Quantum operations acessible by Lindblad semigroups Karol ˙ Zyczkowski Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, PAS, Warsaw in collaboration with Zbigniew Pucha� la (Gliwice), Lukasz Rudnicki (Elrangen / Warsaw), � Fereshte Shahbeigi (Mashhad) Symposium on Mathematical Physics, Toru´ n June 17, 2019 K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 1 / 23
Mixed Quantum States Set M N of all mixed states of size N M N := { ρ : H N → H N ; ρ = ρ † , ρ ≥ 0 , Tr ρ = 1 } example: M 2 = B 3 ⊂ ❘ 3 - Bloch ball with all pure states at the boundary The set M N is compact and convex: ρ = � i a i | ψ i �� ψ i | where a i ≥ 0 and � i a i = 1. The set M N of mixed states has N 2 − 1 real dimensions, M N ⊂ ❘ N 2 − 1 . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 2 / 23
Quantum maps: evolution in discrete time steps Quantum operation: linear, completely positive trace preserving map positivity : Φ( ρ ) ≥ 0, ∀ ρ ∈ M N complete positivity : [Φ ⊗ ✶ K ]( σ ) ≥ 0, ∀ σ ∈ M KN and K = 2 , 3 , ... Enviromental form (interacting quantum system !) ρ ′ = Φ( ρ ) = Tr E [ U ( ρ ⊗ ω E ) U † ] . where ω E is an initial state of the environment while UU † = ✶ . Kraus form ρ ′ = Φ( ρ ) = � i A i ρ A † i , where the Kraus operators satisfy i A † � i A i = ✶ , which implies that the trace is preserved. K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 3 / 23
Classical probabilistic dynamics & Markov chains Stochastic matrices Classical states : N -point probability distribution, p = { p 1 , . . . p N } , where p i ≥ 0 and � N i =1 p i = 1 Discrete dynamics : p ′ i = S ij p j , where S is a stochastic matrix of size N and maps the simplex of classical states into itself, S : ∆ N − 1 → ∆ N − 1 . Frobenius–Perron theorem Let S be a stochastic matrix : a) S ij ≥ 0 for i , j = 1 , . . . , N , b) � N i =1 S ij = 1 for all j = 1 , . . . , N . Then i) the spectrum { z i } N i =1 of S belongs to the unit disk, ii) the leading eigenvalue equals unity, z 1 = 1, iii) the corresponding eigenstate p inv is invariant, S p inv = p inv . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 4 / 23
Quantum stochastic maps (trace preserving, CP) Superoperator Φ : M N → M N A quantum operation can be described by a matrix Φ of size N 2 , ρ ′ = Φ ρ ′ ρ m µ = Φ m µ n ν ρ n ν . or The superoperator Φ can be expressed in terms of the Kraus operators A i , i A i ⊗ ¯ Φ = � A i . Dynamical Matrix D : Sudarshan et al. (1961) obtained by reshuffling of a 4–index matrix Φ is Hermitian, Φ =: Φ R . D Φ = D † D mn µν := Φ m µ n ν , so that Theorem of Choi (1975). A map Φ is completely positive (CP) if and only if the dynamical matrix D Φ is positive , D ≥ 0. K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 5 / 23
Spectral properties of a superoperator Φ Quantum analogue of the Frobenious-Perron theorem Let Φ represent a stochastic quantum map, i.e. a’) Φ R ≥ 0; (Choi theorem) b’) Tr A Φ R = ✶ ⇔ � k Φ kk = δ ij . (trace preserving condition) ij Then i’) the spectrum { z i } N 2 i =1 of Φ belongs to the unit disk, ii’) the leading eigenvalue equals unity, z 1 = 1, iii’) the corresponding eigenstate (with N 2 components) forms a matrix ω of size N , which is positive, ω ≥ 0, normalized, Tr ω = 1, and is invariant under the action of the map, Φ( ω ) = ω . Classical case In the case of a diagonal dynamical matrix , D ij = d i δ ij reshaping its diagonal { d i } of length N 2 one obtains a matrix of size N , where S ij = D ii , jj of size N which is stochastic and recovers the standard F–P theorem. K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 6 / 23
K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 7 / 23
Spectra of One–Qubit Bistochastic Maps Consider one–qubit bistochastic map (Pauli channel + unitary evolution) in the Kraus form , ρ ′ = Φ( ρ ) = � 4 i =1 A i ρ A † i The evolution operator Φ = � 4 i =1 A i ⊗ A † i has spectrum with zero or two complex eigenvalues. In the latter case, { 1 , x , z , ¯ z } with real x ∈ [ − 1 , 1] the following bound holds | z | ≤ 1+ x la, ˙ Rudnicki, Pucha� Zyczkowski , Quantum 2 , (2018). 2 a) x = − 0 . 4 b) x = 0 . 4 Exemplary constraints for the position of complex eigenvalues z and ¯ z of a bistochastic map Φ with real eigenvalue x ( black dot ). K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 8 / 23
Decoherence for quantum states and quantum maps Quantum states → classical states = diagonal matrices Decoherence of a state: ρ → ˜ ρ = diag ( ρ ) Quantum maps → classical maps = stochastic matrices Decoherence of a map: The Choi matrix becomes diagonal, D = diag ( D ) so that the map Φ = D R → ˜ D R → S D → ˜ i A i ρ A † where for any Kraus decomposition defining Φ( ρ ) = � i the corresponding classical map S is given by the Hadamard product , � A i ⊙ ¯ S = A i i i A † If a quantum map Φ is trace preserving, � i A i = ✶ then the classical map S is stochastic , � j S ij = 1 . i A i A † If additionally a quantum map Φ is unital, � i = ✶ then the classical map S is bistochastic , � j S ij = � i S ij = 1 . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 9 / 23
Unistochastic Maps defined by an interaction of the ancilla of the same dimension initially in the maximally mixed state, U ( ρ ⊗ 1 ρ ′ = Φ U ( ρ ) = Tr env � N ✶ N ) U † � Unistochastic maps are unital, Φ U ( ✶ ) = ✶ , hence bistochastic. Is every bistochastic map unistochastic ? No ! One–qubit bistochastic maps = Pauli channels, ρ ′ = Φ p ( ρ ) Φ p ( ρ ) = � 3 i =0 p i σ i ρσ i , where � p i = 1 while σ 0 = ✶ and remaining three σ i denote Pauli matrices. Let λ = ( λ 1 , λ 2 , λ 3 ) be the damping vector containing three axis of the ellipsoid - the image of the Bloch ball by a bistochastic map. Set B 2 of one-qubit bistochastic maps = regular tetrahedron with corners at λ = (1 , 1 , 1), (1 , − 1 , − 1), ( − 1 , 1 , − 1), ( − 1 , − 1 , 1) corresponding to σ i with i = 0 , . . . , 3. K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 10 / 23
One Qubit Unistochastic Maps Unistochastic maps do satisfy following restrictions for the damping vector λ = ( λ 1 , λ 2 , λ 3 ) λ 1 λ 2 ≤ λ 3 , λ 2 λ 3 ≤ λ 1 , λ 3 λ 1 ≤ λ 2 . ( ∗ ) The set U 2 forms a (non-convex!) subset of the tetrahedron B 2 of s, K. ˙ bistochastic maps, Musz, Ku´ Z. , Phys. Rev. A 2012 Example: The following Pauli channel Φ p ( ρ ) = � 3 1 3 σ i ρσ i , i =1 of rank three is not unistochastic . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 11 / 23
K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 12 / 23
Lindblad dynamics in continuous time a) closed system: von Neumann equation d ρ dt = − i [ H , ρ ] leads to unitary dynamics (= reversible, rigid rotation): ρ ′ = U ρ U † = e − iHt ρ e iHt b) open system (of size N )- interaction with environment described by Gorrini - Lindblad - Kossakowski - Sudarshan equation (1976) in terms of jump operators L j , N 2 − 1 d ρ j − 1 j L j ρ − 1 � � � L j ρ L † 2 L † 2 ρ L † = L ( ρ ) = j L j , dt j =1 leads to nonunitary Lindblad dynamics (= irreversible contraction) ρ ( t ) = e L t [ ρ (0)] = Λ t [ ρ (0)] and generates a dynamical semigroup , Λ s Λ t = Λ t + s . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 13 / 23
Lindblad operator some matrix algebra: A product of three matrices, Y = ABC , can also be written as Y = Ψ B , where superoperator reads Ψ = A ⊗ C T i) discrete time: superoperator Φ i A i ρ A † corresponding to an operation in Kraus form Φ( ρ ) = � i reads � A i ⊗ ¯ Φ = A i , i ii) continuous time: Lindblad operator reads L j ⊗ L j − 1 j L j ⊗ I − 1 � � L † � I ⊗ L T L = j L j . 2 2 j j j For a given operation Φ corresponding to { A j } one can take L j = A j to get L = Φ − ■ , but in general this semigroup does not lead to the map Φ. K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 14 / 23
key issue : Fixing a certain set of quantum operations of size N find these operations Φ for which there exists a quantum semigroup Λ t = e t L such that Φ = Λ 1 = e L . In other words we look for a logarithm log Φ = L such that the entire trajectory Λ t = e t log Φ gives a proper quantum channel. related problems a1) classical analogue: for which stochastic matrix S ∈ S N there exists a define L c = log S and check if the entire trajectory Λ t = e t L c semigroup: belongs to the set S N of stochastic matrices of order N a2) which stochastic matrix S has a stochastic square root , S = S 2 2 , where S 2 ∈ S N . a3) ... has a stochastic root of order k : S = S k k where S k ∈ S N b1) which channel Φ is divisible so that there exist Ψ 1 and Ψ 2 � = ■ , so that Φ = Ψ 2 Ψ 1 , Wolf, Cirac (2008) b2) which stochastic transition matrix S is divisible , S = S 2 S 1 . K ˙ Z (IF UJ/CFT PAN ) Lindblad dynamics A June 17, 2019 15 / 23
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