We illustrate a unified frame of quantum systems theory in view of control engineering. In particular, we shift controllability criteria from the well‐known Lie‐algebra rank condition to symmetry conditions that are easy to check yet rigorously rooted in the branching diagrams for simple subalgebras of . Reachable sets of closed bilinear control systems are linked to the theory of ‐numerical ranges. In coherently controlled open Markovian systems, the set of reachable directions (Lindblad generators) forms a Lie wedge that generates a Lie semigroup of quantum maps and helps to approximate reachable sets of open systems. Once reachable sets are known, gradient‐flow algorithms can solve the abstract optimization task on the reachable sets. They thus complement numerical algorithms for concrete optimal control problems on the manifold of admissible control amplitudes presented in a unified programming framework in Chapter 28. Finally, we give an outlook on infinite‐dimensional control of atoms coupled to oscillator modes.

How principles turn into practice has meanwhile emerged in a plethora of examples showing applications in solid‐state devices, circuit‐QED, ion traps, NV‐centers in diamond, quantum dots, spin systems, and ‐level atoms coupled to a (light)field.

27.1 Introduction

This contribution is meant as an invitation to exchange with the vibrant developments in the field of quantum systems and control (1) in view of future technologies (2). These may be triggered by precise controls for, for example, quantum simulation in order to improve the understanding of quantum phase transitions (3) between conducting and superconducting phases, or between ferromagentic and anti‐ferromagnetic phases to name just a few. Needless to say, a thorough picture of these phenomena will boost material design.

An important issue in quantum simulation (4–8) is to manipulate all pertinent dynamical degrees of freedom of a system of interest (which, however, all too often is experimentally not fully accessible) by another quantum system that is in fact well controllable in practice and the dynamics of which are equivalent to those of . We will show how to characterize this situation algebraically in terms of quantum systems theory.

Besides the practical applications and implications, quantum systems should also be of appeal to the (classical) control engineer, because nearly all systems of interest boil down to the standard form of bilinear control systems (9–12)

27.1

Here one may take as linear operators on a (finite‐dimensional) Hilbert space of quantum states . For two‐level spin‐ systems, . More precisely, denotes the system or drift Hamiltonian , whereas the are the control Hamiltonians governed by typically piece‐wise constant control amplitudes . Thus Eq. 27.1 captures all scenarios in Table 27.1, where denotes a unitary operator on (e.g., used as quantum gate). For open systems, is a (linear) quantum map that acts on density operators and whose time evolution is governed by the (super)operator including relaxation.

Setting and task			Drift		Controls
Closed systems:
Pure‐state transfer
Gate synthesis (with specified global phase)
State transfer
Gate synthesis (with free global phase)
Open systems:
State transfer
Quantum‐map synthesis

While linear control systems are fully controllable (13) if the reachability matrix has full rank, bilinear systems of the form Eq. 27.1 are fully controllable on a compact connected Lie group (with Lie algebra so ) if they satisfy the celebrated Lie‐algebra rank condition (14–18)

As in open systems is usually no longer compact, dissipative systems are obviously more subtle and give rise to Lie semigroups in the Markovian case illustrated below.

For closed quantum systems of spins‐ , one has and with illustrating how dynamic degrees of freedom in quantum systems scale exponentially in system size (as opposed to classical systems, where they scale linearly). Thus assessing controllability via an explicit computation of the Lie closure for the rank condition, though mathematically straightforward, becomes dramatically tedious in large quantum systems, and beyond seven qubits, it is mostly prohibitive.

27.2 Systems Theory of Closed Quantum Systems

Hence here we sketch particularly simple and powerful symmetry arguments for assessing controllability of quantum systems avoiding an explicit calculation of the Lie closure. For extending the symmetry arguments to a frame embracing closed and open systems in more detail, see the recent Ref. (19), from whence parts are extracted here.

27.2.1 Controllability and its Symmetry Conditions

It pays to envisage bilinear control systems by graphs as illustrated in Figure 27.1: vertices represent local qubits as controlled by typical control Hamiltonians (represented by Pauli matrices acting on the qubit represented by the respective vertex), edges stand for pair‐wise coupling interactions typically only occurring in the drift term (two‐component tensor products of Pauli matrices as, for example, for Ising interaction or for Heisenberg– interaction. Here the Pauli operators act on the two qubits connected by the respective edge).

As a central notion in the subsequent arguments, we characterize a quantum bilinear control system by its system Lie algebra , which results from the Lie closure of taking nested commutators (until no new linearly independent elements are generated)

as well as by its (potential) symmetries, that is, the centralizer in to the system algebra collecting all terms that commute jointly with all Hamiltonian operators

If there are no symmetries, that is, if the centralizer is trivial, then the system algebra is irreducible. This can easily be checked by determining the dimension of the nullspace to the corresponding commutator superoperators (of dimension ) – so it boils down to solving a system of homogeneous equations in dimensions.

Therefore, a trivial centralizer plus a connected graph imply that the corresponding system algebra is simple. As the largest possible Lie closure is , the system algebra of an irreducible connected qubit system has to be a (proper or improper) irreducible simple subalgebra to . By making heavy use of computer algebra, in Ref. (20), we have classified all these simple subalgebras of for with qubits as summarized by the branching diagrams in Figure 27.2 thus extending the known results from (21,22) to .

This figure also illustrates that every with has two canonical branches, a symplectic branch (upper branch) starting with and an orthogonal branch (lower) commencing with . Actually, for odd , these are the only ones (and we conjecture that this holds true even beyond 15 qubits). In contrast, for even , there are always subalgebras of unitary (spinor) type shown in black plus potential others (observe the instances of ). – Clearly, if the (nontrivial) system algebra of a dynamic system in question can be ruled out to be on any of these three branches, then corresponding control system is indeed fully controllable as will be shown next.

To this end, it is convenient to exclude the symplectic and orthogonal subalgebras in the first place. It is a task that can again be readily accomplished (after having made sure is irreducible) by determining the dimension of the joint null space (over S) to the following equations for each with or in superoperator form where by Schur's Lemma (23). If there is a nontrivial solution for the (+)‐variant, is of orthogonal type, and if there is for the ( )‐variant, is of symplectic type. Therefore, if the solution space for both cases ( ) is zero‐dimensional (corresponding to the only solution being trivial), then is neither of orthogonal nor symplectic type. This can conveniently be decided by solving a system of linear equations as done in Algorithm 3 of Ref. (20).

For odd , this does in fact already ensure full controllability, as only even allows for unitary (spinor‐type) simple subalgebras. Yet we conjecture that these findings also hold for all . Finally, for even , the spinor‐type subalgebras may be excluded by the subsequent theorem of Ref. (20). To prepare for it, observe that for selfadjointness of and entails

hence the projector on is in the commutant of the tensor‐square representation, that is, .

This motivates a closer look at the tensor‐square representation

and its commutant referred to as “quadratic symmetries” of by that give a powerful single necessary and sufficient symmetry condition for full controllability:

To sum up, a bilinear ‐qubit control system as in Eq. 27.1 is fully controllable if and only if all of the following conditions are satisfied

1. the system has no symmetries, that is, is trivial;
2. the system has a connected coupling graph;
3. the system algebra is neither of orthogonal nor of symplectic type; and
4. the system algebra is not of any other type, in particular not of unitary spinor‐type or of exceptional type ( ).

While we gave a rigorous proof in Ref. (20) as already mentioned, above key arguments can easily be made intuitive as follows:

1. symmetries would entail conserved entities (invariant one‐parameter groups) thus precluding full controllability;
2. coupling graphs with several connected components preclude that these components can be coherently coupled, which obviously is necessary for full controllability;
3. orthogonal or symplectic subalgebras are proper subalgebras to (for ) and do not explore all dynamic degrees of freedom of ; and
4. the same holds for unitary spinor‐type or exceptional subalgebras ( ) of .

System type		Fermionic	‘Bosonic’	System algebra
‐spins‐	Levels	— Coupling order —
		2	–
		2	–
for		Up to	–
for		–	Up to
		Up to	Up to

By the branching diagrams in Figure 27.2, it is immediately obvious: establishing full controllability boils down to ensuring the dynamic system is governed by a system algebra that is irreducible (no symmetries), and simple (connected coupling graph) and top of the branch. This shifts the paradigm from the Lie‐algebra rank‐condition to easily verifiable symmetry conditions, which can be checked using only the Hamiltonian generators.

System type		‘Bosonic’	System algebra
spins‐	Levels	Coupling order
		Up to 3
	—”—	—”—	—”—
		Up to 5
	—”—	—”—	—”—
	—”—	—”—	—”—
	—”—	—”—	—”—
	—”—	—”—	—”—
	—”—	—”—	—”—
	—”—	—”—	—”—

27.2.3 Reachable Sets and Expectation Values of Closed Quantum Systems: Link to Relative ‐Numerical Ranges

Once the compact system algebra of a bilinear control system is determined, for example, by symmetry characterization as in the previous section, then the time evolution is brought about by the corresponding group⁰¹ . Therefore, the reachable set for every initial state is given by the corresponding subgroup orbit

In other words, the time evolution of the state is confined to in the sense solves the equation of motion 27.1 under Hamiltonian drift and controls in the absence of relaxation, that is .

In quantum dynamics, the expectation value of a Hermitian observable, or more generally a detection operator , is defined as projection of onto by way of the Hilbert–Schmidt scalar product where . Recall that the field of values of is , whereas for the ‐numerical range of is . Therefore, if is a rank‐1 projector, the expectation value is an element of the field of values , whereas for general , it is an element of the ‐numerical range of , that is, . The latter is a star‐shaped subset of the complex plane (29,30) and it specializes to a real line segment in case and are both Hermitian.

As illustrated in the previous section, different quantum dynamical scenarios come with specific dynamical subgroups generated by the specific system algebras . Typical examples for include or or the subgroup of local unitary operations .

Consequently, in the instances of , the admissible expectation values typically fill but a proper subset of , which hence motivates our definition of a restricted or relative ‐numerical range (31,32) as subgroup orbit projected onto

The particular case of local qubit‐wise actions leads to what we define as local ‐numerical range. As is compact and connected, this obviously extends to . However, note that although being connected, in general turns out to be neither star shaped nor simply connected (32) in contrast to the usual ‐numerical range (29).

The largest absolute value of the relative ‐numerical range is defined as the relative ‐numerical radius

it obviously plays a significant role for optimizations aiming at maximal expectation values.

With these stipulations, we will discuss recent applications of the local ‐numerical range in quantum control.

27.2.4 Constrained Optimization and Relative ‐Numerical Ranges

In quantum control, one may face the problem to maximize the unitary transfer from matrices to subject to suppressing the transfer from to , or subject to leaving another state invariant. For tackling those types of problems, in (33), we asked for a “constrained ‐numerical range of ”

which form it takes and – in view of numerical optimization – whether it is a connected set with a well‐defined boundary. Connectedness is central to any numerical optimization approach, because otherwise one would have to rely on initial conditions in the connected component of the (global) optimum.

Now the constrained ‐numerical range of is a compact and connected set in the complex plane, if the constraint can be fulfilled by restricting the full unitary group to a compact and connected subgroup . In this case, the constrained ‐numerical range is identical to the relative ‐numerical range and hence the constrained optimization problem is solved within it, for example, by the corresponding relative ‐numerical radius .

The new concept of the relative (or restricted) ‐numerical range has meanwhile become a popular tool, for example, for analyzing entanglement properties, see (34,35) (and references therein).

Therefore, next we optimize by gradient flow methods over the underlying ‐orbits.

27.2.5 Optimization by Gradient Flows

A pioneering paper by Brockett (36) extended in books by Helmke and Moore (37) and followed by Bloch (38) triggered to apply gradient flows on the unitary orbit of quantum states (39).

Implementing a gradient method for optimization on a smooth “constrained manifold” – such as an unitary orbit – via the Riemannian exponential map, inherently ensures that the discretized flow remains within that manifold. Therefore, gradient flows on manifolds are intrinsic optimization methods (40), whereas extrinsic optimizations on an embedding space require in general nonlinear projection techniques in order to stay on the constrained manifold. In particular, using the differential geometry of matrix manifolds has become a field of active research. For recent developments, however, without exploiting all the Lie structure, see, for example, Ref. (41,42).

Here we sketch an overview (43) for various optimization tasks in quantum dynamical systems in the common framework of gradient flows on Riemannian manifolds. Let denote a smooth manifold, for example, the unitary orbit of an initial state . A flow is a smooth map such that for all states and times

hence the flow acts as a one‐parameter group, and for positive times as a one‐parameter semigroup of diffeomorphisms on .

Now, let be a smooth quality function on . Recall that the differential of is a mapping (section) of the manifold to its cotangent bundle , while the gradient vector field is a mapping to its tangent bundle . Therefore, the scalar product plays a central role as it allows for identifying with ; this is why the pair has to be a Riemannian manifold with Riemannian metric . Thus one arrives at the gradient flow determined by

27.2

Formally, its solutions are obtained by integrating Eq. 27.2 to give

Observe this ensures that does increase along trajectories of by virtue of following the gradient direction of . – Gradient flows typically run into some local extremum as in Figure 27.3. Therefore, sufficiently many independent initial conditions may be needed to provide confidence into numerical results. Sometimes, local extrema can be ruled out; prominent examples of this type are discussed in (43) for Brockett's double‐bracket flow ( 36, 37) addressed below.

27.2.5.1 Discretized Gradient Flows

In the simplest case, where , gradient flows may be solved by moving along the gradient in the sense of a Steepest Ascent Method

with step size . Here, the manifold coincides with its tangent space containing . Clearly, a generalization is required as soon as and are no longer identifiable. This gap is filled by the Riemannian exponential map

such as to arrive at an intrinsic Euler step method. It is performed by the Riemannian exponential map, so straight line segments used in the standard method are replaced by geodesics on in the Riemannian Gradient Method

27.3

where is a step size ensuring convergence. For matrix Lie groups with bi‐invariant metric, Eq. 27.3 simplifies to the Gradient Method on a Lie Group (43)

where is the usual exponential map.

In either case, the iterative procedure can be pictured as follows: at each point , one evaluates in the tangent space . Then one moves via the Riemannian exponential map in direction to the next point on the manifold so that the quality function improves, , as shown in Figure 27.3.

27.2.5.2 Gradient Flows on Subgroups

For let denote the unitary orbit of . For minimizing the (squared) Euclidean distance between and the unitary orbit of , we give a gradient flow maximizing the target function over with the equivalence Note is a compact and connected naturally reductive homogeneous space isomorphic to where is the stabilizer group of .

Moreover, the double‐bracket flow to just defined is brought about by the gradient where is the skew‐Hermitian part of . Therefore, the corresponding gradient flow

27.4

is an isospectral flow on . The solutions exist for all and converge to a critical point of characterized by . A detailed discussion for the real case can be found in (37); for an abstract Lie algebraic version, see also (44).

In order to obtain a numerical algorithm for maximizing , one can discretize the continuous‐time gradient flow 27.4 as

27.5

with appropriate step sizes . Eq. 27.5 exploits that the adjoint orbit is a naturally reductive homogeneous space and thus the knowledge on its geodesics.

For complex Hermitian (real symmetric) and the full unitary (or orthogonal) group or its respective orbit the gradient flow 27.4 is well understood. However, for non‐Hermitian and , the nature of the flow and in particular the critical points have not been analyzed in depth, because the Hessian at critical points is difficult to come by. Even for Hermitian, a full critical point analysis becomes nontrivial as soon as the flow is restricted to a closed and connected subgroup . Nevertheless, the above techniques can be taken over to establish a gradient flow and a respective gradient algorithm on the orbit in a straightforward manner.

Likewise the gradient flow of Eq. 27.4 restricts to the subgroup orbit by taking the respective orthogonal projection onto the subalgebra of instead of projecting onto the skew‐Hermitian part. Thus With step sizes , the corresponding discrete integration scheme reads

27.6

In view of unifying the interpretation of unitary networks, for example, for the task of computing ground states of quantum mechanical Hamiltonians , the double‐bracket flows for complex Hermitian on the full unitary orbit as well as on the subgroup orbits for partitionings by with have shifted into focus. Thus we gave the foundations for the recursive schemes of Eqs. 27.5 and 27.6 listed with many more worked examples in the comprehensive overview Ref. (43).

In particular, in (43), we addressed gradient flows for constrained optimization problems. The intrinsic constraints can be accommodated by restricting the dynamic group to proper subgroups of the unitary group. Beyond that, gradient flows combining intrinsic constraints by restrictions to proper subgroups with extrinsic constraints can be taken care of by Lagrange‐type penalty parameters. Therefore, Ref. (43) provides a full toolkit of gradient‐flow based optimizations alongside ( 41, 42). It has also been very powerful for best approximations by sums of compact group orbits (45).

27.3 Toward a Systems Theory for Open Quantum Systems

We saw that in closed systems, there is a particularly simple characterization of reachable sets by the compact system algebra generating the Lie group and the corresponding group orbit, that is, In open quantum systems, it is more intricate to estimate the reachable sets. We consider bilinear control systems of open quantum systems for quantum maps following the master equation

In unital systems (those with at least one of the fixed points proportional to ), one finds by the seminal work of (46) and (47) on majorization that

as used recently in (48). However, equality holds only under the assumption that all coherent controls are infinitely fast in the sense of which from the viewpoint of physics is most often hopelessly idealizing. Therefore, this simple inclusion becomes totally inaccurate in all physically more realistic scenarios, where the drift Hamiltonian is necessary for full controllability in the sense of and – even worse – the inaccuracy increases with system size . For these experimentally more realistic relevant cases, we thus recently characterized (49,50) the dynamic system in terms of the underlying Lie wedge generating the dynamic system Lie semigroup of irreversible (Markovian) time evolution. Here the reachable sets can be much more accurately approximated by

where usually few factors suffice to give a good estimate.

This motivates the sketch of basic features of Lie semigroups.

27.3.2 Reachable Sets in Dissipatively Controlled Open Systems

As stated in the introductory part, we have recently characterized coherently controlled bilinear open systems (of spins‐ ) of the form

27.8

(here constant with of the form of Eq. 27.7) by their respective Lie wedges generating the dynamic system Lie semigroup of irreversible (Markovian) time evolution in Ref. (50). This promises that the reachable sets can conveniently be approximated by where with and where usually few factors suffice to give a good estimate. — For the sequel, suppose the unitary part of the above system is fully controllable in the sense

27.9

27.3.2.1 The Magic of Switchable Noise and Coherent Control

We have currently gone a step further such as to include into a coupled network of two‐level (spin‐ ) systems a single qubit the relaxion amplitude of which shall be switchable in a bang‐bang manner between the two values with . The situation corresponds to Eq. 27.8, where and the relaxation term acts locally on a single qubit while all the remaining qubits undergo no relaxation. This paves the way to entirely new domains, since the reachable sets enlarge dramatically: if in addition to unitary control there is nonunital switchable (amplitude damping) noise on a single spin ( for of the form of Eq. 27.7), one finds that the controlled system can act (approximately) transitively on the entire set of density operators, whereas for unital (bit‐flip) switchable noise on a single spin ( ), the reachable set fills all density operators majorized by the initial state.

More precisely, one gets the following:

Needless to say, these physically mild extensions by bang‐bang dissipative control on a single qubit on top of unitary control have a significant impact on numerical optimal control of open quantum systems by implementation into our numerical optimal‐control package DYNAMO (56) (see also Chapter 28) giving explicit control sequences (55) for superconducting qubits coupled to an open transmission line in the explicit experimental setting (GMon) of the Martinis group (57), which (by its short bath correlation) complies well with the Markov conditions.

Next we illustrate the distinction between gradient flows for (i) abstract optimizations on (possibly constrained) reachable sets and (ii) dynamic optimal control via experimentally accessible control amplitudes in a given parameterization.

27.4 Relation to Numerical Optimal Control

While in the previous sections optimizations are treated in an abstract manner, that is, over the dynamic group or over the specific state‐space manifold given by the reachable set (as illustrated in Figure 27.3), quantum engineering takes the optimization problems into the concrete parameterization of the actual experimental setup. More precisely, the parameterization is made in terms of the (discretized) control amplitudes, which then steer the quantum system on the state‐space manifold as an intermediate level. This is illustrated in Figure 27.4 in order to show the distinction from Figure 27.3.

Building upon (58,59), recently we have lined up all the principle numerical algorithms into a unified programming framework DYNAMO (56) matched to solve the underlying bilinear control problems: subject to the equation of motion 27.1 a target function is maximized over all admissible piece‐wise constant control vectors . This turns a control vector (pulse sequence) from an initial guess into an optimized shape by following first‐order gradients (or second‐order increments) to all the time slices of the control vector as shown in Figure 27.3, which may be done sequentially (60–63), or concurrently ( 58, 59) or in the newly unified version DYNAMO allowing hybrids as well as switches on the fly from one scheme to another one (56).

These numerical schemes have been put to good use for steering quantum systems (in the explicit experimental parameter setting) such as to optimize (i) the transfer between quantum states (pure or nonpure) (58), (ii) the fidelity of a unitary quantum gate to be synthesized in closed systems ( 59,64), (iii) the gate fidelity in the presence of Markovian relaxation (65), and (iv) the gate fidelity in the presence of non‐Markovian relaxation (66).

In recent years, examples for spin systems ( 59, 64) as well as Josephson elements (64) have been illustrated in all detail. For optimizing quantum maps in open systems, time‐optimal controls have been compared to relaxation‐optimized controls (65) in the light of an algebraic interpretation (49).

27.5 Outlook on Infinite‐Dimensional Systems

In infinite dimensions, difficulties arise as – by unbounded operators – the group of unitaries on an infinite‐dimensional separable Hilbert space is no Lie group if equipped with the strong topology for studying quantum dynamics. Yet contains infinite‐dimensional subgroups with proper Lie structure – including in particular a Lie algebra consisting of unbounded operators and a well‐defined exponential map, for example, unitaries with abelian ‐symmetry, which in the Jaynes–Cummings model relates to a particle‐number operator.

In (67), this infinite‐dimensional system Lie algebra is exploited for control theory in infinite dimensions in analogy to the finite‐dimensional case. The symmetry of and its Lie group thus excludes full controllability, yet this problem is overcome by complementary methods directly on the group level. The approach is paradigmatic and can be generalized in a natural way to other abelian symmetries (i.e., and representations with ).

For several two‐level atoms interacting with one harmonic oscillator (e.g., a cavity mode or a phonon mode), the methods of (67) allow for extending previous results substantially, mainly in two aspects also summarized in Table 27.4: (i) We answer approximate control and convergence questions for asymptotically vanishing control error. (ii) Our results include not only reachability of states but also its operator lift, that is, simulability of unitary gates. To this end, (67) introduces the notion of strong controllability, and shows that all systems under consideration require only a fairly small set of control Hamiltonians for guaranteeing strong controllability, that is, simulability. – Thus we anticipate the methods of (67) briefly sketched here will find wide application to systematically characterize experimental setups of cavity QED and ion‐traps in terms of pure‐state controllability and simulability.

27.5.1 Controllability and its Symmetry Conditions

More precisely, the control of quantum systems poses considerable mathematical challenges when applied to infinite dimensions. Basically, they arise from the fact that the set of anti‐selfadjoint operators (recall Stone's Theorem (68), VIII.4] to see they are generators of strongly continuous, unitary one‐parameter groups) do neither form a Lie algebra nor even a vector space. On the group level, the group of unitaries equipped with the strong operator topology is a topological group yet not a Lie group. Therefore, whenever strong topology has to be invoked, controllability cannot be assessed via a system Lie algebra. Thus, in these cases, we address the challenges on the group level by employing the controlled time evolution of the quantum system in order to approximate unitary operators, the action of which is measured with respect to arbitrary, but finite sets of vectors. This is formalized in the notion of strong controllability introduced in (67) generalizing the notion of pure‐state controllability in the literature. Central are abelian symmetries: assuming that except one, all Hamiltonians observe such an abelian symmetry, the infinite‐dimensional control system can be analyzed in its block‐diagonalized basis. We obtain strong controllability (beyond pure‐state controllability) if one of the Hamiltonian breaks this abelian symmetry and some further technical conditions are fulfilled.

27.5.1.1 Time Evolution

We treat control problems

27.10

where the with are selfadjoint control Hamiltonians on an infinite‐dimensional, separable Hilbert space and are piecewise‐constant controls. As is infinite‐dimensional, if not bounded the are defined on a dense subspace, that is, the domain .

27.5.1.2 Pure‐state controllability

A key issue is reachability: given two pure states , one looks for a time and controls such that . In infinite dimensions, this condition is too strong, as there are states that can only be reached in infinite time, if at all. Yet, one may find a reachable state arbitrarily “close by.” Hence shall be called reachable from if for all there is a finite time and controls with . Therefore, system 27.10 is called pure‐state controllable, if every pair of pure states can be ‐interconverted⁰⁴.

27.5.1.3 Strong Controllability

Analogously, for unitaries , time , and controls , one can approximate a target in the strong sense by

that is, comparing and only on a finite set of states with worst‐case error bounded by . System 27.10 is called strongly controllable if every unitary can be thus approximated⁰⁵. Clearly, strong controllability implies pure‐state controllability.

27.5.1.4 The Dynamical Group

Strong controllability is conceptually related to the strong operator topology [ (68), VI.1] on the group of unitary operators on : The sets

form a neighborhood base in the strong topology, hence called (strong) ‐neighborhood. Therefore, strong controllability says: any ‐neighborhood of contains a time‐evolution operator for appropriate time and control functions , or in other words: is an accumulation point of the set of all unitaries . The set of all accumulation points of (which contains itself) is a strongly closed subgroup⁰⁶ of , which we will call the dynamical group generated by control Hamiltonians with . For piecewise constant controls with only one different from zero at each time, is just the smallest strongly closed subgroup of that contains all for all and all . It contains in particular all unitaries that can be written as a strong limit s‐ . In finite dimensions, can be calculated via its system algebra generated by the , as each can be written as for an .

In infinite dimensions, one can avoid problems of joint domains by going back to finite‐dimensional Lie algebras with a dense set of analytic vectors (70,71) or to study systems with bounded generators in (72). Yet first way comes at the expense of loosing full controllability, while the second is unphysical. Thus here we take an approach by splitting generators into two classes. The first generators admit an abelian symmetry and can be treated – with Lie‐algebra methods – along the lines outlined next. Secondly, the last generator breaks this symmetry and achieves full controllability by a simple argument.

27.5.1.5 Abelian Symmetries

To study control systems with symmetries, consider the case of a ‐symmetry⁰⁷, that is, a (strongly continuous) unitary representation of the abelian group on . It can be written in terms of a selfadjoint operator with pure point spectrum consisting of (a subset of) as . By the eigenprojection of to the eigenvalue denoted as (allow if is no eigenvalue of ), we get a block‐diagonal decomposition of in the symmetry‐adapted basis

27.11

and we can rewrite again as .

Here two assumptions (with substantial loss of generality) facilitate subsequent discussion:

1. All eigenvalues of shall be of finite multiplicity, that is, the be finite‐dimensional.
2. All eigenvalues of be non‐negative. (This assumption can be relaxed at certain points.)

By finite‐dimensionality of , assumption (1) makes the space of finite particle vectors a “good” domain for basically all unbounded operators in this section and one gets the following theorem:

The idea is to cut off the decomposition 27.11 at sufficiently high without sacrificing the strong approximation, that is, increases with decreasing error. This strategy allows for tracing many calculations back to finite‐dimensional Lie algebras.

Next, consider a subgroup of and its Lie algebra relating unitaries with determinant one to traceless generators. As need neither be bounded nor positive, general definitions of tracelessness and determinant may run into problems circumvented by the block diagonal decomposition of 27.11, where all and are infinite sums of operators⁰⁸ with , , and denoting the projection onto the ‐eigenspace . As all and are operators on finite‐dimensional vector spaces, one can define

is a (strongly closed) subgroup of and is a Lie subalgebra of . The image of under the exponential map coincides with , which is effectively an infinite direct product of groups , not just the “special” subgroup of .

27.5.1.6 Breaking the Symmetry

For a fully controllable system, one has to leave the group “represented” block diagonal in Figure 27.5 a by adding control Hamiltonians that break the symmetry by a complementary direct sum decomposition of , where , with are projections onto the subspaces and should satisfy . For , we thus introduce the nonzero projections , whereas for , the relation shall hold.

We write for the overlap of and , which can (in contrast to ) be equal to zero for all . The are projections onto the subspaces satisfying .

This definition entails an important controllability result:

27.5.2 Application to Jaynes–Cummings Systems

Exploiting controlled dynamics of quantum systems is of increasing importance not only for solving computational tasks but for both quantum communication and simulation ( 4, 8 73–76) including many‐body correlations to create “quantum matter.” Ultra‐cold atoms in optical lattices model large‐scale correlations ( 76,77), where tunability and control over system parameters allow for switching between low‐energy states of different quantum phases ( 3,78) or for following real‐time dynamics such from the super‐fluid to the Mott insulator regime (79). Manipulating several atoms in a cavity is a key step to this end (80) posing challenging infinite‐dimensional control problems. While in finite dimensions controllability can readily be assessed by the Lie‐algebra rank condition ( 14– 18), infinite‐dimensional systems are more intricate (81), as exact controllability seemed daunting ( 70, 71,82,83), before approximate controllability paved the way to realistic assessments (84–86), see also (87) and references therein.

Here we illustrate control systems of two‐level atoms coupled to a cavity mode, that is, the Jaynes–Cummings model (88–91). We build on symmetry arguments ( 20, 26) and apply appropriate operator topologies for assessing (i) to which extent pure states can be interconverted and (ii) unitary gates can be approximated with arbitrary precision thus going beyond previous work (92–95).

System	Control Hamiltonians	Controllability
		System Algebra , Dynamic Group
One atom	, , Eq. 27.12	,	Theorem 3.110
	, , Eq. 27.12	Strongly controllable09
	, Eq. 27.13	with	Theorem 3.210
atoms	,	and
with individual controls of Eq. 27.14			Theorem 3.310
	,	Strongly controllable09
with individual controls Eqs. 27.14 and 27.15		with	Theorem 3.410
atoms	,	and
under collective control of Eq. 27.16			Theorem 3.510
	,	and
under collective control of Eq. 27.18			Theorem 3.610
	,	Strongly controllable09
under collective control of Eq. 27.18		with	Theorem 3.710

Here in the strong topology, no system algebra or exponential map exists.

The theorems are given with reference to (67).

27.5.2.1 One Atom

In one atom, the Hilbert space of the system is given by and the dynamics is described by the well‐known Jaynes–Cummings Hamiltonian (88):

27.12

where with are the Pauli matrices ( ), denote the annihilation and creation operator, and is the number operator. The charge‐type operator (determining the block structure) then takes the form . To get a fully controllable system, one has to add a Hamiltonian that breaks the symmetry, for example, by

27.13

so that transitions in the two‐level system are driven by in the sense of ‐pulses.

27.5.2.2 Many Atoms with Individual Controls

In this case, the Hilbert space of the system is readily generalized to where denotes the number of atoms. The control Hamiltonians become

27.14

where and . As depicted by the dark gray parts in Figure 27.5, all the are invariant under the symmetry defined by the charge operator where denotes again the number operator. To get strong controllability, one has to add again one Hamiltonian, where again a ‐flip of one atom is sufficient (see the light gray parts in Figure 27.5), since

27.15

is complementary to .

27.5.2.3 Many Atoms Under Collective Control

Now one may modify the setup from the last section by considering again atoms interacting with one mode, but assuming that one can control the atoms only collectively rather than individually. Instead of the Hamiltonians and with of Eq. 27.14, one only has their sums

27.16

where and , combined with the free evolution

27.17

of the cavity. The best result so far is to replace the operators from Eqs. 27.16 and 27.17 by

27.18

The operators with commute with and generate the Lie algebra . In addition, we have , which is complementary to .

27.6 Conclusion

We have cast a number of recent results into context to sketch an overarching frame of an emerging quantum systems theory. In particular, the unifying Lie picture comes for bilinear control systems of closed and open systems. This is of eminent importance also for control engineering and steering quantum dynamical systems with high precision. In doing so, we have shown how the emerging quantum systems theory links to many applications in quantum simulation and control without sacrificing mathematical rigor. Beyond addressing optimization tasks on reachable sets and state‐space manifolds, we have pointed out how gradient flows form the missing link to numerical optimal control algorithms for explicit steerings (control amplitudes) for manipulating closed and open (Markovian and non‐Markovian) systems in finite dimensions as, e.g., in Chapter 28.

Finally, we gave an outlook on a Lie picture of a systems and control theory in infinite dimensions and its application to Jaynes–Cummings systems, for example, like atoms in a cavity.

Acknowledgments

This work has been supported in part by the EU program SIQS, the exchange with COQUIT, moreover by the Bavarian excellence network ENB via the International Doctorate Programme of Excellence Exploring Quantum Matter (EXQM) as well as by the Deutsche Forschungsgemeinschaft (DFG) in the collaborative research center SFB 631 as well as the international research group FOR 1482 supported via the grant SCHU 1374/2‐1. Moreover, R.Z. was funded by DFG under grant Gl 203/7‐2.

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