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Dissipation and Quantum Operations Dissipation and Quantum Operations Ignacio García-Mata Dto. de Física, Lab. TANDAR, Comisión Nacional de Energía Atómica Buenos Aires, Argentina International Summer School on Quantum Information, MPI PKS Dresden 2005 Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 1 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 2 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 3 / 21 Intro. Motivations Understanding noise and (trying to) control it. Something to say about how the classical world emerges from the Quantum. Tools Quantum Channels (q. operations, superoperators, QDS’s . ) on an N dimensional Hilbert space, N £ N dimensional phase space (~ = 1=(2¼N)), Kraus operators. Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 4 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 5 / 21 Open Systems Markov (local in time) assumption Lindblad Master equation 1 y y ½_ = (½) = i[H; ½] + [Lk ½; L ] + [Lk ; ½L ] L ¡ 2 k k Xk n o CPTP Quantum Dynamical Semi-group (channel, superoperator, quantum operation,. Lindblad (1976), Gorini et al. (1976)) Kraus operator sum form y S(½) = Mk ½Mk Xk y where k Mk Mk = I for TP. We asume discrete time. P Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 6 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 7 / 21 Open Systems in Phase Space Representation in phase space by the Wigner function. Master eq. Fokker-Planck eq. for the Wigner function ! W_ (q; p) = [Hamiltonian + non-Hermitian Lk ’s] + diffusion Phase space contraction associated to y ¤ [Lk ; Lk ] 2 Lk (q; p); Lk (q; p) = ¡~¡!¡!0 ¡ f g Xk X Dissipation due to non-Hermitian L0 s = (I) = 0 k ) L 6 Non-unital dynamics S(I) = I 6 Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 8 / 21 Open Systems in Phase Space Representation in phase space by the Wigner function. Master eq. Fokker-Planck eq. for the Wigner function ! W_ (q; p) = [Hamiltonian + non-Hermitian Lk ’s] + diffusion Phase space contraction associated to y ¤ ~ [Lk ; Lk ] 2 Lk (q; p); Lk (q; p) = div(drift) < 0 ¡~¡!¡!0 ¡ f g Xk X Dissipation due to non-Hermitian L0 s = (I) = 0 k ) L 6 Non-unital dynamics S(I) = I 6 Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 8 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 9 / 21 Quantum Operations So we have a map like 0 y ½ = S(½) = c®T®½T® with c® 0 ® ¸ X y y TP if ® c®T®T® = I Unital if ® c®T®T® = I UnitalP Diffusion, Dephasing, pure decoherenceP . ´ Bistochastic maps usually Random Unitary Processes Non-unital Dissipation (+ decoherence...) ´ Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 10 / 21 Quantum Operations So we have a map like 0 y ½ = S(½) = c®T®½T® with c® 0 ® ¸ X y y TP if ® c®T®T® = I Unital if ® c®T®T® = I UnitalP Diffusion, Dephasing, pure decoherenceP . ´ Bistochastic maps usually Random Unitary Processes Non-unital Dissipation (+ decoherence...) ´ Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 10 / 21 Quantum Operations So we have a map like 0 y ½ = S(½) = c®T®½T® with c® 0 ® ¸ X y y TP if ® c®T®T® = I Unital if ® c®T®T® = I UnitalP Diffusion, Dephasing, pure decoherenceP . ´ Bistochastic maps usually Random Unitary Processes See for example Bianucci et al, PRE 2002; Nonnenmacher, Nonlinearity 2003, IGM et al., PRL 2003; IGM & Saraceno, PRE 2004; Aolita (“boludo”) et al. PRA 2004; MMO Non-unital Dissipation (+ decoherence...) ´ Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 10 / 21 Quantum Operations So we have a map like 0 y ½ = S(½) = c®T®½T® with c® 0 ® ¸ X y y TP if ® c®T®T® = I Unital if ® c®T®T® = I UnitalP Diffusion, Dephasing, pure decoherenceP . ´ Bistochastic maps usually Random Unitary Processes Non-unital Dissipation (+ decoherence...) ´ Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 10 / 21 Dissipative Quantum Operations Non-unital Dissipation (+ decoherence...) ´ Quantities Tr (S(I=N) I=N)2 ´ = ¡ non-unitality (“distance”) Tr I N 2 £ [( = ) ] ¤ Tr (S(I=N))2 (I=N)2 = ¡ (“@ ” of purity) Tr I N 2 t £ [( = ) ] ¤ 1 y 2 def 1 2 = Tr ( [Mk ; Mk ]) = Tr[¡ ] N " # N Xk In the classical limit ´ is related to the integral over phase space of the Poisson brackets of Lk (q; p)’s global dissipation parameter. ¡! Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 11 / 21 Dissipative Quantum Operations Non-unital Dissipation (+ decoherence...) ´ Quantities Tr (S(I=N) I=N)2 ´ = ¡ non-unitality (“distance”) Tr I N 2 £ [( = ) ] ¤ Tr (S(I=N))2 (I=N)2 = ¡ (“@ ” of purity) Tr I N 2 t £ [( = ) ] ¤ 1 y 2 def 1 2 = Tr ( [Mk ; Mk ]) = Tr[¡ ] N " # N Xk In the classical limit ´ is related to the integral over phase space of the Poisson brackets of Lk (q; p)’s global dissipation parameter. ¡! Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 11 / 21 Dissipative Quantum Operations Non-unital Dissipation (+ decoherence...) ´ Quantities Tr (S(I=N) I=N)2 ´ = ¡ non-unitality (“distance”) Tr I N 2 £ [( = ) ] ¤ Tr (S(I=N))2 (I=N)2 = ¡ (“@ ” of purity) Tr I N 2 t £ [( = ) ] ¤ 1 y 2 def 1 2 = Tr ( [Mk ; Mk ]) = Tr[¡ ] N " # N Xk In the classical limit ´ is related to the integral over phase space of the Poisson brackets of Lk (q; p)’s global dissipation parameter. ¡! Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 11 / 21 Dissipative Quantum Operations Definition y ¡ = [Mk ; Mk ] Xk Hermitian and traceless Representation in phase space (Wigner or Husimi) is real and gives picture of local contraction regions (¡(q; p) < 0) Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 12 / 21 Dissipative Quantum Operations Definition y ~ ¡ = [Mk ; Mk ] div(drift) ¡“~¡¡!¡!0” / Xk Hermitian and traceless Representation in phase space (Wigner or Husimi) is real and gives picture of local contraction regions (¡(q; p) < 0) Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 12 / 21 Outline 1 Intro 2 Open Systems 3 Open Systems in Phase Space 4 Dissipative Quantum Operations 5 Examples Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 13 / 21 Amplitude damping channel 1-qubit Amplitude Damping channel (Preskill, Nielsen & Chuang) Kraus operators 1 0 0 pp M = ; M = 0 0 1 p 1 0 0 µ ¡ ¶ µ ¶ The density matrix transformsp as AD ½00 + p½11 1 p ½01 S (½) = ¡ 1 p ½10 (1 p)½11 µ ¡ p ¡ ¶ Non-unital because p 1 + p 0 SAD(I) = = I 0 (1 p) 6 µ ¡ ¶ AD 2 S ( 0 0 ) = 0 0 ´ = p ¡ = pz j ih j j ih j Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 14 / 21 Amplitude damping channel 1-qubit Amplitude Damping channel (Preskill, Nielsen & Chuang) Kraus operators 1 0 0 pp M = ; M = 0 0 1 p 1 0 0 µ ¡ ¶ µ ¶ The density matrix transformsp as AD ½00 + p½11 1 p ½01 S (½) = ¡ 1 p ½10 (1 p)½11 µ ¡ p ¡ ¶ Non-unital because p 1 + p 0 SAD(I) = = I 0 (1 p) 6 µ ¡ ¶ AD 2 S ( 0 0 ) = 0 0 ´ = p ¡ = pz j ih j j ih j Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 14 / 21 Amplitude damping channel 1-qubit Amplitude Damping channel (Preskill, Nielsen & Chuang) Kraus operators 1 0 0 pp M = ; M = 0 0 1 p 1 0 0 µ ¡ ¶ µ ¶ The density matrix transformsp as AD ½00 + p½11 1 p ½01 S (½) = ¡ 1 p ½10 (1 p)½11 µ ¡ p ¡ ¶ Non-unital because p 1 + p 0 SAD(I) = = I 0 (1 p) 6 µ ¡ ¶ AD 2 S ( 0 0 ) = 0 0 ´ = p ¡ = pz j ih j j ih j Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 14 / 21 Amplitude damping channel 1-qubit Amplitude Damping channel (Preskill, Nielsen & Chuang) Kraus operators 1 0 0 pp M = ; M = 0 0 1 p 1 0 0 µ ¡ ¶ µ ¶ The density matrix transformsp as AD ½00 + p½11 1 p ½01 S (½) = ¡ 1 p ½10 (1 p)½11 µ ¡ p ¡ ¶ Non-unital because p 1 + p 0 SAD(I) = = I 0 (1 p) 6 µ ¡ ¶ AD 2 S ( 0 0 ) = 0 0 ´ = p ¡ = pz j ih j j ih j Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 14 / 21 Simple Model Superoperators of the form S2 = (1 ²)I + ²S1 ¡ Propose y S²®(½) = (1 ²)½ + ² P ½P 0 < ® < 1 ¡ [®i];i [®i];i Xij def where Pij = i j and i = 0; 1; : : : ; N 1 labels momentum eigenbasis j ih j ¡ (pi = i=N) and [®i] = int(®i). ® = 1: Unital, Generalized 1 ® ´ = ²2 ¡ ; ¡ = ²( ®i ®i I) Phase Damping channel ® j ih j ¡ (See Aolita et al., PRE 2004). µ ¶ Xi ® < (N 1)=N Non-Unital ® 0 “mimics” classical friction (AD channel-lik¡ e). Invariant » 0 state: S( 0 0 ) = 0 0 ® q = q j ih j j ih j 8 hpi0 = (1 ²)hpi h i ¡ h i Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 15 / 21 N = 2 case ("1 qubit") In this case N = 2 and the minimum ´ is attained for ® 1=2 (= (N ¡ 1)=N)).
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