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/(2 N)), 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 † † ˙ = ( ) = i[H, ] + [Lk , L ] + [Lk , L ] 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
† S( ) = Mk Mk Xk † 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 † [Lk , Lk ] 2 Lk (q, p), Lk (q, p) = ~ → →0 { } 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 † ~ [Lk , Lk ] 2 Lk (q, p), Lk (q, p) = div(drift) < 0 ~ → →0 { } 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 † = S( ) = c T T with c 0 X † † 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 † = S( ) = c T T with c 0 X † † 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 † = S( ) = c T T with c 0 X † † 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 † = S( ) = c T T with c 0 X † † 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 † 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 † 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 † 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
† = [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 † ~ = [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 √p M = , M = 0 0 1 p 1 0 0 µ ¶ µ ¶ The density matrix transformsp as
AD 00 + p 11 1 p 01 S ( ) = 1 p 10 (1 p) 11 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 = p z | ih | | ih |
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 √p M = , M = 0 0 1 p 1 0 0 µ ¶ µ ¶ The density matrix transformsp as
AD 00 + p 11 1 p 01 S ( ) = 1 p 10 (1 p) 11 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 = p z | ih | | ih |
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 √p M = , M = 0 0 1 p 1 0 0 µ ¶ µ ¶ The density matrix transformsp as
AD 00 + p 11 1 p 01 S ( ) = 1 p 10 (1 p) 11 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 = p z | ih | | ih |
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 √p M = , M = 0 0 1 p 1 0 0 µ ¶ µ ¶ The density matrix transformsp as
AD 00 + p 11 1 p 01 S ( ) = 1 p 10 (1 p) 11 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 = p z | ih | | ih |
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 14 / 21 Simple Model
Superoperators of the form S2 = (1 )I + S1 Propose
† S ( ) = (1 ) + P P 0 < < 1 [ i],i [ i],i Xij def where Pij = i j and i = 0, 1, . . . , N 1 labels momentum eigenbasis | ih | (pi = i/N) and [ i] = int( i).
= 1: Unital, Generalized 1 = 2 , = ( i i I) Phase Damping channel | ih | (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 | ih | | ih | ∀ hpi0 = (1 )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)). Kraus operators
√1 0 √ 0 0 √ M0 = , M1 = , M2 = 0 √1 0 0 0 0 µ ¶ µ ¶ µ ¶ The density matrix transforms as
00 + 11 (1 ) 01 S , ( ) = (1 ) 10 (1 ) 11 µ ¶ Non-unital because 1 + 0 S , (I) = = I 0 (1 ) 6 µ ¶ 2 S , ( 0 0 ) = 0 0 = = (P00 P11) = z | ih | | ih |
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 16 / 21 N = 2 case ("1 qubit")
In this case N = 2 and the minimum is attained for 1/2 (= (N 1)/N)). Kraus operators
√1 0 √ 0 0 √ M0 = , M1 = , M2 = 0 √1 0 0 0 0 µ ¶ µ ¶ µ ¶ The density matrix transforms as
00 + 11 (1 ) 01 S , ( ) = (1 ) 10 (1 ) 11 µ ¶ Non-unital because 1 + 0 S , (I) = = I 0 (1 ) 6 µ ¶ 2 S , ( 0 0 ) = 0 0 = = (P00 P11) = z | ih | | ih |
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 16 / 21 N = 2 case ("1 qubit")
In this case N = 2 and the minimum is attained for 1/2 (= (N 1)/N)). Kraus operators
√1 0 √ 0 0 √ M0 = , M1 = , M2 = 0 √1 0 0 0 0 µ ¶ µ ¶ µ ¶ The density matrix transforms as
00 + 11 (1 ) 01 S , ( ) = (1 ) 10 (1 ) 11 µ ¶ Non-unital because 1 + 0 S , (I) = = I 0 (1 ) 6 µ ¶ 2 S , ( 0 0 ) = 0 0 = = (P00 P11) = z | ih | | ih |
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 16 / 21 N = 2 case ("1 qubit")
In this case N = 2 and the minimum is attained for 1/2 (= (N 1)/N)). Kraus operators
√1 0 √ 0 0 √ M0 = , M1 = , M2 = 0 √1 0 0 0 0 µ ¶ µ ¶ µ ¶ The density matrix transforms as
00 + 11 (1 ) 01 S , ( ) = (1 ) 10 (1 ) 11 µ ¶ Non-unital because 1 + 0 S , (I) = = I 0 (1 ) 6 µ ¶ 2 S , ( 0 0 ) = 0 0 = = (P00 P11) = z | ih | | ih |
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 16 / 21 Some ®gures
(q, p) (N dim. system, quantized torus phase space, Husimi rep.. . . )
p α=0.01 α=0.1 α=0.2 α=0.5 α=0.75 (q, p) = (1−α)
q
† Composition with a unitary map Stot( ) = S (Umap Umap) α 0.015625 0.25 0.328125 0.65625 0.984375 ε 0.1
0.3
0.5
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 17 / 21 Another ®gure ( 0 case)
classical and quantum invariant states
p
q q
classical quantum
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 18 / 21 Summary
Remark differences between unital and nonunital quantum operations.
Independent of master eq. approach.
Give phase space interpretation.
Simple example illustrates main features.
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 19 / 21 Collaboration
Buenos Aires (Quantum Chaos Group, CNEA): Marcos Saraceno María Elena Spina Como, Italy (Universitá degli Studi dell’Insubria): Gabriel Carlo
Propaganda quant-ph/0508190
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 20 / 21 Collaboration
Buenos Aires (Quantum Chaos Group, CNEA): Marcos Saraceno María Elena Spina Como, Italy (Universitá degli Studi dell’Insubria): Gabriel Carlo
Propaganda quant-ph/0508190
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 20 / 21 Muchas Gracias
Nacho García-Mata (Buenos Aires.) Dissipation and Quantum Operations ISSQUI05 21 / 21