A Survey of Quantum Lyapunov Control Methods

Abstract

The condition of a quantum Lyapunov-based control which can be well used in a closed quantum system is that the method can make the system convergent but not just stable. In the convergence study of the quantum Lyapunov control, two situations are classified: nondegenerate cases and degenerate cases. For these two situations, respectively, in this paper the target state is divided into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian. For these four categories, the quantum Lyapunov control methods for the closed quantum systems are summarized and analyzed. Particularly, the convergence of the control system to the different target states is reviewed, and how to make the convergence conditions be satisfied is summarized and analyzed.

1. Introduction

The theory of quantum mechanics is one of the major discoveries in the history of science in the 20th century. It is a very important issue to study the properties of the quantum mechanical systems and their control. According to whether the system is isolated or not, a quantum mechanical system can be a closed system or an open system. In a closed quantum system, the evolution of the state is unitary. There are mainly two methods to describe the evolution of a closed quantum system’s states. They are the Schrödinger equation 𝑖|𝜓̇(𝑡)⟩=(𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡))|𝜓(𝑡)⟩ and the quantum Liouville equation 𝑖𝜌̇(𝑡)=[𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡),𝜌(𝑡)], in which |𝜓(𝑡)⟩ is the quantum state vector, 𝜌(𝑡) is the density operator, 𝐻₀ is the internal Hamiltonian, and 𝐻_𝑘,(𝑘=1,…,𝑟) are the control Hamiltonians. In an open quantum system, the system interacts with the surroundings, thus the loss of the system’s information leads to the non-unitary evolution of the state. The most common method to describe the open system is the Lindblad master equation: 𝑖𝜌̇=[𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡),𝜌]+𝐿_𝐷(𝜌), which is in fact the sum of a closed system and a dissipative term caused by the loss of the information or energy. Obviously, the research of the properties of the closed quantum system and its control is relatively simple. Moreover, there is a more important fact: the research of the closed quantum system is the basis of that of the open quantum system.

Quantum control has attracted much attention in recent years and it has been found the potential applications in many fields such as atomic physics [1–4], molecular chemistry [5–9] and quantum information [10, 12]. Up to now, there have been many quantum methods, such as quantum optimal control [13–15], adiabatic control [16–18], the Lyapunov-based control [19–41], and optimal Lyapunov-based quantum control [42]. For the Lyapunov-based quantum control, it is relatively easy to design an analytical but not numerical control law, and the control system based on this control method is at least stable, so it has been a common control method.

One of the major concerns of the Lyapunov control is choosing an appropriate Lyapunov function to design the control laws. Ordinarily, the control laws and the control effects are different when the Lyapunov functions are distinct. It’s a good idea to choose the Lyapunov function based on the geometrical and physical meanings. Usually, there are mainly three Lyapunov functions to be selected: the Lyapunov function based on the state distance [19–21, 26–32, 39–41], the state error [22, 23, 37, 38], and the average value of an imaginary mechanical quantity [24, 25, 39, 41]. The so-called imaginary mechanical quantity means that it is a linear Hermitian operator to be designed and may be not a physically meaningful observable quantity such as coordinate and energy. Among these three Lyapunov functions, the Lyapunov-based quantum control methods based on the state distance and state error only need to adjust the scale factors of the control laws. These two Lyapunov control methods are relatively simple and easy to grasp. The Lyapunov-based quantum control based on the average value of an imaginary mechanical quantity contains more adjustable parameters. So it is more flexible and also more complex at the same time.

Generally speaking, the Lyapunov-based control method can only ensure that the control system is stable. The probability control in the quantum system requires us to design a control strategy which can make the system convergent, because a stable quantum control method may result in that the control system cannot reach the desired target state. Therefore another major concern of this control strategy for the closed quantum systems is the convergence of the control systems. So far, there have been the following research results on the convergence of the closed quantum systems [19, 23–25, 27–29, 33, 36].(I)For the Schrödinger equation, the convergence conditions are as follows. (i) The internal Hamiltonian is strongly regular; (ii) All the eigenstates, which are different from the target state, are directly coupled to the target state for the Lyapunov control based on the state distance or the state error, or any two eigenstates are coupled directly for the Lyapunov control based on the average value of an imaginary mechanical quantity. (II)For the quantum Liouville equation, the convergence conditions are as follows. (i) The internal Hamiltonian is strongly regular, and (ii) the control Hamiltonians are full connected.

At first, the Lyapunov control method which could only ensure the convergence to the target eigenstate was studied [19, 23, 25]. Then, the target mixed state, which commutes with the internal Hamiltonian, was studied [33, 36]. Later, the convergence to the target superposition state was solved by means of designing the control laws in the interaction picture of the control system, and the convergence to the target mixed state which does not commute with the internal Hamiltonian was solved by using a unitary transformation.

In fact, many actual systems do not satisfy the convergence conditions mentioned above, such as the time domain model of the selective excitation of the stimulated Raman scattering [43], the coupled two spin systems, and one-dimensional oscillators [44]. These systems are called nonideal systems and in the degenerate cases. For the degenerate cases, the convergence of the control systems was solved by introducing a series of implicit function perturbations and choosing an implicit Lyapunov function [37–40]. At first, the convergence to the target eigenstate was only guaranteed [37–40]. Then the convergence to the target mixed state which commutes with the internal Hamiltonian was solved by introducing a series of implicit function perturbations, and the convergence to the target superposition state and the target mixed state which does not commute with the internal Hamiltonian was solved by introducing a series of constant disturbances.

The aim of this paper is to summarize and analyze the existing Lyapunov control methods for the nondegenerate and degenerate cases, respectively. Dividing the target state into four categories: the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian state, we summarize the design methods of the control laws, analyze the convergence to the target state, and investigate how to make these conditions of the convergence be satisfied.

The remainder of this paper is arranged as follows. In Section 2, the research results for the nond-egenerate cases are summarized and analyzed. In Section 3, the research results for the degenerate cases are summarized and analyzed. Some concluding remarks are drawn in Section 4.

2. Non-Degenerate Cases

The design of the control laws and the analysis of the convergence are very important in the Lyapunov control method. The design of the control laws is based on the Lyapunov stability theorem, which is to design the control laws to make the selected positive semi-definite Lyapunov function 𝑉(𝑡) satisfy 𝑉̇(𝑡)≤0. The convergence analysis of this control method is mainly based on the LaSalle invariance principle [44] for the autonomous systems, or the improved Barbalat lemma [45] for the non-autonomous systems.

In this Section, we will summarize and analyze the research results on the convergence of the control system in the non-degenerate cases for the target state being an eigenstate, a mixed state which commutes with the internal Hamiltonian, the superposition state, or/and a mixed state which does not commute with the internal Hamiltonian state, respectively.

2.1. Target Eigenstate

It is convenient to use the bilinear Schrödinger equation to describe the control systems if the target state is a pure state. Consider the 𝑁-level closed quantum system governed by the following bilinear Schrödinger equation: 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡)𝜓(𝑡), (1) where |𝜓(𝑡)⟩ is the quantum state vector, 𝐻₀ is the internal Hamiltonian, 𝐻_𝑘,(𝑘=1,…,𝑟) are control Hamiltonians, and 𝑢_𝑘(𝑡),(𝑘=1,…,𝑟) are scalar and real control laws.

In the following sections, in the case that the target state |𝜓_𝑓⟩ is an eigenstate, that is, 𝐻₀|𝜓_𝑓⟩=𝜆_𝑓|𝜓_𝑓⟩, where 𝜆_𝑓 is the eigenvalue of the internal Hamiltonian 𝐻₀, the research results on the convergence are summarized and analyzed for the Lyapunov control based on the state distance, the state error, and the average value of an imaginary mechanical quantity, respectively.

2.1.1. Lyapunov Control Based on State Distance

Consider the following Lyapunov function based on the state distance: 𝑉₁𝜓=121−𝜓∣𝜓_𝑓². (2) The time derivative of the Lyapunov function (2) is 𝑉̇₁=−^𝑟_𝑘=1𝑢_𝑘(𝑡)⋅Im𝜓∣𝜓_𝑓𝜓_𝑓𝐻_𝑘𝜓. (3) The control laws 𝑢_𝑘(𝑡) which can make 𝑉̇₁≤0 hold can be designed as 𝑢_𝑘(𝑡)=𝐾_𝑘𝑓_𝑘Im𝜓∣𝜓_𝑓𝜓_𝑓𝐻_𝑘𝜓,(𝑘=1,…,𝑟), (4) where 𝐾_𝑘>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=1,…,𝑟) are monotonic increasing functions through the coordinate origin of the plane 𝑥_𝑘−𝑦_𝑘.

The control laws designed in (4) can only ensure the control system (1) to be stable. One needs to do further study on the convergence of the control system. The control system governed by (1) is an autonomous system, whose convergence can be analyzed based on the LaSalle invariance principle [44]. According to the LaSalle invariance principle, as 𝑡→∞, any state trajectory will converge to the largest invariant set contained in the set 𝐸 in which the states satisfy that the first order derivative of the Lyapunov function equals zero. In fact, the set 𝐸 contains not only the target state but also other states, thus the system may converge to other states rather than the target state. The main idea to solve this problem is to add restrictions to make the set 𝐸 as small as possible. Based on the LaSalle invariance principle, the convergence of the control system governed by (1) can be depicted by Theorem 1.

Theorem 1 (see [19]). Consider the control system governed by (1) with control fields 𝑢_𝑘(𝑡) designed in (4). If (i) The system is strongly regular, that is, 𝜔_{𝑖′𝑗′}≠𝜔_𝑙𝑚,(𝑖^′,𝑗^′)≠(𝑙,𝑚), 𝑖^′,𝑗^′,𝑙,𝑚∈{1,2,…,𝑁}, 𝜔_𝑙𝑚=𝜆_𝑙−𝜆_𝑚, where 𝜆_𝑙 is the lth eigenvalue of 𝐻₀ corresponding to the eigenstate |𝜙_l⟩; (ii) for any |𝜙_𝑖⟩≠|𝜓_𝑓⟩,𝑖∈{1,…,𝑁}, there exists at least a 𝑘∈{1,…,𝑟} such that ⟨𝜙_𝑖|𝐻_𝑘|𝜓_𝑓⟩≠0, then any state trajectory will converge toward 𝐸₁={𝑒^𝑖𝜃|𝜓_𝑓⟩,𝜃∈𝑅}.

From Theorem 1, one can see that in the case that the target state is an eigenstate, if the control system governed by (1) satisfies the conditions (i)-(ii), the control system can converge to the equivalent state of the target eigenstate 𝑒^𝑖𝜃|𝜓_𝑓⟩ from any initial pure state. These two conditions are relevant to the internal Hamiltonian and the control Hamiltonians, which are system parameters. Once the control system is determined, the Hamiltonians are fixed and cannot be changed by designing the control laws.

2.1.2. Lyapunov Control Based on State Error

Consider the following Lyapunov function based on the state error: 𝑉₂𝜓=12𝜓−𝜓_𝑓∣𝜓−𝜓_𝑓. (5)

In the case of selecting the Lyapunov function based on the state error defined by (5), in order to facilitate to design the control laws based on the Lyapunov stability theorem, the drift item appeared in the first order time derivative of Lyapunov function, which is caused by the internal Hamiltonian, is needed to be eliminated. The existing solution is to add a global phase control item 𝜔 into the control system governed by (1). This method will not change the population distribution of the control system. Thus the dynamical equation (1) becomes 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡)+𝜔𝐼𝜓(𝑡). (6)

After some deduction, one can obtain the time derivative of the Lyapunov function (5) as 𝑉̇₂=−𝜆_𝑓+𝜔ℑ𝜓_𝑓∣𝜓−^𝑟_𝑘=1ℑ𝜓_𝑓𝐻_𝑘𝜓𝑢_𝑘(𝑡). (7) The control laws which can make 𝑉̇₂≤0 hold can be designed as 𝜔=−𝜆_𝑓+𝑐𝑓₀ℑ𝜓_𝑓,𝑢∣𝜓_𝑘(𝑡)=𝐾_𝑘𝑓_𝑘ℑ𝜓_𝑓𝐻_𝑘𝜓,(𝑘=1,…,𝑟), (8)(9) where 𝐾_𝑘>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=0,…,𝑟) are the monotonic increasing functions through the coordinate origin of the plane 𝑥_𝑘−𝑦_𝑘.

Based on the LaSalle invariance principle, the convergence of the control system governed by (6) can be depicted by Theorem 2.

Theorem 2 (see [19, 23]). Consider the control system governed by (6) with control fields 𝑢_𝑘(𝑡) designed in (9) and 𝜔 designed in (8). If (i) 𝜔_{𝑖′𝑗′}≠𝜔_𝑙𝑚,(𝑖^′,𝑗^′)≠(𝑙,𝑚), 𝑖^′,𝑗^′,𝑙,𝑚∈{1,2,…,𝑁}, 𝜔_𝑙𝑚=𝜆_𝑙−𝜆_𝑚, where 𝜆_l is the lth eigenvalue of 𝐻₀ corresponding to the eigenstate |𝜙_l⟩; (ii) for any |𝜙_𝑖⟩≠|𝜓_𝑓⟩,𝑖∈{1,…,𝑁}, there exists at least a 𝑘∈{1,…,𝑟} such that ⟨𝜙_𝑖|𝐻_𝑘|𝜓_𝑓⟩≠0. Then any state trajectory will converge toward 𝐸₂={𝑒^𝑖𝜃|𝜓_𝑓⟩,𝜃∈𝑅}.

From Theorem 2, one can see that for the case that the target state is an eigenstate, if the control system governed by (6) satisfies the conditions (i)-(ii), the control system can also converge to the equivalent state of the target eigenstate 𝑒^𝑖𝜃|𝜓_𝑓⟩ from any initial pure state.

2.1.3. Lyapunov Control Based on Average Value of an Imaginary Mechanical Quantity

Consider the following Lyapunov function based on the average value of an imaginary mechanical quantity: 𝑉₃𝜓=𝜓𝑃𝜓, (10) where the imaginary mechanical quantity 𝑃 is a positive definite Hermitian operator.

The first order time derivative of the Lyapunov function (10) can be obtained as 𝑉̇₃=𝑖𝜓𝐻₀,𝑃𝜓+𝑖^𝑟_𝑘=1𝜓𝐻_𝑘,𝑃𝜓𝑢_𝑘. (11)

Set [𝐻₀,𝑃]=0 such that the drift term in the right side of (11) can be eliminated. In order to ensure 𝑉̇₃≤0, one can design 𝑢_𝑘(𝑡) as 𝑢_𝑘(𝑡)=−𝐾_𝑘𝑓_𝑘𝑖𝜓𝐻_𝑘,𝑃𝜓,(𝑘=1,…,𝑟), (12) where 𝐾_𝑘>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=1,…,𝑟) are monotonic increasing functions through the coordinate origin of the plane 𝑥_𝑘−𝑦_𝑘.

Then based on the LaSalle invariance principle, all the state trajectories of the system will converge to the invariant set contained in the set 𝐸 in which 𝑉̇=0 holds. Denote the state at the time 𝑡 as |𝜓(𝑡)⟩=∑^𝑁_𝑙=1𝑐_𝑙(𝑡)|𝜙_𝑙⟩, where 𝑐_𝑙(𝑡) is the coefficient corresponding to the 𝑙th eigenstate |𝜙_l⟩. For the Schrödinger equation governed by (1), if the control system is strongly regular and any eigenstate is directly coupled to all other eigenstates, that is, for any 𝑗≠𝑙, 𝑗,𝑙∈{1,2,…,𝑁}, there exists a 𝑘∈{1,…,𝑟} such that ⟨𝜙_𝑗|𝐻_𝑘|𝜙_𝑙⟩≠0, then one can deduce that 𝑉̇₃(𝑡)=0 holds for all 𝑡≥𝑡₀,𝑡₀∈𝑅 is equivalent to 𝑃_𝑙−𝑃_𝑗𝑐_𝑗𝑡₀𝑐^∗_𝑙𝑡₀=0,𝑙,𝑗=1,…,𝑁, (13) where 𝑃_𝑙 and 𝑃_𝑗 are the 𝑙th and 𝑗th eigenvalues of 𝑃, respectively.

If the target state |𝜓_𝑓⟩ is an eigenstate, and all the eigenvalues of 𝑃 are designed mutually different, that is, 𝑃_𝑙≠𝑃_𝑗 for any 𝑙≠𝑗,(𝑙,𝑗=1,2…,𝑁), then (13) is equivalent to 𝑐_𝑗𝑐^∗_𝑙=0,𝑙,𝑗=1,…,𝑁. (14) Equation (14) implies that there is at most one 𝑐_𝑗≠0,(𝑗=1,…,𝑁); that is, the system will converge to an eigenstate with 𝑡→∞. Thus the convergence of the control system (1) can be depicted by Theorem 3.

Theorem 3 (see [19, 24, 25]). Consider the control system governed by (1) with the control fields 𝑢_𝑘(𝑡) designed in (12). If (i) 𝜔_{𝑖′𝑗′}≠𝜔_𝑙𝑚,(𝑖^′,𝑗^′)≠(𝑙,𝑚), 𝑖^′,𝑗^′,𝑙,𝑚∈{1,…,𝑁}, 𝜔_𝑙𝑚=𝜆_𝑙−𝜆_𝑚, where 𝜆_𝑙 is the lth eigenvalue of 𝐻₀ corresponding to the eigenstate |𝜙_l⟩; (ii) for any 𝑖≠𝑗, 𝑖,𝑗∈{1,…,𝑁}, there exsits at least a 𝑘∈{1,…,𝑟} such that ⟨𝜙_𝑖|𝐻_𝑘|𝜙_𝑗⟩≠0; (iii) [𝐻₀,𝑃]=0; (iv) for any 𝑙≠𝑚∈{1,2,…,𝑁}, 𝑃_𝑙≠𝑃_𝑚 holds, where 𝑃_𝑙 is the lth eigenvalue of 𝑃. Then any state trajectory will converge toward 𝐸₃={𝑒^𝑖𝜃|𝜙_𝑖⟩,𝑖∈{1,…,𝑁},𝜃∈𝑅}.

From Theorem 3, one can see that the control system will converge from any initial pure state to an eigenstate which may not be the target eigenstate. In order to make the system converge to the target eigenstate |𝜓_𝑓⟩ from any initial pure state |𝜓₀⟩, as 𝑉̇₃(𝑡)≤0, one can add a restriction as 𝑉₃𝜓_𝑓<𝑉₃𝜓₀<𝑉₃𝜓_other, (15) where |𝜓₀⟩ is the initial state, |𝜓_other⟩ represents any other state in the set 𝐸₃ except the target state.

In such a way, any state trajectory of the system will converge to 𝑒^𝑖𝜃|𝜓_𝑓⟩ from any initial pure state |𝜓₀⟩.

Next, let us analyze how to make these convergence conditions be satisfied. Conditions (i) and (ii) are only relevant to the internal Hamiltonian 𝐻₀ and the control Hamiltonians 𝐻_𝑘,(𝑘=1,…,𝑟) which cannot be changed by designing appropriate control laws. Condition (iii) means that 𝑃 and 𝐻₀ have the same eigenstates. In order to make condition (iii) be satisfied, the eigenvalues of 𝑃 can be designed as 𝑃=^𝑁_𝑗=1𝑃_𝑗𝜙_𝑗𝜙_𝑗. (16) Because 𝑃 should be positive definite, one needs to design 𝑃_𝑖>0,(𝑖=1,…,𝑁). The restriction (15) can be satisfied by means of designing an appropriate 𝑃. For the restriction (15), Grivopoulos and Bamieh proposed a design principle of 𝑃 to make 𝑉₃(|𝜓_𝑓⟩)<𝑉₃(|𝜓_other⟩) hold. This design principle of 𝑃 can be depicted by Proposition 4.

Proposition 4 (see [25]). With the constraint condition ⟨𝜓∣𝜓⟩=1, the set of critical points of the Lyapunov function 𝑉₃(|𝜓⟩)=⟨𝜓|𝑃|𝜓⟩ is given by the normalized eigenvectors of 𝑃. The eigenvectors with the largest eigenvalue are the maxima of 𝑉₃, the eigenvectors with the smallest eigenvalue are the minima and all others are saddle points.

According to Proposition 4, in order to make 𝑉₃(|𝜓_𝑓⟩)<𝑉₃(|𝜓_other⟩) hold, 𝑃_𝑖>𝑃_𝑓,(𝑖=1,…,𝑁,𝑃_𝑖≠𝑃_𝑓) needs to be designed, where 𝑃_𝑓 is the eigenvalue of 𝑃 corresponding to |𝜓_𝑓⟩. Then let us consider the whole restriction (15); it is an attraction problem. If the eigenvalues of 𝑃 except 𝑃_𝑓 are close together, the attraction region will be very large. For the limiting case 𝑃_𝑖=𝑃_𝑗>𝑃_𝑓>0,(𝑖≠𝑗,𝑃_𝑖,𝑃_𝑗≠𝑃_𝑓), the attraction region will be the whole state space. Thus the design principle of 𝑃 is 𝑃_𝑖>𝑃_𝑓,(𝑖=1,…,𝑁,𝑃_𝑖≠𝑃_𝑓) and to make 𝑃_𝑖,𝑃_𝑗≠𝑃_𝑓 close together.

In conclusion, for the target state being an eigenstate, the design principle of 𝑃 is as follows: (i)𝑃_𝑙≠𝑃_𝑚>0 for any 𝑙≠𝑚∈{1,2,…,𝑁}, (ii)𝑃_𝑖>𝑃_𝑓,(𝑖=1,…,𝑁,𝑃_𝑖≠𝑃_𝑓), 𝑃_𝑖,𝑃_𝑗≠𝑃_𝑓 are close together, (iii)Equation (16): 𝑃=∑^𝑁_𝑗=1𝑃_𝑗|𝜙_𝑗⟩⟨𝜙_𝑗|.

From the above analyses, we can conclude that the Lyapunov control method based on the imaginary mechanical quantity proposed in [19, 24, 25] can only ensure the convergence to an eigenstate, but cannot guarantee the convergence to the target eigenstate from any initial pure state. However, if there exists a 𝑃 to make the restriction (15) hold, then any state trajectory of the system will converge to the equivalent state of the target eigenstate 𝑒^𝑖𝜃|𝜓_𝑓⟩ from any initial pure state.

2.1.4. Relations between Three Lyapunov Functions

In the Liouville space, the Hilbert-Schmidt distance between two density operators 𝜌₁ and 𝜌₂ is 𝑑_HS𝜌₁,𝜌₂=tr𝜌₁−𝜌₂². (17)

The inner product of two operators 𝐴 and 𝐵 is defined as ⟨⟨𝐴|𝐵⟩⟩=tr(𝐴^†𝐵), where the operation 𝐴^† refers to the conjugate transpose of 𝐴. Because 𝜌=|𝜓⟩⟨𝜓|, the square of the Hilbert-Schmidt distance between the density operator 𝜌 and the target density operator 𝜌_𝑓 can be deduced as 𝑑²_HS𝜌,𝜌_𝑓=21−𝜓∣𝜓_𝑓². (18)

One can conclude from (18) that the Lyapunov function based on the state distance 𝑉₁(|𝜓⟩)=(1/2)(1−|⟨𝜓∣𝜓_𝑓⟩|²) and the state error 𝑉₂(|𝜓⟩)=(1/2)⟨𝜓−𝜓_𝑓∣𝜓−𝜓_𝑓⟩ are equivalent.

For the Lyapunov function based on the average value of an imaginary mechanical quantity 𝑉₃(|𝜓⟩)=⟨𝜓|𝑃|𝜓⟩, if the imaginary mechanical quantity 𝑃=(1/2)(𝐼−|𝜓_𝑓⟩⟨𝜓_𝑓|), 𝑉₃(|𝜓⟩) will become 𝑉₁(|𝜓⟩) [19]. Therefore 𝑉₁(|𝜓⟩) is formally a special case of 𝑉₃(|𝜓⟩).

In fact, 𝑉₁(|𝜓⟩), 𝑉₂(|𝜓⟩), and 𝑉₃(|𝜓⟩) can be unified in the following quadratic Lyapunov function: 𝑉₄𝜓=𝜓−𝛼𝜓_𝑓𝑄𝜓−𝛼𝜓_𝑓 (19)

Next, we consider three cases as follows:(i)𝛼=0,𝑄=(1/2)(𝐼−|𝜓_𝑓⟩⟨𝜓_𝑓|), 𝑉₄(|𝜓⟩) will reduce to 𝑉₁(|𝜓⟩);(ii)𝛼=1,𝑄=(1/2)𝐼, 𝑉₄(|𝜓⟩) will reduce to 𝑉₂(|𝜓⟩);(iii)𝛼=0, 𝑉₄(|𝜓⟩) will reduce to 𝑉₃(|𝜓⟩).

We can see that 𝑉₁(|𝜓⟩), 𝑉₂(|𝜓⟩), and 𝑉₃(|𝜓⟩) are special cases of 𝑉₄(|𝜓⟩).

From the above analyses, we can conclude that all these three Lyapunov control methods can converge to the equivalent state of the target eigen state from any initial pure state. The Lyapunov control methods based on the state distance and the state error have only one adjustable parameter. So these two methods are very easy to grasp and very simple. The Lyapunov control based on the average value of an imaginary mechanical quantity has more adjustable parameters. So it is more flexible and also more complex at the same time. For the target eigenstate, 𝑉₁(|𝜓⟩) and 𝑉₂(|𝜓⟩) are equivalent, so the Lyapunov control based on the state distance and the state error have similar control effects. Because 𝑉₁(|𝜓⟩) is formally a special case of 𝑉₃(|𝜓⟩), generally, the control effect of the Lyapunov control based on the average value of an imaginary mechanical quantity is better than that of the state distance. At least, it can get the same control effect as the Lyapunov control of the state distance.

2.2. Target Mixed State Which Commutes with the Internal Hamiltonian

The bilinear Schrödinger equation cannot describe the mixed state. Thus for the target mixed state, it needs to use the quantum Liouville equation which can describe the evolution of any state of a closed quantum system. Consider the 𝑁-level closed quantum system governed by the following quantum Liouville equation: 𝑖𝜌̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝑢_𝑘(𝑡),𝜌(𝑡), (20) where 𝜌(𝑡) is the density operator.

Consider the Lyapunov function based on the average value of an imaginary mechanical quantity: 𝑉₅𝜌=tr𝑃𝜌. (21) By means of setting [𝐻₀,𝑃]=0, the first order time derivative of the Lyapunov function (21) can be deduced as 𝑉̇₅=−𝑖tr𝑃,𝐻₀𝜌−𝑖^𝑟_𝑘=1tr𝑃,𝐻_𝑘𝜌𝑢_𝑘. (22)

In order to ensure 𝑉̇₅≤0, one can design 𝑢_𝑘(𝑡) as 𝑢_𝑘(𝑡)=𝑖𝜀_𝑘tr𝑃,𝐻_𝑘𝜌,(𝑘=1,…,𝑟), (23) where 𝜀_𝑘∈𝑅,𝜀_𝑘>0.

Next, let us analyze the convergence to the target state. For the control system governed by (20) in the non-degenerate case, one can deduce that 𝑉̇₅(𝑡)=0 holds for all 𝑡≥𝑡₀,𝑡₀∈𝑅 is equivalent to 𝐻_𝑘𝑗𝑙𝑃_𝑙−𝑃_𝑗𝜌_𝑙𝑗(𝑡)=0,𝑗,𝑙=1,…,𝑁,𝑗<𝑙, (24) where 𝜌_𝑙𝑗(𝑡) is the (𝑙;𝑗)th element of the state 𝜌(𝑡). If the target state 𝜌_𝑓 commutes with 𝐻₀, that is, [𝐻₀,𝜌_𝑓]=0, and all the eigenvalues of 𝑃 are designed mutually different, that is, 𝑃_𝑙≠𝑃_𝑗 for any 𝑙≠𝑗,𝑙,𝑗=1,2,…,𝑁, then (24) is equivalent to 𝜌_𝑙𝑗=0,𝑙,𝑗=1,…,𝑁 (25) which implies that the system will converge to a state which commutes with 𝐻₀. Thus based on the LaSalle invariance principle, the convergence of the control system governed by (20) can be depicted by Theorem 5.

Theorem 5 (see [37]). Consider the control system governed by (20) with the control field 𝑢_𝑘(𝑡) designed in (23). If (i) the internal Hamiltonian is strongly regular, that is, 𝜔_{𝑖′𝑗′}≠𝜔_𝑙𝑚,(𝑖^′,𝑗^′)≠(𝑙,𝑚), 𝑖^′,𝑗^′,𝑙,𝑚∈{1,…,𝑁}, 𝜔_𝑙𝑚=𝜆_𝑙−𝜆_𝑚, where 𝜆_l is the lth eigenvalue of 𝐻₀ corresponding to the eigenstate |𝜙_l⟩; (ii) the control Hamiltonians are full coupled; that is, ∀𝑗≠𝑙, for 𝑘=1,…,𝑟, there exists at least a (𝐻_𝑘)_𝑗𝑙≠0; (iii) [𝐻₀,𝑃]=0; (iv) 𝑃_𝑙≠𝑃_𝑚,𝑙≠𝑚∈{1,2,…,𝑁}, where 𝑃_𝑙 is the lth eigenvalue of 𝑃. Then any state trajectory will converge toward 𝐸₄={𝜌∣(𝜌)_𝑖𝑗=0,𝑖,𝑗∈{1,…,𝑁}}, where (𝜌)_𝑖𝑗 is the (𝑙,𝑙)th element of 𝜌.

From Theorem 5, one can see that if the control system satisfies the conditions (i)–(iv), the control system will converge from any initial state to a state that commutes with the internal Hamiltonian, which may not be the target state. Next, what we need to do is to make the control system converge to the target state.

Denote the state in 𝐸₄ as 𝜌_𝐸4; then [𝜌_𝐸4,𝐻₀]=0 holds which implies that 𝜌_𝐸4 and 𝐻₀ have the same eigenstates. Since the evolution of 𝜌(𝑡) is unitary, 𝜌(𝑡) for 𝑡≥0 are isospectral. So the eigenvalues of 𝜌_𝐸4 are a permutation of the eigenvalues of 𝜌₀. Thus the set 𝐸₄ has countable elements. If the initial state is generic, that is, the eigenvalues of the initial state are mutually different, the set 𝐸₄ will have 𝑁! elements. For the target state 𝜌_𝑓 which commutes with the internal Hamiltonian, that is, [𝜌_𝑓,𝐻₀]=0, in order to make the system converge to the target state 𝜌_𝑓 which commutes with 𝐻₀ from any initial state 𝜌₀, Kuang and Cong proposed a restriction as 𝑉₅𝜌_𝑓<𝑉₅𝜌₀<𝑉₅𝜌_other, (26) where 𝜌_other represents any other state in the set 𝐸₄ except the target state 𝜌_𝑓.

In such a way, if there exists a 𝑃 to make the restriction (26) hold, any state trajectory of the system will converge to the target state 𝜌_𝑓 which commutes with 𝐻₀ from any initial state 𝜌₀.

For how to make conditions (iii)-(iv) to be satisfied, please read Section 2.1.3. One can deduce the design principle of 𝑃 such that 𝑉₅(𝜌_𝑓)<𝑉₅(𝜌_other), the result can be depicted by Proposition 6.

Proposition 6. If (𝜌_𝑓)_𝑖𝑖<(𝜌_𝑓)_𝑗𝑗,1≤𝑖,𝑗≤𝑁 holds, then one can design 𝑃_𝑖>𝑃_𝑗; if (𝜌_𝑓)_𝑖𝑖=(𝜌_𝑓)_𝑗𝑗,1≤𝑖,𝑗≤𝑁 holds, then one can design 𝑃_𝑖≠𝑃_𝑗; else if (𝜌_𝑓)_𝑖𝑖>(𝜌_𝑓)_𝑗𝑗,1≤𝑖,𝑗≤𝑁 holds, then one can design 𝑃_𝑖<𝑃_𝑗, thus 𝑉₅(𝜌_𝑓)<𝑉₅(𝜌_other) holds, where (𝜌_𝑓)_𝑖𝑖 is the (𝑖,𝑖)th element of 𝜌_𝑓.

It is difficult to design 𝑃 such that (26) holds for any initial state 𝜌₀ and any target state 𝜌_𝑓 which satisfies [𝜌_𝑓,𝐻₀]=0. One possible method is to introduce a series of implicit function perturbations into the control laws, this method will be presented in Section 3.3.

For the target state 𝜌_𝑓 which commutes with the internal Hamiltonian, the design principle of 𝑃 is as follows: (i)𝑃_𝑙≠𝑃_𝑚>0 for any 𝑙≠𝑚∈{1,2,…,𝑁}; (ii)Design 𝑃 according to Proposition 6;(iii)Equation (16): 𝑃=∑^𝑁_𝑗=1𝑃_𝑗|𝜙_𝑗⟩⟨𝜙_𝑗|.

In conclusion, if the control system satisfies the conditions (i)–(iv) in Theorem 5 and there exists a 𝑃 to make the restriction (26) hold, any state trajectory of the system will converge to the target mixed state 𝜌_𝑓 which commutes with the internal Hamiltonian from any initial state 𝜌₀.

2.3. Target Superposition State

From Section 2.1.3, one can see that for the control system governed by (1), 𝑉̇(𝑡)=0 holds for all 𝑡≥𝑡₀,𝑡₀∈𝑅 is equivalent to (𝑃_𝑙−𝑃_𝑗)𝑐_𝑗𝑐^∗_𝑙=0,(𝑙,𝑗=1,…,𝑁). If the target state is a superposition state as |𝜓_𝑓⟩=∑^𝑘𝑜_𝑑=𝑘1𝑐_𝑑|𝜙_𝑑⟩,(1≤𝑘1<𝑘𝑜≤𝑁), one can design 𝑃_𝑘1=𝑃_𝑘2=⋯=𝑃_𝑘𝑜 and other eigenvalues of 𝑃 are mutually different, then (𝑃_𝑙−𝑃_𝑗)𝑐_𝑗𝑐^∗_𝑙=0,(𝑙,𝑗=1,…,𝑁) is equivalent to 𝑐_𝑗𝑐^∗_𝑙=0,𝑙,𝑗=1,…,𝑁,𝑙,𝑗≠𝑘1,…,𝑘𝑜. (27) Equation (27) implies that the system will converge to the set 𝐸₅=span{|𝜙_𝑘1⟩,|𝜙_𝑘2⟩,…,|𝜙_𝑘𝑜⟩,|𝜙_𝑙⟩;𝑙∈{1,…,𝑁},𝑙≠𝑘1,…,𝑘𝑜}, which means that the set 𝐸₅ contains infinite elements. Therefore this control method cannot ensure the system converge to the target superposition state by adding the restriction defined by (15). But this control method can ensure that the system will converge to the superposition of the eigenstates corresponding to target state. In fact, once one element of 𝑃 changes, all the populations of the levels will change accordingly. Therefore when there are not very many eigenstates corresponding to target state, the system maybe converge to the target state by regulating the eigenvalues of 𝑃. Otherwise, the system maybe cannot converge to the target state. Consider the extreme situation in which the target state is the superposition of all the eigenstates. According to the design principle of 𝑃, we should design 𝑃=𝑐_𝑝𝐼, where 𝑐_𝑝 is a real number, and 𝐼 is the unit matrix. Obviously, in this special case, the control method proposed in Section 2.1.3 will become invalid.

From Section 2.2, one can see that, for the control system governed by (20), 𝑉̇(𝑡)=0 holds for all 𝑡≥𝑡₀,𝑡₀∈𝑅 is equivalent to (𝐻_𝑘)_𝑗𝑙(𝑃_𝑙−𝑃_𝑗)𝜌_𝑙𝑗(𝑡)=0,𝑗,𝑙=1,…,𝑁,𝑗<𝑙. Consider the target state which does not commute with 𝐻₀, which includes the superposition state and the mixed state which does not commute with 𝐻₀. Without loss of generality, assume (𝜌_𝑓)_3𝑁=(𝜌_𝑓)_𝑁3≠0; one can design 𝑃₃=𝑃_𝑁; thus the system will converge to 𝐸₆={𝜌|(𝜌)_𝑖𝑗=0,(𝑖,𝑗)≠(3,𝑁),(𝑖,𝑗)≠(𝑁,3),𝑖,𝑗∈{1,…,𝑁}}. Because the set 𝐸₆ has infinite elements, this control method also cannot ensure the system will converge to the target state by adding the restriction defined by (26). But when the target state does not have so many nonzero off-diagonal elements, the system maybe converges to the target state by regulating the diagonal elements of 𝑃. When there are many nonzero off-diagonal elements in the target state, the degree of freedom of 𝑃 may be not enough.

From the above analyses, we can conclude that by means of using the control methods proposed in Sections 2.1 and 2.2, the control system may but cannot be ensured to converge to the target superposition state. This problem can be solved by means of designing the control laws in the interaction picture of the control system.

Consider the 𝑁-level control system in the interaction picture as 𝑖𝜕𝜌(𝑡)=𝜕𝑡^𝑟_𝑘=1𝐻_𝑘(𝑡)𝑢_𝑘(𝑡),𝜌(𝑡). (28)

Choose the Lyapunov function based on the average value of an imaginary mechanical quantity defined by (21). The first order time derivative of the Lyapunov function (21) can be obtained as 𝑉̇₅=−^𝑟_𝑘=1𝑢_𝑘(𝑡)tr𝑖𝐻_𝑘(𝑡)𝜌(𝑡),𝑃. (29) To ensure 𝑉̇₅≤0, one can design 𝑢_𝑘(𝑡) as 𝑢_𝑘(𝑡)=−𝐾_𝑘tr𝑖𝐻_𝑘(𝑡)𝜌(𝑡),𝑃,𝐾_𝑘>0, (30) where 𝐾_𝑘>0 and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=1,…,𝑟) are monotonic increasing functions through the coordinate origin of plane 𝑥_𝑘−𝑦_𝑘.

The control system governed by (28) is a non-autonomous system; thus the LaSalle invariance principle cannot be used to analyze the convergence. One can use the improved Barbalat lemma which can be used for the non-autonomous system. According to the improved Barbalat lemma, the convergence of the control system governed by (28) can be depicted by Theorem 7.

Theorem 7. Consider the control system governed by (28) with the control field 𝑢_𝑘(𝑡) designed in (30). If (i) the internal Hamiltonian is strongly regular; (ii) the control Hamiltonians are full connected, then any state trajectory will converge to the limit set ℛ₁≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=𝐷} at 𝑡→∞, where 𝐷 is a diagonal matrix.

For the case that 𝑃 is chosen as a diagonal matrix, the limit set is reduced to ℛ₁≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0}. For the case that 𝑃 is chosen as a nondiagonal matrix, if rank𝐴(𝑃)=𝑛²−𝑛 holds, then the limit set ℛ₁ is regular; namely, ℛ₁≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0}, where 𝐴(𝑃) is the first 𝑛²−𝑛 rows of 𝐴(𝑃), 𝐴(𝑃) is the real (𝑛²−1)∗(𝑛²−1) matrix corresponding to the Bloch representation of 𝐴𝑑_𝑝, and 𝐴𝑑_𝑝 is a linear map from Hermitian or anti-Hermitian matrices into 𝑠𝑢(𝑛). [𝜌_𝑠,𝑃]=0 means that 𝑃 and 𝐻₀ have the same eigenstates. For the target state 𝜌_𝑓=|𝜓_𝑓⟩⟨𝜓_𝑓| being a superposition state, in order to make the target state contain in ℛ₁≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0}, 𝑃 can be designed as 𝑃=𝑃₁𝜓_𝑓𝜓_𝑓+^𝑛_𝑘=2𝑃_𝑘𝜓_𝑘𝜓_𝑘, (31) where |𝜓₁⟩=|𝜓_𝑓⟩ and ⟨𝜓_𝑖|𝜓_𝑗⟩=0, for 𝑖≠𝑗.

Then some deduction shows that 𝜌_𝑠=𝜆_𝑖|𝜓_𝑖⟩⟨𝜓_𝑖|(𝜆_𝑖=1). In order to make the system converge to the target superposition state 𝜌_𝑓 from any initial pure state, one can design 𝑃 such that 𝑉₅𝜌_𝑓<𝑉₅𝜌₀<𝑉₅𝜌_oth, (32) where 𝜌_oth represents any other state in the set ℛ₁ except the target state 𝜌_𝑓.

For the satisfaction of 𝑉₅(𝜌_𝑓)<𝑉₅(𝜌_𝑜th), one can design 𝑃 based on Proposition 6. In order to make (32) holds, one must design 𝑃_𝑖,(𝑖=1,…,𝑁) such that 0<𝑃₁<𝑃₁tr𝜌_𝑓𝜌₀+^𝑁_𝑘=2𝑃_𝑘tr𝜌_𝑘𝜌₀<𝑃_𝑗𝑗≠1. (33)

However, for any initial pure state the target superposition state, there may not exist a 𝑃 such that (33) holds.

In conclusion, the design principle of 𝑃 for the case that the target state 𝜌_𝑓=|𝜓_𝑓⟩⟨𝜓_𝑓| being a superposition state is (31), (33), and Proposition 6. We can also conclude that if the control system satisfies the conditions (i) and (ii) in Theorem 7 and there exists a 𝑃 to make (33) hold, any state trajectory of the system will converge to the target superposition state 𝜌_𝑓 from any initial pure state 𝜌₀.

Remark 8. One can also solve the problem of the convergence of the control system governed by (1) to the target superposition state by means of designing the control laws in the interaction picture of the control system.

2.4. Target Mixed State Which Does Not Commute with the Internal Hamiltonian

From Sections 2.2 and 2.3, one can see that the control method proposed in Section 2.2 cannot guarantee the convergence to the target state which does not commute with the internal Hamiltonian. For the target state 𝜌_𝑓 being a mixed state which does not commute with the internal Hamiltonian, that is, [𝜌_𝑓,𝐻₀]≠0, Cong, Liu and Yang proposed the control system in the interaction picture and used a unitary transformation to solve the convergence of the control system to the target state. The basic idea is to use a unitary transformation 𝜌_𝑓=𝑈^†_𝑓𝜌_𝑓𝑈_𝑓 to make [𝜌_𝑓,𝐻₀]=0 hold. Correspondingly, the control system governed by (28) after this unitary transformation will become 𝑖ℏ𝜌̇(𝑡)=^𝑟_𝑘=1𝐻_𝑘(𝑡)𝑢_𝑘(𝑡),𝜌(𝑡), (34) where 𝜌=𝑈^†_𝑓𝜌𝑈_𝑓, 𝐻_𝑘(𝑡)=𝑈^†_𝑓𝐻_𝑘(𝑡)𝑈_𝑓.

Then control laws can be designed according to Section 2.3. The research results show that the designed control laws and Theorem 7 are also valid with every physical quantity changing accordingly. In order to make 𝜌_𝑓 contain in ℛ₂≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0}, 𝑃 needs to be designed such that [𝐻₀,𝑃]=0 holds. [𝜌_𝑠,𝑃]=0 and [𝐻₀,𝑃]=0 imply that 𝜌_𝑠, 𝑃, and 𝐻₀ have the same eigenstates. Since the evolution of 𝜌(𝑡) is unitary, all the 𝜌(𝑡) for t≥0 are isospectral. So the eigenvalues of 𝜌_𝑠 are a permutation of the eigenvalues of 𝜌₀. Thus the limit set ℛ₂≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0} has countable elements. For sake of the convergence from any initial mixed state 𝜌₀ to any target mixed state 𝜌_𝑓 which does not commute with the internal Hamiltonian 𝐻₀, 𝑃 needs to be designed such that 𝑉₅𝜌_𝑓<𝑉₅𝜌₀<𝑉₅𝜌_oth, (35) where 𝜌_oth represents any other state in the set ℛ₂≡{𝜌_𝑠∶[𝜌_𝑠,𝑃]=0} except 𝜌_𝑓.

In such a way, if there exists a 𝑃 to make (35) hold, any state trajectory of the system will converge to the target mixed state 𝜌_𝑓 which does not commute with the internal Hamiltonian from any initial mixed state 𝜌₀.

Next, let us analyze how to make the restriction defined by (35) hold. Some deductions show that if one designs 𝑃 based on Proposition 6 with changing 𝜌_𝑓 into 𝜌_𝑓 accordingly, then 𝑉₅(𝜌_𝑓)<𝑉₅(𝜌_oth) will hold. Assume 𝜌₀=𝑈𝜌_𝑓𝑈^†. In order to make the whole restriction defined by (35) hold, 𝑃_𝑖,(𝑖=1,…,𝑁) needs to be designed such that (𝑃)_𝑙𝑙>^𝑛_𝑖≠𝑙(𝑃)^𝑛_{𝑖𝑖𝑗≠𝑘}𝜌_𝑓𝑗𝑗−𝜌_𝑓𝑘𝑘𝑈_𝑖𝑗²+(𝑃)_𝑙𝑙⋅^𝑛_{𝑗≠𝑘,𝑗≠𝑙}𝜌_𝑓𝑗𝑗−𝜌_𝑓𝑘𝑘𝑈_𝑙𝑗²×𝜌_𝑓𝑘𝑘−𝜌_𝑓𝑙𝑙⁻¹, (36) where 𝑈_𝑖𝑗 is the (𝑖,𝑗)th element of 𝑈.

In conclusion, the design principle of 𝑃 for the case that the target state 𝜌_𝑓 is a mixed state which does not commute with 𝐻₀ is(i)𝑃_𝑙≠𝑃_𝑚>0 for any 𝑙≠𝑚∈{1,2,…,𝑁}; (ii)Proposition 6 with changing 𝜌_𝑓 into 𝜌_𝑓;(iii)Equations (36) and (16).

We can conclude from the above analysis that for the case that the target state is a mixed state which does not commute with 𝐻₀, if the control system is strongly regular and full connected, and one can seek an imaginary mechanical quantity 𝑃 to make (36) hold, then the control system can converge from any initial mixed state to the target mixed state which does not commute with the internal Hamiltonian.

3. Degenerate Cases

The convergence conditions (i) and (ii) proposed in the Theorem 1 through Theorem 7 are relevant to the internal Hamiltonian 𝐻₀ and the control Hamiltonians 𝐻_𝑘,(𝑘=1,…,𝑟). They are system parameters which cannot be changed. And in practice, many actual systems do not satisfy these convergence conditions. These systems are called in the degenerate cases. In order to solve the convergence of the control systems in degenerate cases, the existing method is to introduce a series of implicit function perturbations into the control laws and choose a Lyapunov function which is an implicit function [37–40].

In this section, we also divide the target state into four categories as: target eigenstate, the target mixed state which commute with the internal Hamiltonian, target superposition state, and the target mixed state which does not commute with the internal Hamiltonian. For these four target state categories, respectively, research results for the degenerate cases are summarized and analyzed.

3.1. Target Eigenstate

In this Section, the convergence of the control system based on the Lyapunov control method in the degenerate cases to any target eigenstate from any initial pure state will be summarized and analyzed.

3.1.1. Implicit Lyapunov Control Based on State Distance

In order to solve the convergence of the control systems in the degenerate cases, several researchers introduced a series of implicit function perturbations into the control laws [37–40]. After a series of perturbations 𝛾_𝑘(𝑡) introducing into the control laws, the dynamical equation (1) becomes 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)𝜓(𝑡), (37) where 𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)=𝑢_𝑘(𝑡),(𝑘=1,…,𝑟) are the total control laws.

Without loss of generality, assume |𝜓_𝑓⟩=|𝜙_𝑔⟩,1≤𝑔≤𝑁. In order to solve the problem of convergence for the degenerate cases, the perturbations 𝛾_𝑘(𝑡) were introduced into the control laws. The basic idea is as follows: Denote the system with the internal Hamiltonian 𝐻₀, the control Hamiltonians 𝐻_𝑘,(𝑘=1,…,𝑟), and the control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡) as system 1, and the system with the internal Hamiltonian 𝐻₀₂=𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘(𝑡), the control Hamiltonians 𝐻_𝑘,(𝑘=1,…,𝑟), and the control laws 𝑣_𝑘(𝑡) as system 2. All these two systems can be depicted by (37). Denote the eigenvalues and eigenstates of 𝐻₀₂ as 𝜆_{1,𝛾1(𝑡),…,𝛾𝑟(𝑡)},𝜆_{2,𝛾1(𝑡),…,𝛾𝑟(𝑡)},…,𝜆_{𝑁,𝛾1(𝑡),…,𝛾𝑟(𝑡)} and |𝜙_{1,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩,|𝜙_{2,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩,…,|𝜙_{𝑁,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩, respectively, which are the functions of the perturbations 𝛾_𝑘(𝑡). Assume |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩=|𝜙_{𝑔,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩. If one can design the perturbations 𝛾_𝑘(𝑡) such that (i) 𝜔_{𝑙,𝑚,𝛾1,…,𝛾𝑟}≠𝜔_{𝑖,𝑗,𝛾1,…,𝛾𝑟},(𝑙,𝑚)≠(𝑖,𝑗),𝑖,𝑗,𝑙,𝑚∈{1,2,…,𝑁}, where 𝜔_{𝑙,𝑚,𝛾1,…,𝛾𝑟}=𝜆_{𝑙,𝛾1,…,𝛾𝑟}−𝜆_{𝑚,𝛾1,…,𝛾𝑟} holds; (ii) for any |𝜙_{𝛾1,…,𝛾𝑟}⟩≠|𝜓_{𝑓,𝛾1,…,𝛾𝑟}⟩, there exists at least a 𝑘∈{1,…,𝑟} satisfying ⟨𝜙_{𝛾1,…,𝛾𝑟}|𝐻_𝑘|𝜓_{𝑓,𝛾1,…,𝛾𝑟}⟩≠0, and select the specific Lyapunov function based on the state distance as 𝑉₆(𝑡)=121−𝜓∣𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}². (38) Then according to Section 2.1.1, system 2 will converge to |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩. And when system 2 converge to |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩, if the perturbations 𝛾_𝑘(𝑡) at |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩ are designed to equal zero, then system 2 will become system 1, and |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩ will become |𝜓_𝑓⟩. Then the convergence of system 1 to |𝜓_𝑓⟩ will be ensured. In fact, the evolution of system 1 can be viewed as a composite of two evolution processes. One is system 2 converges to |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩ from the initial state |𝜓₀⟩, another one is 𝛾_𝑘(𝑡) converges to 0. In order to make the introduced perturbations take effect to make system 1 in the non-degenerate case converge to |𝜓_𝑓⟩, the speed of 𝛾_𝑘(𝑡),(𝑘=1,…,𝑟) converging to 0 must be slower than the speed at which system 2 converges toward |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩. For convenience, the control system in the following section means system 1.

The existing design method of 𝛾_𝑘(𝑡) is to design it to be a monotonically increasing functional of 𝑉(𝑡), that is, 𝛾_𝑘(𝑡)=𝜃_𝑘(𝑉(𝑡))=𝜃_𝑘121−𝜓∣𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}²,(𝑘=1,…,𝑟), (39) where the function 𝜃_𝑘(⋅) satisfies 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0 and 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0 and 𝑠 is the independent variable of the function 𝜃_𝑘(⋅).

The right side of (39) contains |𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩ which is a functional of the perturbations 𝛾_𝑘(𝑡),(𝑘=1,…,𝑟). One can see that the relation between 𝛾_𝑘 and the time 𝑡 is defined by 𝑟 equations and cannot be expressed by an explicit expression, so 𝛾_𝑘 is the implicit function of the time 𝑡. From (38) and (39), one can also see that the Lyapunov function 𝑉₆ is the implicit function of the time 𝑡. The existence of 𝛾_𝑘(𝑡) can be depicted by Lemma 9.

Lemma 9 (see [37, 40]). Let 𝜃_𝑘∈𝐶^∞(𝑅⁺;[0,𝛾^∗_𝑘]),𝑘=1,…,𝑟,(𝛾^∗_𝑘∈𝑅,𝛾^∗_𝑘>0) satisfy 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0, 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0, ‖𝜃^′_𝑘‖_∞<1/𝐶^∗, and 𝐶^∗=1+max{‖(𝜕𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}/𝜕𝛾_𝑘)|_{(𝛾10,…,𝛾𝑟0)}‖;𝛾_𝑘0∈[0,𝛾^∗_𝑘]}, where 𝛾_𝑘0=𝛾_𝑘(0). Then for every state |𝜓⟩∈𝑆^2𝑁−1={𝑥∈𝐶^𝑁;‖𝑥‖=1}, there exists a unique 𝛾₁,𝛾₂,…, and 𝛾_𝑟 with 𝛾_𝑘∈𝐶^∞(𝛾_𝑘∈[0,𝛾^∗_𝑘]),(𝑘=1,…,𝑟) satisfying 𝛾_𝑘(𝑡)=𝜃_𝑘((1/2)(1−|⟨𝜓∣𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩|²)),(𝑘=1,…,𝑟),𝛾_𝑘(|𝜓_𝑓⟩)=0.

By (38) and Lemma 9, one can see that for every state |𝜓⟩∈𝑆^2𝑁−1={𝑥∈𝐶^𝑁;‖𝑥‖=1}; there also exists a unique 𝑉.

Assume 𝜕𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}𝜕𝛾₁=𝜕𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}𝜕𝛾₂𝜕𝜓=⋯=^{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}𝜕𝛾_𝑟. (40) The first order time derivative of the Lyapunov function (38) is 𝑉̇₆=−∑^𝑟_𝑘=1𝑣_𝑘(𝑡)ℑ𝜓∣𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝐻_𝑘𝜓1+ℜ𝜓∣𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝜕𝜓_{𝑓,𝛾1,…,𝛾𝑟}/𝜕𝛾_𝑘∣𝜓∑^𝑟_𝑗=1𝜃^′_𝑗. (41) In order to make 𝑉̇₆(𝑡)≤0, one can design 𝛾_𝑘(𝑡) such that ‖𝜃^′_𝑗‖_∞<1/(𝑟𝐶^∗) to make 1/(1+ℜ(⟨𝜓∣𝜙_{𝛾1,…,𝛾𝑟}⟩⟨(𝜕𝜙_{𝛾1,…,𝛾𝑟}/𝜕𝛾_𝑘)∣𝜓⟩)∑^𝑟_𝑗=1𝜃^′_𝑗)>0 hold and design 𝑣_𝑘(𝑡) as 𝑣_𝑘(𝑡)=𝐾_𝑘𝑓_𝑘ℑ𝑒^{𝑖∠⟨𝜓∣𝜓1𝑟𝑓,𝛾,…,𝛾⟩}𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝐻_𝑘𝜓, (42) where 𝐾_𝑘>0, and 𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) are monotonic functions through the coordinate origin and in the first quadrant and the third quadrant.

Based on the LaSalle invariance principle, the convergence of the control system governed by (37) can be depicted by Theorem 10.

Theorem 10 (see [37, 40]). Consider the control system (37) with control fields 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡), 𝛾_𝑘(𝑡) designed in (39), Lemma 9 and ‖𝜃^′_𝑗‖_∞<1/(𝑟𝐶^∗), and 𝑣_𝑘(𝑡) designed in (42). Assume that the target state |𝜓_𝑓⟩ is an eigenstate of H₀. If the control system satisfies (i) 𝜔_{1𝑟𝑙,𝑚,𝛾,…,𝛾}≠𝜔_{1𝑟𝑖,𝑗,𝛾,…,𝛾},(𝑙,𝑚)≠(𝑖,𝑗), 𝑖,𝑗,𝑙,𝑚∈{1,2,…,𝑁}, 𝜔_{1𝑟𝑙,𝑚,𝛾,…,𝛾}=𝜆_{𝑙,𝛾1,…,𝛾𝑟}−𝜆_{𝑚,𝛾1,…,𝛾𝑟}, where 𝜆_{𝑙,𝛾1,…,𝛾𝑟} is an eigenvalue of (𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘), (ii) For any |𝜙_{𝛾1,…,𝛾𝑟}⟩≠|𝜓_{𝑓,𝛾1,…,𝛾𝑟}⟩, there exists at least a 𝑘∈{1,…,𝑟} satisfying ⟨𝜙_{𝛾1,…,𝛾𝑟}|𝐻_𝑘|𝜓_{𝑓,𝛾1,…,𝛾𝑟}⟩≠0; then the largest invariant set is 𝑆^2𝑁−1∩𝐸₇, where 𝐸₇={|𝜓⟩=𝑒^𝑖𝜃|𝜓_𝑓⟩,𝜃∈𝑅}. And the control system will converge toward |𝜓_𝑓⟩𝑒^𝑖𝜃,(𝜃∈𝑅).

Conditions (i) and (ii) in Theorem 10 are associated with 𝐻₀, 𝐻_𝑘,(𝑘=1,…,𝑟) and 𝛾_𝑘(𝑡). By designing appropriate 𝛾_𝑘(𝑡), these two conditions can be satisfied in most cases. one can see that if one designs appropriate control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) to make the conditions (i) and (ii) hold, the control system depicted by (37) can converge to the equivalent state of the target eigenstate |𝜓_𝑓⟩ from any initial pure state.

3.1.2. Implicit Lyapunov Control Based on the State Error

The basic idea of this method is similar to that of the state distance. In the Lyapunov control based on the state error, a global phase control item 𝜔 is added into the control system to facilitate the design of the control laws. Thus the control system can be depicted by 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)+𝜔𝐼𝜓(𝑡), (43) where 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) and 𝜔 is the control laws which need to design.

Consider the Lyapunov function based on the state error as 𝑉₇(𝑡)=12𝜓−𝜓_{𝑓,𝛾1,…,𝛾𝑟}∣𝜓−𝜓_{𝑓,𝛾1,…,𝛾𝑟}. (44) Thus the implicit function perturbation 𝛾_𝑘(𝑡) needs to be designed as 𝛾_𝑘(𝑡)=𝜃_𝑘(𝑉(𝑡))=𝜃_𝑘12𝜓−𝜓_{𝑓,𝛾1,…,𝛾𝑟}∣𝜓−𝜓_{𝑓,𝛾1,…,𝛾𝑟}, (45) where functions 𝜃_𝑘(⋅) satisfy 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0, 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0.

The existence of 𝛾_𝑘(𝑡) can be depicted by Lemma 11.

Lemma 11 (see [38, 39]). Let 𝜃_𝑘∈𝐶^∞(𝑅⁺;[0,𝛾^∗_𝑘]),𝑘=1,…,𝑟,(𝛾^∗_𝑘∈𝑅,𝛾^∗_𝑘>0) satisfy 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0, 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0, ‖𝜃^′_𝑘‖_∞<1/𝑟𝐶^∗, and 𝐶^∗=1+max{‖(𝜕|𝜓_{𝑓,𝛾1,…,𝛾𝑟}⟩/𝜕𝛾_𝑘)|_{(𝛾10,…,𝛾𝑟0)}‖;𝛾_𝑘0∈[0,𝛾^∗_𝑘]}. Then for every state |𝜓⟩∈𝑆^2𝑁−1={𝑥∈𝐶^𝑁;‖𝑥‖=1}, there exists a unique 𝛾₁,𝛾₂,…, and 𝛾_𝑟 with 𝛾_𝑘∈𝐶^∞(𝛾_𝑘∈[0,𝛾^∗_𝑘]),(𝑘=1,…,𝑟) satisfying 𝛾_𝑘(|𝜓⟩)=𝜃_𝑘((1/2)⟨𝜓−𝜓_{𝑓,𝛾1(|𝜓⟩),…,𝛾𝑟(|𝜓⟩)}∣𝜓−𝜓_{𝑓,𝛾1(|𝜓⟩),…,𝛾𝑟(|𝜓⟩)}⟩), 𝛾_𝑘(|𝜓_𝑓⟩)=0.

Assume (40) holds, the first order time derivative of the Lyapunov function (44) is 𝑉̇₇=−11+ℜ𝜕𝜓_{𝑓,𝛾1,…,𝛾𝑟}/𝜕𝛾_𝑘∣𝜓∑^𝑟_𝑗=1𝜃^′_𝑗⋅𝜆_{𝛾1,…,𝛾𝑟}+𝜔ℑ𝜓_{𝑓,𝛾1,…,𝛾𝑟}+∣𝜓_𝑟𝑘=1ℑ𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝐻_𝑘𝜓𝑣_𝑘(𝑡). (46) In order to make 𝑉̇₇(𝑡)≤0, 𝜔 and 𝑣_𝑘(𝑡) need to be designed as 𝜔=−𝜆_{𝛾1,…,𝛾𝑟}+𝑐𝑓₀ℑ𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝑣∣𝜓,_𝑘(𝑡)=𝐾_𝑘𝑓_𝑘ℑ𝜓_{𝑓,𝛾1,…,𝛾𝑟}𝐻_𝑘𝜓,(𝑘=1,…,𝑟), (47)(48) where 𝐾_𝑘>0, 𝑐>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=0,1,…,𝑟) are monotonic functions through the coordinate origin and in the first quadrant and the third quadrant.

Based on the LaSalle invariance principle, the convergence of the control system governed by (44) can be depicted by Theorem 12.

Theorem 12 (see [38, 39]). Consider the control system (43) with control fields 𝛾_𝑘(𝑡) designed in (45) and Lemma 11, 𝑣_𝑘(𝑡) designed in (48), and 𝜔 designed in (47). Assume that the target state |𝜓_𝑓⟩ is an eigenstate of 𝐻₀. If the control system satisfies (i) 𝜔_{𝑙,𝑚,𝛾1,…,𝛾𝑟}≠𝜔_{𝑖,𝑗,𝛾1,…,𝛾𝑟},(𝑙,𝑚)≠(𝑖,𝑗), 𝑖,𝑗,𝑙,𝑚∈{1,…,𝑁}, 𝜔_{𝑙,𝑚,𝛾1,…,𝛾𝑟}=𝜆_{𝑙,𝛾1,…,𝛾𝑟}−𝜆_{𝑚,𝛾1,…,𝛾𝑟},𝜆_{𝑙,𝛾1,…,𝛾𝑟} is the 𝑙th eigenvalue of (𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘(𝛾_𝑘+𝜂_𝑘)) corresponding to the eigenstate |𝜙_{l,𝛾1,…,𝛾r}⟩; (ii) There is at least a 𝑘∈{1,…,𝑟}, ⟨𝜙_{𝛾1(𝑡),…,𝛾𝑟(𝑡)}|𝐻_𝑘|𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩≠0, for |𝜙_{𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩≠|𝜓_{𝑓,𝛾1(𝑡),…,𝛾𝑟(𝑡)}⟩; then the largest invariant set is 𝑆^2𝑁−1∩𝐸₈, where 𝐸₈={|𝜓⟩=𝑒^𝑖𝜃|𝜓_𝑓⟩,𝜃∈𝑅}. And the control system will converge toward the equivalent state of the target state: |𝜓_𝑓⟩𝑒^𝑖𝜃,(𝜃∈𝑅).

By designing appropriate perturbations 𝛾_𝑘(𝑡), conditions (i) and (ii) in Theorem 12 can be satisfied in most cases. From Theorem 12, one can see that if one design appropriate control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) and 𝜔 to make the conditions (i) and (ii) in Theorem 12 hold, the control system depicted by (43) can converge to the equivalent state of the target eigenstate |𝜓_𝑓⟩ from any initial pure state.

3.1.3. Implicit Lyapunov Control Based on Average Value of an Imaginary Mechanical Quantity

For the implicit Lyapunov control based on the average value of an imaginary mechanical quantity, one can consider the control system depicted by (37). In this control method, the introduced implicit function perturbations 𝛾_𝑘(𝑡) mainly have two tasks. One task is to solve the convergent problem of the control system in the degenerate cases. The basic idea is similar to that of the state distance. Another one is to choose a simpler restriction 𝑉(|𝜓_𝑓⟩)<𝑉(|𝜓_other⟩), which can be satisfied for any initial state and any target state by designing the imaginary mechanical quantity. In order to ensure the system converge to the target state by adding 𝑉(|𝜓_𝑓⟩)<𝑉(|𝜓_other⟩), we can design all the perturbations 𝛾_𝑘(𝑡)=0 hold for 𝑘=1,…,𝑟 only at |𝜓_𝑓⟩, that is, (1) 𝛾_𝑘(|𝜓_𝑓⟩)=0,(𝑘=1,…,𝑟), and (2) for |𝜓⟩≠|𝜓_𝑓⟩, there exists at least one 𝑘 such that 𝛾_𝑘(|𝜓⟩)≠0. For sake of completing these two tasks, we can design 𝛾_𝑘(𝑡) as a monotonically increasing functional of 𝑉(𝑡): 𝛾_𝑘𝜓=𝐶_𝑘⋅𝜃_𝑘𝑉𝜓−𝑉𝜓_𝑓, (49) where 𝐶_𝑘≥0, and for 𝑘=1,…,𝑟, there exists at least a 𝐶_𝑘>0, and the function 𝜃_𝑘(⋅) satisfies 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0 and 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0, 𝑠 is the independent variable of the function 𝜃_𝑘(⋅).

The specific Lyapunov function based on the average value of an imaginary mechanical quantity can be selected as: 𝑉₈𝜓=𝜓𝑃_{𝛾1,…,𝛾𝑟}𝜓, (50) where 𝑃_{𝛾1,…,𝛾𝑟}=𝑓(𝛾₁(𝑡),…,𝛾_𝑟(𝑡)) is a functional of 𝛾_𝑘(𝑡) and positive definite.

The existence of 𝛾_𝑘(𝑡) can be depicted by Lemma 13.

Lemma 13. If 𝐶_𝑘=0, 𝛾_𝑘(|𝜓⟩)=0. Else if 𝐶_𝑘>0, 𝜃_𝑘∈𝐶^∞(𝑅⁺;[0,𝛾^∗_𝑘]),𝑘=1,…,𝑟(𝛾^∗_𝑘 is a positive constant) satisfy 𝜃_𝑘(0)=0, 𝜃_𝑘(𝑠)>0 and 𝜃^′_𝑘(𝑠)>0 for every 𝑠>0, and |𝜃^′_𝑘|<1/(2𝐶_𝑘𝐶^∗), 𝐶^∗=1+𝐶, 𝐶=max{‖𝜕𝑃_{𝛾1,…,𝛾𝑟}/𝜕𝛾_𝑘‖_∞,𝛾_𝑘∈[0,𝛾^∗_𝑘]}, then for every |𝜓⟩∈𝑆^2𝑁−1, there is a unique 𝛾_𝑘∈𝐶^∞(𝛾_𝑘∈[0,𝛾^∗_𝑘]) satisfying 𝛾_𝑘(|𝜓⟩)=𝐶_𝑘⋅𝜃_𝑘(⟨𝜓|𝑃_{𝛾1,…,𝛾𝑟}|𝜓⟩−⟨𝜓_𝑓|𝑃_{𝛾1,…,𝛾𝑟}|𝜓_𝑓⟩).
For the sake of simplicity, 𝛾_𝑘(𝑡) can be designed zero for some 𝑘, and other 𝛾_𝑘(𝑡) equal, denoted by 𝛾(𝑡); that is, set 𝛾_𝑘(𝑡)=𝛾(𝑡)=𝜃𝜓𝑃_𝛾𝜓−𝜓_𝑓𝑃_𝛾𝜓_𝑓,𝑘=𝑘₁,…,𝑘_𝑚;𝛾_𝑘(𝑡)=0,𝑘≠𝑘₁,…,𝑘_𝑚1≤𝑘₁,…,𝑘_𝑚≤𝑟, (51) where 𝜃(⋅)=𝜃_𝑘1(⋅)=⋯=𝜃_𝑘𝑚(⋅) and 𝑃_𝛾 are functionals of 𝛾(𝑡).
The time derivative of the selected Lyapunov function is 𝑉̇₈=^𝑟_𝑘=1𝑖𝑣_𝑘(𝑡)𝜓𝐻_𝑘,𝑃_𝛾𝜓⋅1+𝜃^′𝜓_𝑓𝜕𝑃_𝛾/𝜕𝛾𝜓_𝑓1−𝜃^′𝜓𝜕𝑃_𝛾/𝜕𝛾𝜓−𝜓_𝑓𝜕𝑃_𝛾/𝜕𝛾𝜓_𝑓. (52) In order to ensure 𝑉̇₈(𝑡)≤0, 𝑣_𝑘(𝑡) can be designed as 𝑣_𝑘(𝑡)=−𝐾_𝑘𝑓_𝑘𝑖𝜓𝐻_𝑘,𝑃_𝛾𝜓,(𝑘=1,…,𝑟), (53) where 𝐾_𝑘 is a constant and 𝐾_𝑘>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=1,2,…,𝑟) are monotonic increasing functions through the coordinate origin of the plane 𝑥_𝑘−𝑦_𝑘.

Based on LaSalle’s invariance principle, the convergence of the control system governed by (37) can be depicted by Theorem 14.

Theorem 14. Consider the control system governed by (37) with control fields 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟), where 𝛾_𝑘(𝑡) is defined by Lemma 13 and (51) and 𝑣_𝑘(𝑡) is defined by (53). If the control system satisfies (i) 𝜔_{𝑙,𝑚,𝛾}≠𝜔_{𝑖,𝑗,𝛾},(𝑙,𝑚)≠(𝑖,𝑗), 𝑖,𝑗,𝑙,𝑚∈{1,2,…,𝑁}, 𝜔_{𝑙,𝑚,𝛾}=𝜆_𝑙,𝛾−𝜆_𝑚,𝛾, where 𝜆_𝑙,𝛾 is the 𝑙th eigenvalue of 𝐻₀+∑^𝑘𝑚_𝑛=𝑘1𝐻_𝑛𝛾(𝑡) corresponding to the eigenstate |𝜙_𝑙,𝛾⟩; (ii) for any 𝑖≠𝑗, 𝑖,𝑗∈{1,2,…,𝑁}, there exists at least one 𝑘 such that (𝐻_𝑘)_𝑙𝑚≠0, where (𝐻_𝑘)_𝑙𝑚 is the (𝑙,𝑚)th element of 𝐻_𝑘=𝑈^†₁𝐻_𝑘𝑈₁, 𝑈₁=(|𝜙_1,𝛾⟩,…,|𝜙_𝑁,𝛾⟩); (iii) [𝑃_𝛾,𝐻₀+∑^𝑘𝑚_𝑛=𝑘1𝐻_𝑛𝛾(𝑡)]=0; (iv) (𝑃_𝛾)_𝑙𝑙≠(𝑃_𝛾)_𝑚𝑚,𝑙≠𝑚, where (𝑃_𝛾)_𝑙𝑙 is the (𝑙,𝑙)th element of 𝑈^𝐻₁𝑃_𝛾𝑈₁, then any trajectory will converge toward 𝐸₉={|𝜓_𝑡0⟩|𝑒^𝑖𝜃𝑙|𝜙_{𝑙,𝛾(|𝜓𝑡0⟩)}⟩;𝜃_𝑙∈𝑅,𝑙∈{1,…,𝑁}}.

From Theorem 14, one can see that if the target state |𝜓_𝑓⟩ is an eigenstate, |𝜓_𝑓⟩ is contained in 𝐸₉. In order to make the system converge to the target eigenstate from any initial pure state |𝜓₀⟩, on the one hand, 𝑃_𝛾 needs to be designed to make 𝑉₈𝜓_𝑓<𝑉₈𝜓_other (54) hold, where |𝜓_other⟩ represents any other state in the set 𝐸₉ except the target state. On the other hand, because 𝜕𝛾/𝜕𝑉>0,𝑉̇≤0,𝛾≥0 holds, when 𝑣_𝑘(𝑡)=0,𝛾(𝑡)=𝛾≠0 holds for some time, we can design 𝛾=𝛾−𝛼,(0<𝛼≪𝛾) to make the state trajectory evolve but not stay in 𝐸₉ until |𝜓_𝑓⟩𝑒^𝑖𝜃𝑙 is reached.

Next, let us analyze how to realize the convergence conditions in Theorem 14 and restriction defined by (54). By designing appropriate 𝛾_𝑘(𝑡), conditions (i) and (ii) in Theorem 14 can be satisfied in most cases. In order to make condition (iii) be satisfied, the eigenvalues of 𝑃_𝛾 can be designed constant, denoted by 𝑃₁,𝑃₂,…,𝑃_𝑁, and 𝑃_𝛾 can be designed as 𝑃_𝛾=^𝑁_𝑗=1𝑃_𝑗𝜙_𝑗,𝛾𝜙_𝑗,𝛾. (55)

To make condition (iv) satisfied, 𝑃_𝑙≠𝑃_𝑗(∀𝑙≠𝑗;1≤𝑙,𝑗≤𝑁) can be designed. The results on how to make (54) hold can be depicted by Theorem 15.

Theorem 15. If one designs 𝑃_𝑖>𝑃_𝑓,(𝑖=1,…,𝑁,𝑃_𝑖≠𝑃_𝑓), then 𝑉₈(|𝜓_𝑓⟩)<𝑉₈(|𝜓_other⟩) holds, where 𝑃_𝑓 is the eigenvalue of 𝑃_{𝛾(|𝜓𝑓⟩)} corresponding to |𝜓_𝑓⟩.

We can conclude from the above analyses that by using the implicit Lyapunov control based on the imaginary mechanical quantity, if one designs appropriate control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) to make the conditions (i)–(iv) in Theorem 15 and (54) hold, the control system governed by (37) in the degenerate cases can converge from any initial pure state to the target eigenstate. The design principle of the imaginary mechanical quantity 𝑃_𝛾 is Theorem 15 and (55).

3.2. Target Superposition State

The method proposed in Section 3.1 cannot guarantee the control system governed by (37) or (43) in the degenerate cases converge from any initial pure state to the target superposition state. In order to solve this problem, one can introduce a series of constant disturbances 𝜂_𝑘 into the control laws. Thus the mechanical equation (37) becomes 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝜂_𝑘+𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)𝜓(𝑡). (56) And the mechanical equation (43) in Section 3.1.2 becomes 𝑖𝜓̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)+𝜂_𝑘+𝜔𝐼𝜓(𝑡), (57) where 𝜂_𝑘∈𝑅.

The basic idea of solving the convergence to the target superposition state is to design 𝜂_𝑘 to make the target state |𝜓_𝑓⟩ be an eigenstate of 𝐻^′₀=𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝜂_𝑘⋅𝐻^′₀ can be viewed as the new internal Hamiltonian of the control system. If the number of the control Hamiltonians 𝑟 is large enough, by designing appropriate 𝜂_𝑘, (𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝜂_𝑘)|𝜓_𝑓⟩=𝜆^′_𝑓|𝜓_𝑓⟩ can be satisfied in most cases, where 𝜆^′_𝑓 is the eigenvalue of 𝐻₀+∑^𝑟_𝑘=1𝐻_𝑘𝜂_𝑘 corresponding to |𝜓_𝑓⟩. Then one can design the control laws and analyze the convergence according to the method for the target eigen state cases. Research results show that every conclusion in Section 3.1 also holds with changing 𝐻₀ into 𝐻^′₀.

3.3. Target Mixed State Which Commutes with the Internal Hamiltonian

Consider the 𝑁-level closed quantum system governed by the following quantum Liouville equation: 𝑖𝜌̇(𝑡)=𝐻₀+^𝑟_𝑘=1𝐻_𝑘𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),𝜌(𝑡), (58) where 𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡)=𝑢_𝑘(𝑡) are the total control laws.

The design ideas are similar to those of Section 3.1. Consider the Lyapunov function based on the imaginary mechanical quantity as 𝑉₉𝜌=tr𝑃_{𝛾1,…,𝛾𝑟}𝜌, (59)𝛾_𝑘(𝑡) can be designed as 𝛾_𝑘(𝑡)=𝛾(𝑡)=𝜃𝑉₉𝜌−𝑉₉𝜌_𝑓,𝑘=𝑘₁,…,𝑘_𝑚;𝛾_𝑘(𝑡)=0,𝑘≠𝑘₁,…,𝑘_𝑚1≤𝑘₁,…,𝑘_𝑚≤𝑟. (60)

The existence of 𝛾(𝑡) can be depicted by Lemma 16.

Lemma 16. If 𝜃∈𝐶^∞(𝑅⁺;[0,𝛾^∗]),𝑘=1,…,𝑟 (𝛾^∗ is a positive constant) satisfy 𝜃(0)=0, 𝜃(𝑠)>0 and 𝜃^′(𝑠)>0 for every 𝑠>0, and |𝜃^′|<1/(2𝐶^∗), 𝐶^∗=1+𝐶, 𝐶=max{‖𝜕𝑃_𝛾/𝜕𝛾‖_𝑚1,𝛾∈[0,𝛾^∗]}, then for every 𝜌, there is a unique 𝛾∈𝐶^∞(𝛾∈[0,𝛾^∗]) satisfying 𝛾(𝜌)=𝜃(tr(𝑃_𝛾𝜌)−tr(𝑃_𝛾𝜌_𝑓)).

According to the Lyapunov stabilty Theorem, 𝑣_𝑘(𝑡) can be designed as 𝑣_𝑘(𝑡)=𝐾_𝑘𝑓_𝑘𝑖tr𝑃_𝛾,𝐻_𝑘𝜌,(𝑘=1,…,𝑟), (61) where 𝐾_𝑘 is a constant and 𝐾_𝑘>0, and 𝑦_𝑘=𝑓_𝑘(𝑥_𝑘),(𝑘=1,2,…,𝑟) are monotonic increasing functions through the coordinate origin of the plane 𝑥_𝑘−𝑦_𝑘.

Based on the LaSalle invariance principle, the convergence of the control system can be depicted by Theorem 17.

Theorem 17. Consider the control system depicted by (58) with control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡), where 𝛾_𝑘(𝑡) is defined by Lemma 16 and (60), and 𝑣_𝑘(𝑡) is defined by (61). If the control system satisfies (i) 𝜔_{𝑙,𝑚,𝛾}≠𝜔_{𝑖,𝑗,𝛾},(𝑙,𝑚)≠(𝑖,𝑗),𝑖,𝑗,𝑙,𝑚∈{1,2,…,𝑁}, 𝜔_{𝑙,𝑚,𝛾}=𝜆_𝑙,𝛾−𝜆_𝑚,𝛾, where 𝜆_𝑙,𝛾 is the 𝑙th eigenvalue of (𝐻₀+∑^𝑘𝑚_𝑛=𝑘1𝐻_𝑛𝛾(𝑡)) corresponding to the eigenstate |𝜙_l,𝛾⟩; (ii) ∀𝑗≠𝑙, for 𝑘=1,…,𝑟, there exists at least a (𝐻_𝑘)_𝑗𝑙≠0, where (𝐻_𝑘)_𝑗𝑙 is the (𝑗,𝑙)th element of 𝐻_𝑘=𝑈^𝐻₂𝐻_𝑘𝑈₂ with 𝑈₂=(|𝜙_1,𝛾⟩,…,|𝜙_𝑁,𝛾⟩); (iii) [𝑃_𝛾,𝐻₀+∑^𝑘𝑚_𝑛=𝑘1𝐻_𝑛𝛾(𝑡)]=0,1≤𝑘₁,…,𝑘_𝑚≤𝑟; (iv) for any 𝑙≠𝑗,(1≤𝑙,𝑗≤𝑁), (𝑃_𝛾)_𝑙𝑙≠(𝑃_𝛾)_𝑗𝑗 holds, where (𝑃_𝛾)_𝑙𝑙 is the (𝑙,𝑙)th element of 𝑃_𝛾=𝑈^𝐻₂𝑃_𝛾𝑈₂, then the control system will converge toward 𝐸₁₀={𝜌_𝑡0|(𝑈^†₂𝜌_𝑡0𝑈₂)_𝑖𝑗=0,𝛾=𝛾(𝜌_𝑡0),𝑡₀∈𝑅}.

For the target state 𝜌_𝑓 which commutes with the internal Hamiltonian; that is, [𝜌_𝑓,𝐻₀]=0, 𝜌_𝑓 is contained in 𝐸₁₀. In order to make the system converge to the target state 𝜌_𝑓 which commutes with 𝐻₀ from any initial stat; on the one hand, 𝑃_𝛾 needs to be designed to make 𝑉₉𝜌_𝑓<𝑉₉𝜌_other (62) hold, where 𝜌_other represents any other state in the set 𝐸₁₀ except the target state. On the other hand, 𝛾 was designed as 𝛾=𝛾−𝛼,(0<𝛼≪𝛾) when 𝑣_𝑘(𝑡)=0,𝛾(𝑡)=𝛾≠0 holds for some time to make the state trajectory evolve but not stay in 𝐸₁₀ until 𝜌_𝑓 is reached.

For the satisfaction of conditions (i)–(iv) in Theorem 17, one can follow that of Section 3.1.3. In order to make (62) hold, one can design 𝑃_𝛾 based on Proposition 6. The total design principle of 𝑃_𝛾 is (55) and Proposition 6.

We can conclude from the above analyses that for the target state which commutes with the internal Hamiltonian, by using the implicit Lyapunov control method based on the imaginary mechanical quantity, if one designs appropriate control laws 𝑢_𝑘(𝑡)=𝛾_𝑘(𝑡)+𝑣_𝑘(𝑡),(𝑘=1,…,𝑟) to make the conditions (i)–(iv) in Theorem 17 and (62) hold, the control system governed by (58) in the degenerate cases can converge from any initial state to the target state which commutes with the internal Hamiltonian, which contains the target eigenstate and the target mixed state which commutes with the internal Hamiltonian.

3.4. Target Mixed State Which Does Not Commute with Internal Hamiltonian

For the target mixed state which does not commute with the internal Hamiltonian, the design idea is similar to Section 3.2. The difference is to design 𝜂_𝑘 to make [𝜌_𝑓,𝐻^′₀]=0. If the number of the control Haimltonians 𝑟 is large enough, by designing appropriate 𝜂_𝑘, [𝜌_𝑓,𝐻^′₀]=0 can be satisfied in most cases. Then one can design the control laws and analyze the convergence according to the method mentioned in Section 3.3. Research results show that every conclusion in Section 3.3 also holds with changing 𝐻₀ into 𝐻^′₀.

4. Conclusion

In this paper, for the non-degenerate and degenerate cases, the existing quantum Lyapunov control based on the state distance, state error, and average value of an imaginary mechanical quantity for the control systems have been summarized and analyzed. For the target state being the eigenstate, the mixed state which commutes with the internal Hamiltonian, the superposition state, and the mixed state which does not commute with the internal Hamiltonian, respectively, the design methods of the control laws have been summarized; the convergence to the target state has been summarized and analyzed. Research results show that the Lyapunov-based quantum control method can make the control system converge from any initial state to the target state in both non-degenerate and degenerate cases. After ten years of development, the quantum control theory based on the Lyapunov stability theorem has been established.

Acknowledgments

This work was supported in part by the National Key Basic Research Program under Grant no. 2011CBA00200 and the National Science Foundation of China under Grant no. 61074050.