New Estimates and Bounds on the Reachable Sets of Controlled - PowerPoint PPT Presentation

New Estimates and Bounds on the Reachable Sets of Controlled Lindblad-Kossakowski Equations G. Dirr I. Kurniawan and U. Helmke Institute of Mathematics University of Würzburg, Germany QC-Workshop, Paris 2010 Dirr (Würzburg) QC-Workshop, Paris 2010 1 / 33

Table of Contents Preliminaries on Open Quantum Systems 1 Two Recent Results by H. Yuan 2 Result I: The Majorization Theorem Result II: Reachable sets of 2-level systems Refined Majorization Theorem 3 Unitary invariant norms and the von Neumann entropy The Statement The Proof Generalized Reachability Result 4 Comments on Yuan’s proof Simple double commutator case General double commutator case Two helpful auxiliary results Appendix: Some remarks on “dissipation” in open quantum systems 5 Dirr (Würzburg) QC-Workshop, Paris 2010 2 / 33

Preliminaries: The Lindblad-Kossakowski Equation (LKE) Let ρ ∈ D n be the density matrix of a quantum system with underlying state space C n , i.e. ρ ∈ C n × n , ρ = ρ † ≥ 0, and tr ρ = 1. Its dynamics are given by (Σ) ρ = − i [ H , ρ ] + L ( ρ ) . ˙ (Lindblad-Kossakowski-Equation) Here, H denotes the system’s Hamiltonian and L allows for interactions with the environment ( = open quantum system). Two equivalent representations of the interaction term L : N � 2 L k ρ L † k − L † k L k ρ − ρ L † L ( ρ ) := k L k , (Lindblad-Form) k = 1 wherer L 1 , . . . , L N ∈ C n × n and N ∈ N are arbitrary or, equivalently, ... Dirr (Würzburg) QC-Workshop, Paris 2010 3 / 33

The Lindblad-Kossakowski Equation (cont’d) Two equivalent representations of the interaction term L : N � 2 L k ρ L † k − L † k L k ρ − ρ L † L ( ρ ) := k L k , (Lindblad-Form) k = 1 where L 1 , . . . , L N ∈ C n × n and N ≤ n 2 − 1, or equivalently, n 2 − 1 � � � [ G j , ρ G † k ] + [ G j ρ, G † L ( ρ ) := a jk k ] , (GKS-Form) j , k , = 1 where G 1 , . . . , G n 2 − 1 form an orthonormal basis of sl C ( n ) and the GKS-matrix A := ( a jk ) j , k = 1 ,..., n 2 − 1 is positive semi-definite. The vector of coherence representation which transfers (Σ) to a (bilinear control) system in R n 2 − 1 will not be used in this talk. Dirr (Würzburg) QC-Workshop, Paris 2010 4 / 33

Preliminaries: Completely positive maps (CP-maps) A linear map P : C n × n → C n × n is positive if and only if ρ = ρ † ≥ 0 P ( ρ ) = P ( ρ ) † ≥ 0 , = ⇒ i.e. the set of all positive semidefinite matrices is invariant under P . A linear map P : C n × n → C n × n is completely positive if and only if I p ⊗ P : C p × p ⊗ C n × n → C p × p ⊗ C n × n is positive for all p ∈ N . A linear map P : C n × n → C n × n is unital if and only if P ( I n ) = I n . A linear map P : C n × n → C n × n is trace-preserving if and only if for all ρ ∈ C n × n . tr P ( ρ ) = tr ρ Remark Positive maps which are unital and trace-preserving are often called doubly stochastic. Dirr (Würzburg) QC-Workshop, Paris 2010 5 / 33

Completely positive maps (cont’d) Facts: Any completely positive map P : C n × n → C n × n is given by N � V k ρ V † P ( ρ ) := k , (diagonal Kraus-Form) k = 1 where V 1 , . . . , V N ∈ C n × n and N ≤ n 2 The forward flow of the LKE yields a one-parameter semigroup of completely positive, trace-preserving maps. Dirr (Würzburg) QC-Workshop, Paris 2010 6 / 33

Preliminaries: Controlled LKEs Introducing Hamiltonian controls to the LKE (in a semiclassical way) yields m � � � (Σ c ) ρ = − i ˙ H 0 + u k ( t ) H k , ρ + L ( ρ ) , (controlled LKE) k = 1 where t �→ u k ( t ) ∈ U ⊂ R m are arbitrary measurable, locally bounded control functions. U = R m General assumption: Further notation: R ≤ T ( ρ 0 ) := reachable set of ρ 0 up to time T > 0. R ( ρ 0 ) := entire reachable set of ρ 0 Dirr (Würzburg) QC-Workshop, Paris 2010 7 / 33

Preliminaries: The majorization preorder y be decreasing rearrangements of x ∈ R n and y ∈ R n , respectively. Let ˆ x and ˆ Then, x is majorized by y (denoted by x ≺ y ) if and only if n n � � x 1 ≤ ˆ ˆ y 1 , x 1 + ˆ ˆ x 2 ≤ ˆ y 1 + ˆ y 2 , . . . , ˆ x k = y k . ˆ k = 1 k = 1 For density matrices we define ρ ≺ ρ ′ ( λ 1 , . . . , λ n ) ≺ ( λ ′ 1 , . . . , λ ′ : ⇐ ⇒ n ) , where λ 1 , . . . , λ n and λ ′ 1 , . . . , λ ′ n are the eigenvalues of ρ and ρ ′ , respectively. Dirr (Würzburg) QC-Workshop, Paris 2010 8 / 33

Result I: The Majorization Theorem Theorem I [H. Yuan, IEEE 2010] Let (Σ) be any LKE. Then the following statements are equivalent: (a) (Σ) is unital, i.e. L ( I n ) = 0. (b) The flow Φ t of (Σ) is monotonically majorization-decreasing, i.e. Φ t ( ρ 0 ) ≺ Φ t ′ ( ρ 0 ) for all t ≥ t ′ and all ρ 0 ∈ D n . L ( I n ) = 0 ⇐ ⇒ Φ t ( I n ) = I n for all t ∈ R . Note: Corollary Let (Σ c ) be any controlled unital LKE. Then one has � � cl R ( ρ 0 ) ⊂ { ρ ∈ D n | ρ ≺ ρ 0 } = conv O ( ρ 0 ) for all ρ 0 ∈ D n . Dirr (Würzburg) QC-Workshop, Paris 2010 9 / 33

Result II: Reachable sets of 2-level systems Theorem II[H. Yuan, IEEE 2010 1 ] Let n = 2 and consider a controlled unital LKE of the form � � (Σ c ) ρ = − i ˙ σ z + u x ( t ) σ x + u y ( t ) σ y , ρ + L ( ρ ) , where σ x , σ y and σ z denote the Pauli matrices. Then, for generic L one has � � cl R ( ρ 0 ) = { ρ ∈ D 2 | ρ ≺ ρ 0 } = conv O ( ρ 0 ) for all ρ 0 ∈ D 2 . 1 This result is not explicitly stated, but implicitly contained in the cited paper. Dirr (Würzburg) QC-Workshop, Paris 2010 10 / 33

Refined Majorization Theorem: Unitary invariant norms and the von Neumann entropy A norm � · � on C n × n is unitarily invariant if and only if for all X ∈ C n × n and all U , W ∈ U ( n ) � UXW � = � X � Standard examples: � 1 / 2 = � � tr | X | 2 � 1 / 2 tr X † X � X � 2 := (Hilbert/Schmidt norm) More general, let p ∈ [ 1 , ∞ ) and set √ � tr | X | p � 1 / 2 , X † X � X � p := | X | := (p-norm) Let ρ be a density matrix. The von Neumann entropy of ρ is defined by N ( ρ ) := − tr ( ρ log ρ ) . Dirr (Würzburg) QC-Workshop, Paris 2010 11 / 33

Refined Majorization Theorem: Statement Theorem A Let (Σ) be any LKE. Then the following statements are equivalent: (a) (Σ) is unital, i.e. L ( I n ) = 0. (b) The flow Φ t of (Σ) is monotonically majorization-decreasing, i.e. Φ t ( ρ 0 ) ≺ Φ t ′ ( ρ 0 ) for all t ≥ t ′ and all ρ 0 ∈ D n . (c) The flow Φ t of (Σ) is monotonically norm-decreasing for all unitarily invari- ant norms � · � , i.e. � Φ t ( ρ 0 ) � ≤ � Φ t ′ ( ρ 0 ) � for all t ≥ t ′ and all ρ 0 ∈ D n . (d) The flow Φ t of (Σ) is monotonically norm-decreasing for at least one strictly convex, unitarily invariant norm, i.e. � Φ t ( ρ 0 ) � ≤ � Φ t ′ ( ρ 0 ) � for all t ≥ t ′ and all ρ 0 ∈ D n . (e) The flow Φ t of (Σ) is monotonically increasing with respect to the von Neu- � � � � for all t ≥ t ′ and all ρ 0 ∈ D n . mann entropy, i.e. N Φ t ( ρ 0 ) ≥ N Φ t ′ ( ρ 0 ) Dirr (Würzburg) QC-Workshop, Paris 2010 12 / 33

Refined Majorization Theorem: Consequence Corollary The semi-flow Φ t ≥ 0 of (Σ) is a (weak) contraction semi-group with respect to all unitarily invariant norm if and only if (Σ) is unital. Remark Theorem A generalizes several known results from the literature, e.g. Φ t is purity decreasing if and only if (Σ) is unital. [Lidar, et al.] The semi-flow Φ t ≥ 0 of (Σ) is a (weak) contraction semi-group with respect to all p -norm if and only if (Σ) is unital. [Wolf, et al.] Dirr (Würzburg) QC-Workshop, Paris 2010 13 / 33

Refined Majorization Theorem: Proof The essential ingredient to the proof of Theorem A is the following result: Theorem (Uhlmann 1971, Ando 1989) The following statements are equivalent: (a) ρ ≺ ρ ′ . (b) ρ ∈ conv O ( ρ ′ ) , where O ( ρ ′ ) denotes the unitray orbit of ρ ′ . (c) There exists a unital, trace-preserving, CP-map P with P ( ρ ′ ) = ρ . (d) There exists a unital, trace-preserving, positive map P ( = doubly stochastic map) with P ( ρ ′ ) = ρ . Dirr (Würzburg) QC-Workshop, Paris 2010 14 / 33

Refined Majorization Theorem: Proof “(a) = ⇒ (b):” Ando (Thm. 7.1) “(b) = ⇒ (a):” “Monotonicity” implies Φ t ( I n / n ) ≺ I n / n for all t ≥ 0. However, I n / n ≺ ρ for all density operators ρ . Hence, Φ t ( I n / n ) = I n / n for all t ≥ 0 and thus L ( I n / n ) = 0. “(a) = ⇒ (c):” Ando (Corollary 7.8) “(c) = ⇒ (d):” � “(a) = ⇒ (e):” Ando (Thm. 7.4), Yuan (Prop. 5) “(e) = ⇒ (a):” � � ≥ N ( I n / n ) for all t ≥ 0. “Monotonicity” implies N Φ t ( I n / n ) However, N ( I n / n ) ≥ N ( ρ ) for all ρ ∈ D n and N ( I n / n ) > N ( ρ ) if ρ � = I n / n . Hence, Φ t ( I n / n ) = I n / n for all t ≥ 0 and thus L is unital. Dirr (Würzburg) QC-Workshop, Paris 2010 15 / 33