Open quantum dynamics Dariusz Chru sci nski Institute of Physics, - PDF document

Open quantum dynamics Dariusz Chru´ sci´ nski Institute of Physics, Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University Grudziadzka 5, 87–100 Torun, Poland 1 Introduction: closed systems vs. open systems 1.1 Closed systems Consider a quantum system S and let H be the corresponding system’s Hilbert space. The evolution of the closed system is fully governed by the system Hamiltonian H via the Schr¨ odinger equation i ˙ ψ t = Hψ t , ( � = 1) , (1) and hence ψ − → ψ t = U t ψ, (2) where the unitary operator U t is defined by U t = e − iHt , (3) and ψ ∈ H is an initial ( t = 0) state. Mixed states represented by density operators evolve according to von Neumann equation ρ t = − i [ H, ρ t ] . ˙ (4) 1. pure state evolves into pure state 2. mixed state ρ evolves → ρ t = U t ( ρ ) := U t ρU † ρ − t , (5) 3. entropy S ( ρ ) = − Tr( ρ log ρ ) satisfies S ( ρ t ) = S ( ρ ) , (6) 4. purity Tr ρ 2 t is constant, 5. the evolution U t is reversible , that is, U − 1 = U − t . t 1

1.2 Open systems Consider now a quantum system S interacting with another system E – environment – and let H = H S ⊗ H E be the corresponding ‘ S + E ’ Hilbert space. The Hamiltonian of the total closed ‘ S + E ’ system reads H = H 0 + H int = H S ⊗ 1 l E + 1 l S ⊗ H E + H int . (7) Note, that the splitting is not unique. Let the initial state of ‘ E + S ’ be as follows ρ SE = ρ ⊗ ρ E , (8) that is, initially (at t = 0) S and E are not correlated. Since ‘ S + E ’ is a closed system its evolution reads as follows ρ ⊗ ρ E U SE † → ρ SE ( t ) := U SE ρ SE − , (9) t t where U SE = e − iHt . t Question : what is the evolution of the system S itself? The state of the system S evolves according to ρ t := Tr E ρ SE ( t ) (10) and it is called reduced evolution of the system S . The map � � ρ ⊗ ρ E U SE † U SE ρ → Λ t ( ρ ) := Tr E (11) t t enjoys the following properties: • completely positive (CP) • trace-preserving (TP) • Λ t =0 = id. Λ t is called a dynamical map . 1.3 Positive and completely positive maps Let L ( H ) be a space of linear operators in H (in this notes I assume that dim H = d < ∞ ). 2

Definition 1 A linear map (super-operator) Φ : L ( H ) → L ( H ) is called • positive iff X ≥ 0 = ⇒ Φ( X ) ≥ 0 . • n -positive if id n ⊗ Φ : M n ( C ) ⊗ L ( H ) → M n ( C ) ⊗ L ( H ) is positive • completely positive if it is n -positive for n = 1 , 2 , 3 , . . . . A linear map Φ : L ( H ) → L ( H ) is • trace-preserving if TrΦ( X ) = Tr X for all X ∈ L ( H ) • unital if Φ(1 l) = 1 l. Note, that fixing an othonormal basis | k � in H one may define a matrix T ij := Tr( P i Φ( P j )) (12) If Φ is positive and trace-preserving, then T ij is stochastic. Frobenius-Perron theorem — some remarks (classical vs. quantum). Let T be a stochastic matrix. Consider T x = λ x . There exists a leading eigenvalue λ 1 = 1 and the corresponding eigenvector (after appropriate normalization) corresponds to probability distribution. All other eigenvalues belong to the unit disc in the complex plane, that is, λ k | ≤ 1. Similar result hold for positive trace-preserving map Φ: Φ( X ) = λX. There exists a leading eigenvalue λ 1 = 1 and the corresponding eigenvector (after appropriate normalization) corresponds to density operator. All other eigenvalues belong to the unit disc in the complex plane, that is, λ k | ≤ 1. Let E ij be a matrix unit in M n ( C ). Any operator X ∈ M n ( C ) ⊗ L ( H ) has a following form n � X = E ij ⊗ X ij , X ij ∈ L ( H ) . i,j =1 One has 3

n � (id n ⊗ Φ)( X ) := E ij ⊗ Φ( X ij ) . (13) i,j =1 Proposition 1 Φ is CP iff it is d -positive. Corollary 1 One has CP = P d ⊂ P d − 1 ⊂ . . . ⊂ P 1 = Positive . (14) Theorem 1 (Stinespring, 1955) Φ : A → L ( H ) is CP ( A is a C ∗ -algebra) iff there exist • a Hilbert space K • a ∗ -homomorphism π : A → B ( K ) • a linear operator V : K → H such that Φ( a ) = V π ( a ) V † . (15) for all a ∈ A . Theorem 2 Φ is CP iff the Choi matrix d � C Φ := E ij ⊗ Φ( E ij ) ≥ 0 . (16) i,j =1 Theorem 3 (Stinespring,Sudarshan,Kraus) A map Φ is CP if and only if � K i XK † Φ( X ) = (17) i i where K i ∈ L ( H ) are called Kraus operators . The map Φ represented in (17) is • trace-preserving if � K † i K i = 1 l . (18) i • unital if � K i K † i = 1 l . (19) i 4

Example 1 Some examples of positive but not CP maps – they are important in entanglement theory! Basic properties of quantum channels: E : L ( H ) → L ( H ) • �E ( X ) � 1 ≤ � X � 1 • S ( E ( ρ ) ||E ( σ )) ≤ S ( ρ || σ ) • F ( E ( ρ ) , E ( σ )) ≥ F ( ρ, σ ) � Tr[ ρ (log ρ − log σ ]) , if supp ρ ⊆ supp σ D ( ρ || σ ) = . (20) + ∞ , otherwise and � √ ρ σ √ ρ � � 2 F ( ρ, σ ) = Tr . (21) Example 2 (Pure decoherence) Consider d -level system S coupled to the environment � H = H S ⊗ 1 l E + 1 l S ⊗ H E + P k ⊗ B k (22) k where � H S = E k P k . (23) k One has � H = P k ⊗ Z k ; Z k = E k 1 l S + H E + B k . (24) k One finds � U t = e − iHt = P k ⊗ e − iZ k t , (25) k and hence � Λ t ( ρ ) = C kl ( t ) P k ρP l (26) k,l with � e − iZ k t ρ E e iZ l t � C kl ( t ) = Tr (27) . The evolution of the density operator is very simple: 5

ρ kl − → C kl ( t ) ρ kl , that is, it is defined by the Hadamard product of C ( t ) and ρ . Recall, that ( A ◦ B ) kl := A kl B kl . (28) The map Φ C ( X ) := C ◦ X (29) is CP if and only if C ≥ 0 . 2 Markovian semigroup The simplest evolution is provided by the following master equation ρ t = L ( ρ t ) , ˙ (30) which generalizes von Neumann equation ρ t = − i [ H, ρ t ] =: L H ( ρ t ) , ˙ (31) that is, the super-operator L H : L ( H ) → L ( H ) is defined by L H ( ρ ) := − i [ H, ρ ] . (32) The solution to (30) has the following form Λ t = e t L . (33) Theorem 4 (Gorini,Kossakowski,Sudarshan,Lindblad) A linear map L : L ( H ) → L ( H ) generates legitimate dynamical map if and only if � � � k − 1 V k ρV † 2 { V † L ( ρ ) = − i [ H, ρ ] + γ k k V k , ρ } (34) k where { A, B } = AB + BA , and γ k > 0 . 2.1 Examples of Markovian semigroups Example 3 (Qubit decoherence) L ( ρ ) = γ 2( σ z ρσ z − ρ ); γ > 0 . (35) Note that 6

L ( E 11 ) = 0 L ( E 22 ) = 0 L ( E 12 ) = − γE 12 L ( E 21 ) = − γE 21 and hence Λ t ( E 11 ) = E 11 Λ t ( E 22 ) = E 22 e − γt E 12 Λ t ( E 12 ) = e − γt E 21 Λ t ( E 21 ) = Now finds the following Kraus representation Λ t ( ρ ) = 1 + e − γt ρ + 1 − e − γt σ z ρσ z . (36) 2 2 Another way is a direct computation of e L t . Example 4 (Qubit dissipation) Let us consider � � Φ( ρ ) = 1 γ + L + + γ − L − (37) 2 where where L 1 ( ρ ) = [ σ + , ρσ − ] + [ σ + ρ, σ − ] , L 2 ( ρ ) = [ σ − , ρσ + ] + [ σ − ρ, σ + ] , (38) L + corresponds to pumping (heating) process, L − corresponds to relaxation (cooling). To solve the master equation ˙ ρ t = Lρ t let us parameterize ρ t as follows ρ t = p 1 ( t ) P 1 + p 2 ( t ) P 2 + α ( t ) σ + + α ( t ) σ − , (39) with P k = | k �� k | . Using the following relations L ( P 1 ) = − γ + σ 3 , L ( P 2 ) = γ − σ 3 , L ( σ + ) = γ σ + , L ( σ − ) = γ σ − , where γ = γ + + γ − . 2 7

one finds the following Pauli master equations for the probability distribution ( p 1 ( t ) , p 2 ( t )) p 1 ( t ) ˙ = − γ + p 1 ( t ) + γ − p 2 ( t ) , (40) p 2 ( t ) ˙ = γ + p 1 ( t ) − γ − p 2 ( t ) , (41) together with α ( t ) = e − γt α (0) . The corresponding solution reads � 1 − e − ( γ + + γ − ) t � p 1 (0) e − ( γ + + γ − ) t + p ∗ p 1 ( t ) = , (42) 1 � 1 − e − ( γ + + γ − ) t � p 2 (0) e − ( γ + + γ − ) t + p ∗ p 2 ( t ) = (43) , 2 where we introduced γ + γ + p ∗ p ∗ 1 = 2 = (44) , . γ + + γ − γ + + γ − Hence, we have purely classical evolution of probability vector ( p 1 ( t ) , p 2 ( t )) on the diagonal of ρ t and very simple evolution of the off-diagonal element α ( t ) . Note, that asymptotically one obtains completely decohered density operator � p ∗ � 0 1 − → ρ t . p ∗ 0 2 In particular if γ + = γ − a state ρ t relaxes to maximally mixed state (a state becomes completely depolarized). 3 Beyond Markovian semigroup – non-Markovian evolution Consider now ˙ Λ t = L t Λ t , Λ 0 = id , (45) with time dependent generator L t . The formal solution reads �� t � � t � t � t 2 Λ t = T exp L u du = id + L u du + dt 1 L t 2 L t 1 + . . . . (46) dt 2 0 0 0 0 If [ L t , L u ] = 0, then �� t � �� t � 2 � t L u du + 1 Λ t = exp L u du = id + L u du + . . . . (47) 2 0 0 0 Evolution Λ t is called divisible if Λ t = V t,s Λ s ; t ≥ s. (48) It is called • P-divisible if V t,s is PTP • CP-divisible if V t,s is CPTP 8