A geometric analysis of the Markovian evolution of open quantum systems

Transcription

1 A geometric analysis of the Markovian evolution of open quantum systems University of Zaragoza, BIFI, Spain Joint work with J. F. Cariñena, J. Clemente-Gallardo, G. Marmo Martes Cuantico, June 13th, 2017

2 Outline of the talk 1 Introduction: open quantum systems and Markovian evolution

3 Contents Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation 1 Introduction: open quantum systems and Markovian evolution

4 Open quantum systems Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation Open quantum systems evolve under more complex dynamics than the Schrödinger equation. The interaction with their environment causes phenomenon such as decoherence and lose of purity.

5 Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation Consider a system of two interacting spins. H = H A H B, H A = H B = C 2. Each spin alone can be described as an open quantum system, with the other spin being the environment. Even if the whole system evolves unitarily, it is possible that the state of the total system is an entangled state, for example: ψ = 1 2 ( ) H. It is not possible to describe the state of any of the composing spins as an element of the Hilbert space associated to the subsystem. It is however possible to determine expectation value functions of observables on any of the subsystems. For example: O Herm(H A ), O A = ψ O A I B ψ.

6 Open quantum systems Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation In general, open quantum systems appear as parts of composite systems. The whole system evolves under the Schrödinger equation. However, as in general the environment cannot be completely characterised, it is not possible to analyse the whole composite system and then trace out the environment. Instead, the analysis of open quantum systems involves proposing dynamical evolutions different from the unitary one.

7 Gleason s theorem Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation Wave functions are not enough to describe open quantum systems. Instead, states are described as probabilistic measurements on the algebra Herm(H) of observables of the system. Theorem (Gleason) For any probability measure A on O, there exists a unique trace class operator ρ such that A = Tr(ρA). As a consequence, it is enough to consider the set S of density matrices of an open quantum system, a convex set defined as S = {ρ Herm(H) Trρ = 1; Tr(ρA 2 ) 0, A Herm(H)}.

8 Markovian evolution Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation Dynamics of open quantum systems is thus described in terms of evolutions on the set S of density matrices of the system. Among all the possible dynamics, consider in particular the Markovian evolution. The evolution of a system is said to be Markovian if it depends only on the present state of the system, i.e. it has no memory of previous states. Proposition (Breuer, Petruccione (2002). The Theory of Open Quantum Systems) It is possible to describe the Markovian evolution of a quantum system by a one-parameter family of transformations {Φ t : Herm(H) Herm(H), t 0} with a semigroup structure: Φ t Φ s = Φ t+s.

9 The Kossakowski-Lindblad operator Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation By imposing on the one-parameter family of transformations the preservation of positivity, it is possible to describe the generator of the family by the Kossakowski-Lindblad equation. Theorem (Gorini, Kossakowski, Sudarshan (1976); Lindblad (1976)) A linear operator L : Herm(H) Herm(H), with dim H = n, is the generator of a completely positive dynamical semigroup of Herm(H) if it can be expressed in the (non unique) form L(ρ) = i[h, ρ] n 2 1 j,k=1 c jk ( [Fj, ρf k ] + [F jρ, F k ]), where H = H, Tr(H) = Tr(F j ) = 0, Tr(F j F k ) = δ jk and (c jk ) is a complex positive matrix.

10 The Kossakowski-Lindblad equation Open quantum systems Markovian evolution: the Kossakowski-Lindblad equation Alternative forms of the Kossakowski-Lindblad operator can be obtained. By diagonalising the (c jk ) matrix and redefining the operators, it is possible to obtain the following expression: L(ρ) = H, ρ 1 n V ρ + V j ρv j, j=1 n 2 1 V = V j V j, j=1 with H, ρ = i(hρ ρh) and V ρ = V ρ + ρv. The differential equation governing the evolution is d dt ρ = L(ρ). A geometric interpretation as a vector field on the manifold S of states of open quantum systems is straighforward.

11 Contents Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball 1 Introduction: open quantum systems and Markovian evolution

12 Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball The origin of the relevance of geometry in physics Aiming to solve the n-body problem, Henry Poincaré introduced new mathematical methods and developed a qualitative analysis of Mechanics. This was the starting point of the application of differential geometry and topology to the resolution of mechanical problems. A qualitative description is concerned with the description of the phase space as a whole. Its topological properties are essential in order to understand the behaviour of systems. Dynamics is represented by vector fields and their integral curves. Their analysis allows to describe stability, integrability and other intrinsic properties of the systems.

13 Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball Geometric formalism for density matrices A geometric approach can be taken in the analysis of states of open quantum systems. For an n-level system, with Hilbert space H: The set S of density matrices is an (n 2 1)-dimensional manifold with boundary, with elements of the form: ρ = 1 I + n n 2 1 j=1 x j σ j, with {σ 0 = I, σ 1,..., σ n 2 1} an orthonormal basis for Herm(H). Observables A Herm(H) are represented as functions f A (ρ) = Tr(Aρ).

14 The 2-level system Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball For a 2-level system, H = C 2. A basis for Herm(H) is ( ) ( ) ( i σ 0 =, σ =, σ = i 0 ), σ 3 = ( ) By imposing Trρ = 1 and Tr(ρ 2 ) 1, the coordinate expression of a density matrix of the 2-level system is ρ = 1 2 σ x jσ j = 1 ( ) 1 + x3 x 1 ix 2, x 2 x 1 + ix 2 1 x x2 2 + x As a consequence, the set S of density matrices of a 2-level quantum system is parametrised by the points in a 3-dimensional ball of radius 1, known as the Bloch ball.

15 The Bloch ball Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball ( ( 0 1 Basis : 0 =, 1 = 1) 0) ( ) ρ 0 = 0 0 =, 0 1 ( ) ρ 1 = 1 1 =, 0 0

16 The Bloch ball Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball ( ( 0 1 Basis : 0 =, 1 = 1) 0) ( ) ρ 0 = 0 0 =, 0 1 ( ) ρ 1 = 1 1 =, ( ) ρ A = ( 0 + i 1 ) ρ B = 1 2 ρ M = 1 2 ρ ρ 1 = 1 2 ( ) 1 1, 1 1 ( 1 i i 1 ( ) ),

17 Bloch ball stratification Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball Points on the surface of the ball (that is, vectors with radius 1) parametrise states that are rank-1 projectors, as ρ 2 = ρ; these are the pure states of the system. The interior of the ball parametrises mixed states. The set of states is stratified as S = S 1 S 2, with S 1 the surface of the Bloch ball and S 2 its interior. It is thus immediate to identify the sphere S 1 with the manifold of pure states of the system.

18 Unitary evolution Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball Consider the unitary evolution of states, governed by von Neumann equation: d ρ = i[h, ρ] = H, ρ, dt with H the Hamiltonian observable of the system. In coordinates: H = 3 B µ σ µ H, ρ = 1 2 µ=0 3 B j x k σ j, σ k = j,k=1 3 j,k,l=1 ε jkl B j x k σ l. It is immediate to consider a vector field X H on S with value at each point given by H, ρ. In coordinates (with 3-dimensional vector notation): X H ρ = 2 B x. Integral curves of X H are rotations around the vector B.

19 Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball Hamiltonian vector field for B=(0,0,1)

20 Non-unitary evolution Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball Non-unitary evolutions include additional terms. The Kossakowski-Lindblad operator, L(ρ) = H, ρ 1 n V ρ + V j ρv j, j=1 n 2 1 V = V j V j, j=1 involves a terms with an anticommutator and a Kraus map. Evolution by anticommutator generators has been proposed as model for dissipations (Kaufman (1984), Morrison (1986)): dρ dt = S ρ = Sρ + ρs.

21 Non-unitary evolution Geometry and Mechanics Open quantum systems: the 2-level system Dynamics on the Bloch ball However, a careful description is needed for these models. Observe that the set S, as a subset of Herm(H), is not preserved by these evolutions: d (Trρ) = 2Tr(Sρ) 0. dt It is necessary to take a step back and obtain a precise description of geometric structures on S.

22 Contents Representation of the set of states The reduction procedure Geometric structures on the set of states 1 Introduction: open quantum systems and Markovian evolution

23 Representation of the set of states The reduction procedure Geometric structures on the set of states GOAL: To provide useful expressions, i.e. recipes, for the vector field associated to the Kossakowski-Lindblad operator. PROCESS: 1 To determine the problem, by a full description of the manifold of states as a subset of Herm(H). 2 To provide a method to properly map objects from Herm(H) to S: the reduction procedure. 3 To apply the reduction procedure to the existing structures.

24 Representation of the set of states Representation of the set of states The reduction procedure Geometric structures on the set of states States are positive normalised linear elements in V = Herm(H): S = {ρ Herm(H) f I (ρ) = 1, f A 2(ρ) 0} V = Herm(H). On the whole linear space V, commutators and anticommutators of observables define vector fields with values at each point: X H ξ = H, ξ, Y A ξ = A ξ. Vector fields have a dual interpretation: as families of tangent vectors and as derivations of functions. They define the following tensor fields, which act on the differentials of functions: Λ(df A, df B )(ξ) = X A ξ (f B ) = f A,B (ξ) =: {f A, f B }, R(df A, df B )(ξ) =Y A ξ (f B ) = f A B (ξ) =: (f A, f B ),

25 Restriction to the set of states Representation of the set of states The reduction procedure Geometric structures on the set of states A correct description of the whole algebraic structure, in particular of the Jordan structure, presents difficulties, as the set of states is not preserved by gradient vector fields. X H (f I ) =Λ(df H, df I ) = 0, Y A (f I ) =R(df A, df I ) = 2f A 0. Geometric objects are to be mapped from the whole space onto S. A projection and a reduction procedure are needed.

26 Group action on the dual space Representation of the set of states The reduction procedure Geometric structures on the set of states Consider the group action on V 0 = V {0}: φ : R + V 0 V 0, φ(a, ξ) = aξ. This action is generated by the dilation vector field = Y I. As the orbits are regular, the image by the projection π : V 0 V 0 /R + is a differentiable manifold. This quotient manifold can be embedded as the unit sphere in V 0. By relating points in the same orbits of the group action, it is possible to define a bijection ϖ : (V 0 /R + ) Π S

27 Representation of the set of states The reduction procedure Geometric structures on the set of states

28 Projection onto S Representation of the set of states The reduction procedure Geometric structures on the set of states A projection of geometric objects is done by identifying their values at different points of the orbits. However, the relevant tensor fields and functions are not invariant under the group action: L Λ 0, L R 0, L f A 0. A different approach has to be taken.

29 Representation of the set of states The reduction procedure Geometric structures on the set of states Reduction of Lie-Jordan algebras of functions Theorem (Falceto, Ferro, Ibort, Marmo, 2013) Consider an algebra (C (M), ) of smooth functions on the manifold M. If the set G of invariant functions with respect to the Lie group action φ : G M M is a subalgebra, then the restriction of the composition law to G defines an algebraic structure on the set of smooth functions on the orbit set M/G. This property can be considered in a tensorial way. Consider a tensor field Ξ associated with a composition law of smooth functions: Ξ(df, df ) = f f, f, f C (M). The condition given in the theorem is equivalent to the assumption that Ξ is π-projectable by π : M M/G. Thus, the new composition law of smooth functions on M G is determined by a tensor field Ξ G on M/G which is π-related with Ξ.

30 Invariant functions Representation of the set of states The reduction procedure Geometric structures on the set of states Invariant functions under the group action are expectation value functions e A (ξ) = f A(ξ) f I (ξ) = Tr(Aξ) Tr(ξ). They do not close an algebra with the existing tensor fields Λ and R: { fa {e A, e B } =, f } B = f A,B f I f I fi 2 = 1 e A,B, f I ( fa (e A, e B ) =, f ) B = 1 f I f I fi 2 (f A, f B ) f A fi 3 (f I, f B ) f B fi 3 (f A, f I ) + f Af B fi 4 (f I, f I ) = = f A B f 2 I 2f Af B f 3 I = 1 f I (e A B 2e A e B ).

31 Reduction procedure Representation of the set of states The reduction procedure Geometric structures on the set of states There exists a pair of tensor fields Λ = f I Λ and R = f I R that can be reduced by the group action φ : R + V 0 V 0. Consider the map ϖ π : V S and the notation: e A = (ϖ π) (ɛ A ), Λ S = (ϖ π) ( Λ), R S = (ϖ π) ( R). The action of the tensor fields on S on expectation value functions are Theorem Λ S (dɛ A, dɛ B ) = ɛ A,B, R S (dɛ A, dɛ B ) = ɛ A B 2ɛ A ɛ B. The set of expectation value functions on S is a Lie-Jordan algebra with respect to the products {ɛ A, ɛ B } S :=Λ S (dɛ A, dɛ B ) = ɛ A,B, (ɛ A, ɛ B ) S :=R S (dɛ A, dɛ B ) + 2ɛ A ɛ B = ɛ A B.

32 Contents Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay 1 Introduction: open quantum systems and Markovian evolution

33 Tensor fields on the 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Coordinate functions on S are given by the expectation value functions associated to the Pauli matrices: x j = ɛ σj (ρ). With the products σ j, σ k = 2ε jkl σ l, σ j σ k = 2δ jk σ 0, the values of Λ S and R S on the coordinate functions are the following Λ S (dx j, dx k )(ρ) = ɛ σj,σ k (ρ) = 2ε jkl x l, R S (dx j, dx k )(ρ) = ɛ σj σ k (ρ) 2ɛ j (ρ)ɛ k (ρ) = 2δ jk 2x j x k. The coordinate expressions for tensor fields Λ S and R S are 3 Λ S = ε jkl x l, x j x k R S = 2 j,k,l=1 3 j=1 x j x j 3 2x j x k. x j x k j,k=1

34 Unitary evolution Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Consider the unitary evolution of states, governed by von Neumann equation: d ρ = i[h, ρ] = H, ρ, dt with H the Hamiltonian observable of the system. Unitary evolution is described in geometric terms by the integral curves of the Hamiltonian vector field X H associated to the expectation value function ɛ H of the Hamiltonian H. In coordinates: H = 3 B µ σ µ ɛ H = µ=0 X H = Λ S (dɛ H, ) = 2 3 j,k,l=1 3 B j x j, j=1 ε jkl B j x l. x k

35 Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Hamiltonian vector field for B=(0,0,1)

36 Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Properties of Hamiltonian vector fields Hamiltonian vector fields preserve the stratification of S. Vectors are always tangent to the surface. In the interior, integral curves of X H never reach the surface. The surface S 1 is a compact manifold. Thus, a complete vector field necessarily has fixed points. In this case, fixed points correspond with eigenstates of the Hamiltonian. In the given coordinates, fixed points are (0, 0, 1) and (0, 0, 1). The vector field is linear. This implies that (convex) linear combinations of fixed points are also fixed points. As a result, the whole x 3 -axis is fixed under unitary dynamics.

37 Dissipative evolution Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Evolution associated with the equation d ρ = S ρ, dt is described in geometric terms by the integral curves of the gradient vector field ỸS associated to the expectation value function ɛ S of the observable S. In coordinates: S = 3 S µ σ µ ɛ S = µ=0 Ỹ S = R S (dɛ S, ) = 2 3 j=1 S j 3 S j x j, j=1 2ɛ S x j 3 j=1 x j. x j

38 Gradient vector field for S=(0,0,1) Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay

39 Properties of gradient vector fields Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Gradient vector fields preserve the stratification of S. Vectors are always tangent to the surface. In the interior, integral curves of ỸS never reach the surface (in finite time). The surface S 1 is a compact manifold. Thus, a complete vector field necessarily has fixed points. In this case, fixed points correspond with eigenstates of the observable S. In the given coordinates, fixed points are (0, 0, 1) and (0, 0, 1). The vector field is non-linear. The only fixed points are the extremal points of the expectation value function ɛ S : d dt ɛ S = Y S (ɛ S ) = 2(ɛ 2 S S 2 ) 0, equality holding only for x = ±S.

40 Kraus maps Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Hamiltonian and gradient vector fields are generators of the action of the general linear group GL(H) on S (T, ρ) GL(H) S T ρt S, T = exp(a+ib), A, B Herm(H). The action of non-invertible operators is instead generated by Kraus maps (Grabowski, Kuś, Marmo (2006)): K(ρ) = r j=1 V j ρv j. Geometrically, the action of this map is represented by the vector field: Z K (ɛ A )(ρ) = ɛ K (A)(ρ) ɛ K (I )(ρ)ɛ A (ρ)

41 Generic Markovian evolution Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Consider two observables H, S and a Kraus map K : V V. The vector field X H + Y S + Z K is tangent to the manifold of states, and its action on expectation value functions is (X H + Y S + Z K )(ɛ A )(ρ) =ɛ H,A (ρ) + ɛ S A (ρ) + ɛ K (A)(ρ) 2ɛ S (ρ)ɛ A (ρ) ɛ K (I )(ρ)ɛ A (ρ). The vector field is non-linear in general. Linearity is recovered if S = 1 r V j 2 V j. In this case, the vector field X H + Y S + Z K is precisely associated to the Kossakowski-Lindblad operator: j=1 L(ρ) = H, ρ 1 n V ρ + V j ρv j, j=1 n 2 1 V = V j V j, j=1

42 Phase damping of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Consider the Kossakowski-Lindblad operator L(ρ) = γ(ρ + σ 3 ρσ 3 ) = 1 2 (γi ) ρ + ( γσ 3 )ρ( γσ 3 ), which is an appropriate operator, as ( γσ 3 ) ( γσ 3 ) = γi. There is no Hamiltonian term, and the gradient term only includes the identity operator: ɛ I = 1 Y I = R(dɛ I, ) = 0. Therefore, only the Kraus term is necessary for the description of the dynamics: Z L = Z K = 2γx 1 2γx 2. x 1 x 2

43 Phase damping of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay

44 Phase damping of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay The stratification of S is no longer preserved. If the initial state is on the boundary and it is not a fixed point, then it evolve into the inner part of the sphere. No integral curve, however, goes from inside onto the boundary. Fixed points are along the x 3 -axis. Physically, these are all the possible statistical (not linear) combinations of the states on the north and south pole. Additionally, the set of fixed points is also the limit manifold of the evolution. This vector field models the measurement process and the decoherence phenomenon: quantum probabilities (linear combinations of states, on the surface) evolve onto classical probabilites (in the interior).

45 Decay of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay The decay of a two level system is modelled by the following Kossakowski-Lindblad operator L(ρ) = 1 ( ) 2 γ(j J) ρ + γjρj 0 0, J = 1 0

46 Decay of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay There is no Hermitian term. The gradient and Kraus terms lead to the following vector fields on S: Y V = 1 2 γx 1x x 1 2 γx 2x x 2 2 γ(x 3 2 1), x 3 Z K = γx 1 2 (1 + x 3) γx 2 x 1 2 (1 + x 3) γ x 2 2 (1 + x 3) 2. x 3 The total Kossakowski-Lindblad vector field is linear: Z L = Y V + Z K = γx 1 2 γx 2 x 1 2 γ(1 + x 3 ). x 2 x 3

47 Decay of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay Gradient term Kraus term

48 Decay of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay

49 Decay of a 2-level system Unitary evolution Dissipative evolution Kraus maps and Markovian evolution Phase damping Decay The stratification of S is no longer preserved. If the initial state is on the boundary and it is not a fixed point, then it evolve into the inner part of the sphere. No integral curve, however, goes from inside onto the boundary in finite time. There is a single fixed point, which is also the limit point of the evolution. Neither the gradient nor the Kraus terms are linear. However, they combine in such a way that the final vector field is linear. This could be interpreted as follows. In the Kossakowski-Lindblad operator, the Kraus term is responsible for the non-unitary evolution. The gradient term is necessary in order to cancel the non-linear summands. A generalisation of the Kossakowski-Lindblad operator could involve considering generic gradient vector fields on its expression.

50 Contents 1 Introduction: open quantum systems and Markovian evolution

51 Different dynamics have been considered for 2-level system. Markovian dynamics provides a useful model for the description of phenomena such as decoherence and dissipation. A geometric analysis also provides a deep characterisation of the relevant structures on the set.

52 Different dynamics have been considered for 2-level system. Markovian dynamics provides a useful model for the description of phenomena such as decoherence and dissipation. A geometric analysis also provides a deep characterisation of the relevant structures on the set. Higher-dimensional systems can be considered. In those cases, the structure of the manifold S is more complex. Not every state on the boundary is a pure state, and other types of the dynamics (such as the decay to a face of the boundary) can be considered.

53 Different dynamics have been considered for 2-level system. Markovian dynamics provides a useful model for the description of phenomena such as decoherence and dissipation. A geometric analysis also provides a deep characterisation of the relevant structures on the set. Higher-dimensional systems can be considered. In those cases, the structure of the manifold S is more complex. Not every state on the boundary is a pure state, and other types of the dynamics (such as the decay to a face of the boundary) can be considered. Differential geometry presents an interesting framework for the analysis of dynamical properties. The existence and characteristics of limit manifolds is related with Lyapunov stability. Tensor fields Λ S and R S are not preserved by the evolution, which can be understood as a contraction of the algebra of observables of the quantum system.

54 Bibliography J. F. Cariñena, A. Ibort, G. Marmo, G. Morandi. Geometry from Dynamics: Classical and Quantum. Springer (2015). J. F. Cariñena, J. Clemente-Gallardo, J. A. Jover-Galtier, G. Marmo. Tensorial dynamics on the space of quantum states. arxiv: J. Grabowski, M. Kuś, G. Marmo. Geometry of quantum systems: density states and entanglement. J. Phys. A 38, (2005). J. Grabowski, M. Kuś, G. Marmo. Symmetries, group actions and entanglement. Open Syst. Inf. Dyn. 13, (2006). J. A. Jover-Galtier. Open quantum systems: geometric description, dynamics and control. PhD Thesis (University of Zaragoza, 2017). F. Strocchi. An Introduction to the Mathematical Structure of Quantum Physics. World Scientific (2005).

55 Thank you for your attention