A Convergent Control Strategy for Quantum Systems

Theorem 1

([10]) Suppose system (2) is transferred, from an given initial state to the desired, target state ρf, whose coherent vector isRρf=(f1,f2,⋯,fn2−1)T.In order to make the target state ρ_f be the minimum point of the energy function (3), the coherent vectors of Ρ and, ρ_f must have opposite directions, i.e.,

in whichρf=I/n+∑jfjXjandP=c0I+∑kckXk,where {– iX_l} is the basis of su(n),

which satisfiestr(XXlXXj)=δlj;x_l is a real number. The vector R_ρ formed, by x₁,x₂,··· , andxn2−1is called, the coherent vector of density matrix ρ, i.e., Rρ=(x1,x2,⋯,xn2−1)T.

Theorem 1 implies that Ρ is not unique. For instance, the simplest way to get observable operator Ρ is to let Ρ = –ρ_f. For a two-level system with the target state ρ_f = |0〉〈0|, R_P and Rρf have opposite directions when one takes Ρ = a |0〉〈0| + b |1〉〈1| with a < b. Generally, for pure states, the following conclusion holds:

Corollary 1

Suppose the system state and the target state are both pure states, i.e., ρ = |ψ〉〈ψ| and, ρ_f = |ψ_f〉〈ψ_f|. In order for ρf to be the minimum point of (3), observable operator Ρ can be constructed by an orthonormal basis containing target state ρ_f:

Remark 1

It can be proven from (7) that energy function (3) reaches its minimum if and only if |ψ〉 = |ψ_f〉 for a two-level system with pure states. According to Theorem 1, the coherent vector of Ρ in (7) has an opposite direction with that of ρ_f, i.e. Ε (ρ) = –tr(ρρ_f) ≤ 0. This energy function is not a standard Lyapunov function with Ε (ρ) ≥ 0 any more because Ρ is no longer positive definite. That is the reason why we define a bounded energy function (3).

Theorem 1 and Corollary 1 are the construction methods and the expression of observable operator Ρ for general states including the pure states. Such an observable operator guarantees that target state ρ_f is the minimum point of energy function (3) and control law (5) is stable. In order to make such controls be convergent, some restrictions on control Hamiltonians should be placed. To this end, three assumptions are given in advance.

1. System (1) is strongly regular, i.e., all the transition frequencies are different, viz. Δ_jk ≠ Δpq, (j, k) ≠ (p, q), where Δ_jk = λ_j – λ_j and e_j is an eigenvalues of H₀.

2. Control Hamiltonian H_l has a particular structure: Hl∈{ℏhjk|hjk=|j〉〈k|+|k〉〈j|,j>k}, where |j〉 is the eigenstate associated with λ_j.

3. Initial state and target state are unitarily equivalent, i.e., there exists a unitary transform U such that ρ_f = UρU^†.

Remark 2

Assumption 1 indicates that transition frequencies between all the energy levels are distinct, so that each transition can be distinguished and selected. This enables Assumption 2 to hold by considering rotating wave approximation. Operator h_jk denotes that the transition between energy levels j and k is admitted. If two transitions are the same, e.g., Δ₁₂ = Δ₃₄, then h₁₂ and h₃₄ should be incorporated as h = h₁₂ + h₃₄·

Now we’ll propose a sufficient and necessary condition that guarantees the convergence of control system (1):

Theorem 2

Based on Assumptions 1, 2 and 3, for any given initial and the target states, observable operator Ρ is constructed according to (6). Then, a sufficient and necessary condition that system (1) with control law (5) converges to the target state is: for∀j,k,∃l,s.t.Hl=ℏhjk.

Lemma 1

If scalar function E(x,t) satisfies: (1) E(x,t) is lower bounded; (2) Ė(x,t) is negative semi-definite; (3) Ė(x,t) is uniformly continuous in time. Then, Ė(x,t) → 0 as t → ∞.

For the pure state, the energy function E(ρ)=tr(Pρf)=−tr(ρρf)≥−tr(ρf2) is lower bounded. Its first derivative is negative semi-definite under control law (5). The second derivative is

Eq. (8) is bounded when the inputs are bounded. Thus, Ė(ρ,t) is uniformly continuous in time. According to Lemma 1, the first order time derivative of the energy function converges to zero, viz., Ė(ρ(∞) ,∞) = 0.

Let R be the set of critical points on any dynamic trajectory, viz.,

Then, the controlled system converges to R.

Moreover, system (2) is homogeneous, and the states in R make their controls be zero according to (5). So, the largest invariant set in R is itself.

Theorem 3

System (2) with control law (5) converges to the set R defined by (9).

Wang and Schirmer[6] indicated that kinematically critical points ρ_s can be defined by [ρ_s,ρ_f] = 0, and if the system is controllable, then all the critical points except the maximum and minimum points are kinematically critical stable. However, kinematical analysis implicates that the direction fields of state evolution are unrestricted. In fact, the direction fields determined by the Hamiltonians are usually non-arbitrary. This means that some trajectories in kinematics are forbidden in dynamics. That is to say, the results in kinematics and in dynamics may be different. So, dynamical stability of critical points in R will be further analyzed as follows.

First, we show that if the conditions in Theorem 2 are satisfied, then critical points in kinematics and in dynamics are the same.

Lemma 2

If the condition in Theorem 2 is satisfied. Then, ρ_s can be defined by [ρ_s, P] = D, viz., the invariant set R in Theorem 3 can be redefined by:

where D is a diagonal matrix.

Lemma 3

([12]) If rankrankA~(P→)=n2−nholds, then the invariant set R is regular, viz. R = {ρ_s : [ρ_s,P] = 0}, where one denotes[ρ,P]=Adp(ρ→),and Ad_p is a linear map from Hermitian or anti-Hermitian matrices into su(n). LetA(P→)be the real (n² – l) * (n² – l) matrix corresponding to the Bloch representation of Ad_p. Denote su(n) =T ⊕ C andRn2−1=ST⊕SC, where S_C and S_T are real subspaces corresponding to the Cartan and non-Cartari subspaces, C and T, respectively. And thenA~(P→) be the first n² – n rows ofA(P→).

For target state ρ_f with diagonal type, the inviariant set with Ρ = –ρ_f in (10) can be simplified as:

For non-diagonal target state ρ_f included some mixed states and superposition states, Lemma 3 is employed. If ρ_f satisfies Lemma 3 automatically, one obtains (11) with Ρ = –ρ_f.Otherwise, taking superposition states for example, Ρ is chosen as Eq. (7). We can always choose eigenvalues p_i to meet the condition of Lemma 3, then (11) is obtained. To sum up, for most diagonal target states contained diagonal target states, the superposition states and some mixed-states satisfy Lemma 3, there are two ways to construct P: one is Ρ =–ρ_f and the other one is (7), by which the invariant set can be reduced to (11). All the convergence analysis will be carried out based on (11).

Next, we will give a sufficient and necessary condition on control Hamiltonians such that all the states in R except the maximum and minimum points are dynamically critical stable under Assumptions 1 and 2.

For a dynamically minimum point, Ë(ρ_s,t) ≥ 0 holds for any time and any control u = (u₁, u₂,⋯u_N). Otherwise, the state will be critical stable. That is to say, it is enough to consider the second derivative of the energy function.

For tr([ρs,P]A˙j(t))=iℏtr([ρs,P][H0,Aj])=0, the following equation holds:

If Ρ = –ρ_f, one has:

If Ρ is chosen as (7), where we denote ρ_h = |ψ_h〉〈ψ_h|, one obtains:

Obviously, ρ_f is a stable point since Ë(ρ_f, t) ≥ 0 holds for ∀t. Further analysis implicates the following Lemma:

Lemma 4

For any diagonal target state ρ_f a sufficient and necessary condition such that all the states in R except the minimum point are dynamically critical stable is: for ∀l₀,r₀, ∃A_q (t), such that(Aq(t))l0r0≠0.

Proof

Necessity. It is equivalent to prove that if ∃l₀,r₀, such that (Aq(t))l0r0=0 for A_q (t), (A_q (t))_lr ≠ 0, then ∃ρ_f, ρ_s ≠ ρ_f such that Ë(ρ_s,t) ≥ 0 holds for ∀t and any controls.

We can denote diagonal matrices ρ_f and ρ_s as follows. Let the two smallest elements of ρ_f are (ρf)r0r0and(ρf)l0l0. Then, ρ_s can be selected by exchanging the l₀thand r₀th elements of ρ_f, i.e., (ρs)l0l0=(ρf)r0r0,and(ρs)r0r0=(ρf)l0l0. Obviously, ρ_f and ρ_s are commutable, viz., ρ_s ∊ R.

According to Assumptions 1 and 2, one has:

Denote H_q by H_lr and u_q by u_lr. If (A_q (t))_lr ≠0, it can be obtained from (12) that

For ∀j≠l0,j≠r0,(ρs)jj=(ρf)jj,((ρs)rr−(ρs)ll)((ρf)rr−(ρf)ll)≥0 holds. Therefore, holds for Ë(ρ_s, t) ≥ 0 and any controls. The necessity is proven.

Sufficiency. It is enough to prove that ∃u, Ë(ρ_s) < 0.

For [ρ_s, ρ_f] = 0, ρ_s is a diagonal matrix, too. Since ρ_f and ρ_s are unitarily equivalent, ρ_s has the same spectrum with ρ_f, that is, the diagonal elements of ρ_s is a permutation of that of ρ_f. Suppose the largest element of the permutated ones is (ρf)l0l0or(ρs)r0r0, then we have (ρs)r0r0−(ρs)l0l0>0and(ρf)r0r0−(ρf)l0l0<0. According to (15), if ul0r0≠0 and other controls are equal to zero, then Ë(ρ_s) < 0. The sufficiency is proven. Lemma 4 is proven. □

For a general ρ_f, the proof of the necessity is the same as Lemma 4, one needs only to prove the sufficiency.

Now, suppose ρf=∑jcj|ψj〉〈ψj|,where|ψj〉=∑kαjk|k〉. There must be U=∑j|j〉〈ψj| to diagonalize ρ_f and ρ_s simultaneously, then

where D_s, D_f are two diagonal matrices. Especially, if ρ_f is a superposition state, Ρ is chosen as (7). Eq. (16) is also obtained from (13). Calculate ∑jujUAj(t)U† one gets

in which, the first sum on the right hand side is a diagonal matrix, which commutes with D_s, D_f. So, it can be ignored. Let

Then

Eq. (18) can be rewritten as:

If the square matrix in (20) is noted as Μ, it is easy to verify that if the condition in Theorem 2 is satisfied, M has full rank. Because if there exists a linear combination of some columns in the square matrix being zero, then for ∀_j, k, there is

Consider arbitrary two states |ψ′j〉and|ψ′j〉,equation (21) holds only for β_lr = 0, viz. the matrix M is full rank.

In the following we will find the control u that makes Ë(ρ_s.u) < 0. Equation (21) can be rewritten as:

in which K is a diagonal matrix and defined by:

Matrix Κ has at least one negative element, because the two matrices D_s and D_f satisfy D_f ≠ D_s, and their diagonal elements are ranges of the same numbers. In other words, matrix Κ is non-positive definite. Therefore the matrix M^†KM is non-positive definite, too. And its eigenvalues include at least one real negative element. Viz.

where Q is a real negative diagonal matrix in its k₀th element; Τ is an unitary matrix whose k₀th row is supposed to be z^†.The matrix Τ and the vector z can be separated into real and imaginary parts, respectively, as:

and

Because Ë(ρ_s.u)is a real number, so the imaginary part of the matrix M⁺KM has no contribution to Ë(ρ_s.u), thus equation (22) can be written as:

If the control u is replaced by z:

Then one can see that at least one of the two items x′(A′QA+B′QB)x and y′(A′QA+B′QB)y is negative. It is no harm to suppose that x′ (A′QA + B′QB)x < 0, then if only the control is the real part of the vector z, one can has Ë(ρ_s, Re (z)) < 0. So if the condition in Theorem 2 is satisfied, all the states in the largest invariant set R except the target state are dynamically critical stable, i.e., Theorem 2 can be obtained.

Theorem 2 means that any restriction on the evolution direction will produce new dynamically stable points, which may have a strong impact on control design. The condition in Theorem 2 is obtained based on Assumption 2, which in fact defines a basic orthonormal basis for permissible state evolving directions. This basic basis spans a full space of evolving direction fields when the condition in Theorem 2 is satisfied. In fact, Assumption 2 is not necessary, if another basis spans the same space of evolving direction fields, then all the states in invariant set R except the target state are dynamically critical stable, and then control law (5) is still convergent.