Control landscapes for two-level open quantum systems
A quantum control landscape is defined as the physical objective as a function of the control variables. In this paper the control landscapes for two-level open quantum systems, whose evolution is described by general completely positive trace preserving maps (i.e., Kraus maps), are investigated in details. The objective function, which is the expectation value of a target system operator, is defined on the Stiefel manifold representing the space of Kraus maps. Three practically important properties of the objective function are found: (a) the absence of local maxima or minima (i.e., false traps); (b) the existence of multi-dimensional sub-manifolds of optimal solutions corresponding to the global maximum and minimum; and (c) the connectivity of each level set. All of the critical values and their associated critical sub-manifolds are explicitly found for any initial system state. Away from the absolute extrema there are no local maxima or minima, and only saddles may exist, whose number and the explicit structure of the corresponding critical sub-manifolds are determined by the initial system state. There are no saddles for pure initial states, one saddle for a completely mixed initial state, and two saddles for partially mixed initial states. In general, the landscape analysis of critical points and optimal manifolds is relevant to explain the relative ease of obtaining good optimal control outcomes in the laboratory, even in the presence of the environment.
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
Institute of Spectroscopy, Troitsk, Moscow Region 142190, Russia
A common goal in quantum control is to maximize the expectation value of a given target operator by applying a suitable external action to the system. Such an external action often can be realized by a tailored coherent control field steering the system from the initial state to a target state, which maximizes the expectation value of the target operator [1, 2, 3, 4, 5, 6, 7, 8, 9]. Tailored coherent fields allow for controlling Hamiltonian aspects (i.e., unitary dynamics) of the system evolution. Another form of action on the system could be realized by tailoring the environment (e.g., incoherent radiation, or a gas of electrons, atoms, or molecules) to induce control through non-unitary system dynamics . In this approach the control is the suitably optimized, generally non-equilibrium and time dependent distribution function of the environment; the optimization of the environment would itself be attained by application of a proper external action. Combining such incoherent control by the environment (ICE) with a tailored coherent control field provides a general tool for manipulating both the Hamiltonian and dissipative aspects of the system dynamics. A similar approach to incoherent control was also suggested in  where, in difference with , finite-level ancilla systems are used as the control environment. The initial state of the field and the interaction Hamiltonian as the parameters for controlling non-unitary dynamics was also suggested in . Non-unitary controlled quantum dynamics can also be realized by using as an external action suitably optimized quantum measurements which drive the system towards the desired control goal [13, 14, 15, 16, 17]. General mathematical definitions for the controlled Markov dynamics of quantum-mechanical systems are formulated in .
In this paper we consider the most general physically allowed transformations of states of quantum open systems, which are represented by completely positive trace preserving maps (i.e., Kraus maps) [19, 20, 21, 22]. A typical control problem in this framework is to find, for a given initial state of the system, a Kraus map which transforms the initial state into the state maximizing the expected value of a target operator of the system. Practical means to find such optimal Kraus maps in the laboratory could employ various procedures such as adaptive learning algorithms [3, 23], which are capable of finding an optimal solution without detailed knowledge of the dynamics of the system. Kraus maps can be represented by matrices satisfying an orthogonality constraint (see Sec. II), which can be naturally parameterized by points in a Stiefel manifold , and then various algorithms may be applied to perform optimization over the Stiefel manifold (e.g., steepest descent, Newton methods, etc. adapted for optimization over Stiefel manifolds) [25, 26].
The quantum control landscape is defined as the objective expectation value as a function of the control variables. The efficiency of various search algorithms (i.e., employed either directly in the laboratory or in numerical simulations) for finding the minimum or maximum of a specific objective function can depend on the existence and nature of the landscape critical points. For example, the presence of many local minima or maxima (i.e., false traps) could result in either permanent trapping of the search or possibly dwelling for a long time in some of them (i.e., assuming that the algorithm has the capability of extricating the search from a trap) thus lowering the search efficiency. In such cases stopping of an algorithm at some solution does not guarantee that this solution is a global optimum, as the algorithm can end the search at a local maximum of the objective function. A priori information about absence of local maxima could be very helpful in such cases to guarantee that the search will be stopped only at a global optimum solution. This situation makes important the investigation of the critical points of the control landscapes. Also, in the laboratory, evidence shows that it is relatively easy to find optimal solutions, even in the presence of an environment. Explanation of this fact similarly can be related with the structure of the control landscapes for open quantum systems.
The critical points of the landscapes for closed quantum systems controlled by unitary evolution were investigated in [27, 28, 29, 30, 31], where it was found that there are no sub-optimal local maxima or minima and only saddles may exist in addition to the global maxima and minima. In particular, it was found that for a two-level system prepared initially in a pure state the landscape of the unitary control does not have critical points except for global minima and maxima.
The capabilities of unitary control to maximize or minimize the expectation value of the target operator in the case of mixed initial states are limited, since unitary transformations can only connect states (i.e., density matrices) with the same spectrum. In going beyond the latter limitations, the dynamics may be extended to encompass non-unitary evolution by directing the controls to include the set of Kraus maps (i.e., dual manipulation of the system and the environment). Quantum systems which admit arbitrary Kraus map dynamics are completely controllable, since for any pair of states there exists a Kraus map which transforms one into the another .
In this paper the analysis of the landscape critical points is performed for two-level quantum systems controlled by Kraus maps. It is found that the objective function does not have sub-optimal local maxima or minima and only saddles may exist. The number of different saddle values and the structure of the corresponding critical sub-manifolds depend on the system initial state. For pure initial states the landscape has no saddles; for a completely mixed initial state the landscape has one saddle value; for other (i.e., partially mixed) initial states the landscape has two saddle values. For each case we explicitly find all critical sub-manifolds and critical values of the objective as functions of the Stokes vector of the initial density matrix. An investigation of the landscapes for multi-level open quantum systems with a different method may also be performed . The absence of local minima or maxima holds also in the general case although an explicit description of the critical manifolds is difficult to provide for multi-level systems. The absence of false traps practically implies the relative ease of obtaining good optimal solutions using various search algorithms in the laboratory, even in the presence of an environment.
It should be noted that the property of there being no false traps relies on the assumption of the full controllability of the system, i.e., assuming that an arbitrary Kraus map can be realized. Restrictions on the set of available Kraus maps can result in the appearance of false traps thus creating difficulties in the search for optimal solutions. Thus, it is important to consider possible methods for engineering arbitrary Kraus type evolution of a controlled system. One method is to put the system in contact with an ancilla and implement, on the coupled system, specific unitary evolution whose form is determined by the structure of the desired Kraus map  (see also Sec. II). Lloyd and Viola proposed another method of engineering arbitrary Kraus maps, based on the combination of coherent control and measurements . They show that the ability to perform a simple single measurement on the system together with the ability to apply coherent control to feedback the measurement results allows for enacting arbitrary Kraus map evolution at a finite time.
A level set of the objective function is defined as the set of controls which produce the same outcome value for . We investigate connectivity of the level sets of the objective functions for open quantum systems and show that each level set is connected, including the one which corresponds to the global maximum/minimum of the objective function. Connectivity of a level set implies that any two solutions from the same level set can be continuously mapped one into another via a pathway entirely passing through this level set. The proof of the connectivity of the level sets is based on a generalization of Morse theory. Experimental observations of level sets for quantum control landscapes can be practically performed, as it was recently demonstrated for control of nonresonant two-photon excitations .
In summary, the main properties of control landscapes for open quantum systems are: (a) the absence of false traps; (b) the existence of multi-dimensional sub-manifolds of global optimum solutions, and (c) the connectivity of each level set. The proof of the properties (a)–(c) is provided in the next sections for the two-level case. Figure 1 illustrates the properties (a), (b), and connectivity of the manifold of global maximum solutions; the figure does not serve to illustrate other properties such as connectivity of each level set. It is evident that the function drawn on figure 1 does not have local minima or maxima and the set of solutions for the global maximum is a connected sub-manifold (a curve in this case). A simple illustration is chosen for the figure since an exact objective function for an -level quantum system depends on real variables (such that for ) and therefore can not be drawn.
The present analysis is performed in the kinematic picture which uses Kraus maps to represent evolution of quantum open systems. An important future task is to investigate the structure of the control landscape in the dynamical picture, which can be based on the use of various dynamical master equations to describe the dynamics of quantum open systems [22, 40, 41, 42, 43, 44]. Such analysis may reveal landscape properties for quantum open systems under (possibly, restricted) control through manipulation by a specific type of the environment (e.g., incoherent radiation).
In addition to optimizing expected value of a target operator, a large class of quantum control problems includes generation of a predefined unitary (e.g., phase or Hadamard)  or a non-unitary  quantum gate (i.e., a quantum operation). This class of control problems is important for quantum computation and in this regard a numerical analysis of the problem of optimal controlled generation of unitary quantum gates for two-level quantum systems interacting with an environment is available [34, 35].
Although the assumption of complete positivity of the dynamics of open quantum systems used in the present analysis is a generally accepted requirement, some works consider dynamics of a more general form [45, 46]. Such more general evolutions may result in different controllability and landscape properties. For example, for a two-level open quantum system positive and completely positive dynamics may have different accessibility properties . In this regard it would be interesting to investigate if such different types of the dynamics have distinct essential landscape properties.
In Sec. 2 the optimal control problem for a general -level open quantum system is formulated. Section 3 reduces the consideration to the case of a two-level system. In Sec. 4 a complete description is given of all critical points of the control landscape. The connectivity of the level sets is investigated in Sec. 5.
2 Formulation for an -level system
Let be the linear space of complex matrices. The density matrix of an -level quantum system is a positive component in , , with unit trace, (Hermicity of follows from its positivity). Physically allowed evolution transformations of density matrices are given by completely positive trace preserving maps (i.e., Kraus maps) in . A linear Kraus map satisfies the following conditions :
Complete positivity. Let be the identity matrix in . Complete positivity means that for any integer the map acting in the space is positive.
Trace preserving: , .
where the Kraus operators satisfy the relation . For an -level quantum system it is sufficient to consider at most Kraus operators .
Let be the Hilbert space of the system under control. An arbitrary Kraus map of the form (1) can be realized by coupling the system to an ancilla system characterized by the Hilbert space , and generating a unitary evolution operator acting in the Hilbert space of the total system as follows . Choose in a unit vector and an orthonormal basis , . For any let . Such an operator can be extended to a unitary operator in and for any one has . Therefore the ability to dynamically create, for example via coherent control, an arbitrary unitary evolution of the system and ancilla allows for generating arbitrary Kraus maps of the controlled system.
Let be the initial system density matrix. A typical optimization goal in quantum control is to maximize the expectation value of a target Hermitian operator over an admissible set of dynamical transformations of the system density matrices. For coherent unitary control this expectation value becomes
where is a unitary matrix, , which describes the evolution of the system during the control period from the initial time until some final time and implicitly incorporates the action of the coherent control field on the system.
In the present paper we consider general non-unitary controlled dynamics such that the controls are Kraus maps, for which the parametrization by Kraus operators is used. The corresponding objective function specifying the control landscape has the form
where the Kraus operators describe evolution of the open quantum system from an initial time until some final time . The control goal is to maximize the objective function over the set of all Kraus operators satisfying , thereby forming a constrained optimization problem.
Let be a field of real or complex numbers, i.e., or . A Stiefel manifold over , denoted , is the set of all orthonormal -frames in (i.e., the set of ordered -tuples of orthonormal vectors in ). The case (respectively, ) corresponds to a real (complex) Stiefel manifold.
Let be the matrix defined as , where is the transpose of matrix and is the number of Kraus operators. Consider vectors with components , i.e., vector is the -th row of the matrix . The constraint in terms of the vectors takes the form , where is the Kronecker delta symbol. This constraint defines the complex Stiefel manifold . Therefore optimization of the objective function defined by Eq. (2) can be formulated as optimization over the complex Stiefel manifold .
3 Two-level system
In the following we consider the case of a two-level system in detail. Any density matrix of a two-level system can be represented as
where is the vector of Pauli matrices and is the Stokes vector, . Thus, the set of density matrices can be identified with the unit ball in , which is known as the Bloch sphere.
Any Kraus map on can be represented using at most four Kraus operators
as , where the Kraus operators satisfy the constraint
Let be the initial system density matrix with Stokes vector , where , and let be a Hermitian target operator. The objective functional for optimizing the expectation value of has the form . The control goal is to find all quadruples of Kraus operators which maximize (or minimize, depending on the control goal) the objective functional . The goal of the landscape analysis is to characterize all critical points of , including local extrema, if they exist.
The analysis for an arbitrary Hermitian matrix can be reduced to the case
which we will consider in the sequel. This point follows, as an arbitrary Hermitian operator has two eigenvalues and and can be represented in the basis of its eigenvectors as
where . One has and
Therefore, the objective function for a general observable operator depends linearly on the objective function defined for . We denote . In the trivial case the landscape is completely flat and no further analysis is needed.
4 The critical points of the objective function landscape
The Kraus operators for a two-level system can be parameterized by a pair of vectors of the form and , where , , , and . The objective function in terms of these vectors has the form
where , and denote the standard inner product and the norm in (here the numbers are the components of the Stokes vector of the initial density matrix , see Sec. 3). The constraint (3) in terms of the vectors and has the form , and determines the Stiefel manifold . The matrix constraint (3) in terms of the vectors and has the form
If , then the objective function is diagonalized by introducing new coordinates in according to the formulas
where and . The objective function in these coordinates has the form
Let be a real vector such that and let . For any such , the global maximum and minimum values of the objective function are
The critical sub-manifolds and other critical values of in are the following:
Case 1. (the completely mixed initial state). The global minimum sub-manifold is . The global maximum sub-manifold is . The objective function has one saddle value with the corresponding critical sub-manifold . The Hessian of at any point at has positive, negative, and zero eigenvalues.
Case 2. (a partially mixed initial state). The global minimum sub-manifold is . The global maximum sub-manifold is . The objective function has two saddle values:
The corresponding critical sub-manifolds are and . The Hessian of at any point at (resp., ) has positive, negative (resp., positive, negative), and zero eigenvalues.
Case 3. (a pure initial state). The global minimum sub-manifold is . The global maximum sub-manifold is . The objective function has no saddles.
Proof. The objective function has the form , where is the diagonal matrix element of the density matrix. Therefore and the value (resp., ) corresponds to the global minimum (resp., maximum).
The constraints can be included in the objective function (10) by adding the term , where the two real and one complex Lagrange multipliers , and correspond to the two real and one complex valued constraints , and , respectively. Critical points of the function on the manifold are given by the solutions of the following Euler-Lagrange equations for the functional :
Case 2. . Consider in the open subset . Let us prove that the set of all critical points of in is the set of all points of such that .
Suppose that there are critical points in such that or . For such points the following identity holds
In , and therefore . This equality together with (16) gives
This system of equations implies which is in contradiction with the assumption for the present case. If , then it follows from (14), (15) that . In this case equations (12) and (13) have only the solution .
Points in with form the global minimum manifold , which is a Stiefel manifold and hence is connected. In some small neighborhood of zero we can choose and as normal coordinates. So is non degenerate. Similar treatment of the region gives the global maximum manifold .
Now consider the region . In this region the objective function has the form
Using the analysis for the region , we conclude that the objective function has no critical points such that in . Therefore all critical points in are in the sub-manifold . The restriction of to has the form
Note that is a subset of all sets of vectors satisfying the constraints
It is clear from this representation of that if and only if . This gives the critical sub-manifold . The objective function has the value on this manifold.
To show that this is a saddle manifold, and not a local maximum or minimum, we calculate the Morse indices of the objective function on and show that both positive and negative Morse indices are different from zero (the Morse indices are the numbers of positive, negative and zero eigenvalues of the Hessian of and positive and negative Morse indices determine the number of local coordinates along which the function increases or decreases, respectively). With regard to this goal, consider the manifold . Let . Below we introduce some coordinates in a neighborhood of on .
For any such that we define the unit vector . Let be some coordinate system on (embedded in as a unit sphere with the origin at zero) in some neighborhood of and be some coordinate system on in some neighborhood of . We will use the following functions defined in some neighborhood of on ():
Let be the maximal complex subspace of the tangent space of . For each let be coordinates on and for each be coordinates on .
Let and be functions on defined as follows.
Let be in a small enough neighborhood of . By definition is the projection from to and is the projection from to . By definition
Now let and be the complex-valued functions defined on by the formulas
Thus, the functions , where and , are coordinates on in some neighborhood of the point . Locally the manifold is a sub-manifold of defined by the constraint . In our coordinates this constraint has a form
Therefore , where and are the coordinates on in some neighborhood of . The second differential of at the point in this coordinates has the form
Since for the present case, the Morse indices of this point are (note that is a complex coordinate).
Similar treatment of the region shows the existence of the critical sub-manifold . This sub-manifold corresponds to the critical value and its Morse indices are . Since , this concludes the proof for the case .
Case 1. w=0. Consider in the open subset .
Let . Then in the region Eqs. (14) and (15) imply that . Equations (12) and (13) for such have only the solution which defines the global minimum manifold . Now let and or . In this case Eqs. (12)–(15) give and , which imply and . Then Eqs. (12) and (15) have the solution
where we used the notation and the relation . Note that for a given pair , can be any non-zero complex number such that . The solutions of the form (20) constitute the critical set . A similar treatment of the region shows that the objective function in this region has as critical points only the global maximum manifold and the set .
Now consider the region .
Let . The treatment is similar to the treatment of the case for the region and gives the critical set . A similar treatment of the region shows that the set of critical points of the objective function in this region is .
Combining together the results for the regions , , , and , we find that the critical manifolds are the global minimum manifold , the global maximum manifold , and the set . Since , these manifolds are all critical manifolds of the objective function for the case . A simple computation using the constraints (5)–(7) shows that the value of the objective function at any point equals to , i.e.,