 1 a 2 ¸ b 2 I m ( A ) = 0 i b 0 ¡ i b 00 000 Example:jÁ 1i=j 2i=j¡i; ¾ 3j§i= §j§i j u i = 1 i 0 h u j I m ( A )j u i)2= 4 b2> 0 If, gets entangled for small timesb6= 0 j ¡ ih ¡ j ­ j + ih + j

16. Two atoms in a scalar thermal field in equilibrium at inverse temperature qubit 1 and qubit 2 linearly coupled to qubit 1 and qubit 2 linearly coupled to full Hamiltonian: full Hamiltonian: H 0 1 = H 0 2 = ! 0 2 3 X i = 1 n i ¾ i F ( x ) h F i ( x ) F j ( y )i = ± ij G ( x ¡ y ) = ± ij d 4 k ( 2 ¼ ) 3 µ ( k 0 ) ± ( k 2 )( e ¡ i k ( x ¡ y ) 1 ¡ e ¡ ¯k 0 + e + i k ( x ¡ y ) e ¯k 0 ¡ 1 ) + ¸ 3 X i = 1 (( ¾ i ­ 1 2 ) + ( 1 1 ­ ¾ i )) ­ F i ( f ) F i ( f ) =RR 3 d x f ( x ) F i ( x ) f ( x ) = 1 ¼ 2 " = 2 x 2 + ( " = 2 ) 2 H S + E = H 0 1 + H 0 2 + H E ¯

17. Kossakowski matrix D explicitly calculable A = a + cn 2 1 cn 1 n 2 ¡ i b n 3 cn 1 n 3 + i b n 2 cn 1 n 2 + i b n 3 a + cn 2 2 cn 2 n 3 ¡ i b n 1 cn 1 n 3 ¡ i b n 2 cn 2 n 3 + i b n 1 a + cn 2 3 D = AA AA c = 1 2 ¼ ¯ ¡ ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! a = ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! b =!4 ¼

18. Can entanglement created irreversibly survive decoherence? YES Moreover, entangled states can remain entangled asymptotically and even become more entangled F.B., R. Floreanini (Int. J. Quant. Inf. 2006)

19. Quantifying Entanglement: Concurrence 2 qubits entanglement content (Wootters 1998) : 2 qubits entanglement content (Wootters 1998) : spectrum(R) = spectrum(R) = concurrence: concurrence: ¸21 ¸ ¸22 ¸ ¸23 ¸ ¸24 C ( ½ ) : = max f 0 ; ¸ 1 ¡ ¸ 2 ¡ ¸ 3 ¡ ¸ 4 g ½ 7! ^ ½: = ( ¾ 2 ­ ¾ 2 ) ½ ¤ ( ¾ 2 ­ ¾ 2 ) 7! R = ½ ^ ½

20. ½ 1 = 1 4 ( 1 1 ­ 1 2 + 3 X i = 1 ¿ ¡ R 2 ( 1 ¡ ( ¿ + 3 ) n 2 i ) 2 ( 3 + R 2 ) ¾ i ­ ¾ i + X i 6 = j R 2 ( ¿ + 3 ) n i n j 2 ( 3 + R 2 ) ¾ i ­ ¾ j ) 0 · R = b a = 1 ¡ e ¡ ¯ ! 1 + e ¡ ¯ ! · 1 ½ = 1 4 ( 1 1 ­ 1 2 + 3 X i = 1 ½ 0 i 1 1 ­ ¾ i + 3 X i = 1 ½ i 0 ¾ i ­ 1 2 + 3 X i ; j = 1 ½ ij ¾ i ­ ¾ j ) Any initial state goes into ¿ = 3 X i = 1 T r ( ½¾ i ­ ¾ i ) ¡ 3 X i = 1 R n i ( ¿ + 3 ) 3 + R 2 ( 1 1 ­ ¾ i + ¾ i ­ 1 2 ) R(¯ = 0)= 0

21. Asymptotic Concurrence: initial state: initial state: concurrence: concurrence: asymptotic gain: asymptotic gain: C ( ½ ) = 1 ¡ 3 s 2 ; s < 2 3 C ( ½ 1 ) ¡ C ( ½ ) = 3 R 2 s 3 + R 2 ½ =s4 1 S 1 ­ 1 S 2 + ( 1 ¡ s )j ª 01 ih ª 01 j C ( ½ 1 ) = 3 ¡ R22 ( 3 + R 2 ) [ 5 R2¡ 3 3 ¡ R 2 ¡ ¿ ]

22. Two atoms separated by a distance L interaction Hamiltonian: interaction Hamiltonian: smearing functions: smearing functions: f 1 ( x ) = 1 ¼ 2 " = 2 x 2 + ( " = 2 ) 2 f 2 ( x ) = f ( x + L ) H i n t =3X i = 1 ( ¾ ( 1 ) i ­ F i ( f 1 ) + ¾ ( 2 ) i ­ F i ( f 2 ))

23. Kossakowski Matrix b = ! 4 ¼ c = 1 2 ¼ ¯ ¡ ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! A = a + cn 2 1 cn 1 n 2 ¡ i b n 3 cn 1 n 3 + i b n 2 cn 1 n 2 + i b n 3 a + cn 2 2 cn 2 n 3 ¡ i b n 1 cn 1 n 3 ¡ i b n 2 cn 2 n 3 + i b n 1 a + cn 2 3 a = ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! D = AA0A 0 A A 0 = a 0 + cn 2 1 c 0 n 1 n 2 ¡ i b 0 n 3 c 0 n 1 n 3 + i b 0 n 2 c 0 n 1 n 2 + i b 0 n 3 a 0 + cn 2 2 c 0 n 2 n 3 ¡ i b 0 n 1 c 0 n 1 n 3 ¡ i b 0 n 2 c 0 n 2 n 3 + i b 0 n 1 a 0 + c 0 n 2 3 b 0 = ! 4 ¼ s i n ( ! L ) ! L c 0 = 1 2 ¼ ¯ ¡ ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! s i n ( L ) ! L a 0 = ! 4 ¼ 1 + e ¡ ¯ ! 1 ¡ e ¡ ¯ ! s i n! L ! L

24. Controlling Entanglement Creation separable initial state: separable initial state: sufficient condition: sufficient condition: with it becomes with it becomes separable if separable if ½ =j¡i¡j­j+ih+j jui=jvi=(1 ; ¡ i ; 0) R 2 + S 2 > 1 S = s i n(! L)! L R = b a = 1 ¡ e ¡¯! 1 + e ¡ ¯ ! h u j A j u ih v j ATj v i < jh u j R e ( A 0 )j v ij 2 ½1 L > 0 T =1¯ = 1