The time optimal control of quantum systems is a rapidly expanding area of research, with many new results emerging as a result of this interesting aspect of variational calculus. The application of open-loop control methods is desirable in quantum systems, as the delicate nature of measurement has difficulties associated with the use of feedback systems that operate in the same way as in a classically controlled system. This paper discusses several highly technical aspects of this emerging science of time optimal control.

With many resources in quantum information science devoted towards the understanding of single-qubit systems and their composition via tensor products, it is important to consider alternative methodologies. We take the view that qutrits offer a broader description of the wider variety of quantum phenomena we can expect to observe in qudit systems of arbitrary dimension, not necessarily reducible to a decomposition over qubits. This paper shows how one can arrive at similar results to those found in Gell-Mann’s papers [1] and the collective motion models of the isotropic oscillator through the use of the quantum brachistochrone equation and time optimal control theory.

We shall briefly outline the context of the brachistochrone problem, and its analogous counterpart in quantum mechanics. In classical theory, the brachistochrone problem solves for the shortest time path between any two points, given constraints. This question, originally formulated by Bernoulli (1696) may be stated classically in the form of such questions as “Given two points, A & B, in a vertical plane, what is the curve traced out by a point acted on by gravity, which starts at A and reaches B in the shortest time?”. The answer, as was found by Bernoulli, is the cycloid that connects the two points A and B.

Let us now discuss the quantum generalisation of this question. The simplest way in which to pose a similar problem occurred in [2,3], where the authors determined for any two reachable quantum states the Hamiltonian operator that drives the state from one configuration to another in least time, given constraints. Other variants on this problem include the construction of the unitary operator associated to this Hamiltonian. In either sense, we can think of both these problems as being somewhat equivalent to Fermat’s principle of optics, which states that the physical path of light through a medium is the path which takes the least time, and also Huygen’s principle, where it has been known classically that the two principles are equivalent in every nature. Young (1809) was of the opinion that “The principle of Fermat, although it was assumed by that mathematician on hypothetical... grounds, is in fact a fundamental law with respect to undulatory motion, and is explicitly the basis of every determination in the Huygenian theory.” [4]. From such considerations, one may go on to develop such formulations of ray optics as Snell’s law of refraction. Of course, it was known to Hamilton, Laplace and others that the eikonal equation provides the connection between the two seemingly disparate perspectives. In quantum terms, we may coin a quantum Fermat principle, and state that the quantum state travels on the complex projective space such that the time between any two points is a minimum, given constraints on our objective which are represented as Lagrange multipliers. The question of the correct formulation of this yields naturally to variational calculus, as initially explored in [2,3]. We shall not pursue their Lagrangian method in this work, as we shall explore the complementary Hamiltonian mechanics from a different perspective. Our originating principles shall overlap, but generally we shall use the von Neumann approach to quantum mechanics and some insights from projective geometry in order to analyse time optimal control problems in the quantum realm.

The following paper is divided into two major sections. We shall review the relevant literature for each section in turn below.

The sections are as follows; firstly, we outline the basic machinery required from matrix mechanics required in this proof, including the basic brachistochrone equations, constraints and definition of the quantum Fermat principle. The following proof shows how the von Neumann perspective of quantum mechanics has particular applicability to this problem, and demonstrates how one may set up the problem of time optimal control using some simple constraints and relationships from matrix operator theory. We then apply this technique to the simple toy problem for a qubit as given in [2,3] by a more complicated method, following with a brief review of known advanced results from qutrit operator theory as given in [18]. The final section of the paper is devoted to an exposition of spherical operator theory as it is applied to the problem of an isotropic oscillator, including its relation to SU(3), the analysis of group relations related to spherical harmonics, and radial Whittaker functions.

Let us now define the basic structures we shall encounter in this proof. In general, we shall be dealing with quantum states that exist on a complex vector space, normalised to probability 1 and of finite dimension. The state vector shall be written as a column vector of possibly complex functions of time: with Hermitian conjugate defined by the adjoint:

The inner product is such that we are constrained to the surface of a sphere via: with outer product given by the density matrix:

Finally, the expectation value and trace of an operator are defined through:

These are the basic vector and matrix operators that we shall require in this proof. The Dirac notation in this case is abused to merely mean finite state vectors. However, as we shall show, it has particular utility in allowing a compact formulation of this theory of time optimality.

The Euler-Lagrange equation in the variable $\left|\varphi \right.〉$ for the action principle gives

To define the state, we need access to the form of the Hamiltonian operator as a function of time. One simple system where a solution exists is the case of finite isotropic energy, and a linear constraint on the accessible degrees of freedom. In this situation we may write the isotropic condition as: and the linear constraint on the Hamiltonian operator in the form:

A lengthy calculation following the prescription in [2,3] yields the resulting differential equations for the Hamiltonian operator, but there is a direct method that hinges on the application of the von Neumann perspective. In this case, we take as our basis axiom the von Neumann equation for a matrix operator: for any Hermitian matrix $\widehat{A}={\widehat{A}}^{†}$. Choose a particular $\widehat{A}=\tilde{H}+\tilde{F}$, substituting above, obtain

This is now a set of coupled differential equations in the matrix operators. The Euler-Lagrange equation in the variable $\tilde{H}$ will contain a term and with some effort it is possible to show that the dynamical equations can be re-arranged to give the same result as the ansatz obtained from the von Neumann equation. We call this the quantum brachistochrone equation. The system of equations is then completely specified in this situation by the following set of matrix differential equations:

Consider the smallest non-trivial problem in time optimal quantum control. If we take the spinor space defined over the Pauli matrices $\left\{{\widehat{\sigma }}_x,{\widehat{\sigma }}_y,{\widehat{\sigma }}_z\right\}$, we have:

If we consider the geometric algebra, we have spinor geometry defined by product formula:

Choosing then a Hamiltonian matrix through $\tilde{H}\left(t\right)={\lambda }_x\left(t\right){\widehat{\sigma }}_x+{\lambda }_y\left(t\right){\widehat{\sigma }}_y=\mathbf{a}\cdot \sigma$, we are able to specify the time optimal control problem. In terms of the isotropic constraint, we have

We can see that this forces the vector which defines the Hamiltonian matrix to be of finite length. The linear constraint gives

The second term $\mathbf{Tr}\left(i(\mathbf{a}\times \mathbf{b})\cdot \sigma \right)=i\mathbf{Tr}\left(\mathbf{c}\cdot \sigma \right)=0$, so we must have $\mathbf{a}\cdot \mathbf{b}=0$, meaning that we have $\tilde{F}=\mathsf{\Omega }\left(t\right){\widehat{\sigma }}_z$. The matrix operators with functions to be determined are then with $\epsilon \left(t\right)={\lambda }_x\left(t\right)-i{\lambda }_y\left(t\right)$, and

We can show that we satisfy the linear constraint using elementary matrix operations: whereas the isotropic condition reads as

The quantum brachistochrone equation is then defined by $i{\displaystyle \frac{d}{dt}}\left(\tilde{H}+\tilde{F}\right)=[\tilde{H},\tilde{F}]$, plugging in the form of the matrices as from above, we find: i.e. $\dot{\mathsf{\Omega }}=0$, which implies $\mathsf{\Omega }\left(t\right)=\mathsf{\Omega }$ (a constant). The remaining expressions in vector-matrix form

Using $\epsilon \left(0\right)=r{e}^{i\varphi }$:

We shall now outline a simple method where one may construct a set of states with analogous properties to the finite matrix systems considered above. Working now in the representation theory, our states will be given by basis functions over the Hilbert space. The Schrödinger equation is defined by: where $\mathsf{\Phi }=\mathsf{\Phi }(\mathbf{x},t)$ and $\widehat{H}$ is some Hermitian differential operator. We shall be concerned with the differential equation given by: where $\psi =\psi \left(\mathbf{x}\right)$, and $\mathsf{\Phi }(\mathbf{x},t)=\phi \left(t\right)\psi \left(\mathbf{x}\right)$ is a separable solution to the differential equation, which is the form a time independent solution will obey. In terms of the potential energy, we can write $V\left(r\right)={\displaystyle \frac{1}{2}}{\omega }^2{r}^2$, which is isotropic as it contains no angular dependence. If we think back to Hooke’s law for a spring, in this situation we will have $\ddot{x}=-kx$, hence $V\left(x\right)={\displaystyle \frac{1}{2}}k{x}^2$, so in this sense the system is an isotropic oscillator. Indeed, the fundamental eigenfunction equation may be written: and following the analysis in [11,16,17], the basic identities for the angular momentum are readily evaluated as: where ${ϵ}_{ijk}$ is the Levi-Civita symbol and ${\widehat{p}}_k=-i{\partial }_k=-i{\displaystyle \frac{\partial }{\partial x_k}}$, e.g. ${\widehat{L}}_x=i(z{\partial }_y-y{\partial }_z)$ and so on, the commutation brackets being given by identities such as:

& cyclic permutations thereof, the Laplacian given by: where ${\widehat{L}}^2={\widehat{L}}_x^2+{\widehat{L}}_y^2+{\widehat{L}}_z^2$. Finally, the constants of motion and angular momentum tensor may be found through: with the raising and lowering operators given by the standardised expressions: with $\left[{\widehat{a}}_j,{\widehat{a}}_k^{†}\right]={\delta }_{jk}$ and $\left[{\widehat{a}}_j,{\widehat{a}}_k\right]=\left[{\widehat{a}}_j^{†},{\widehat{a}}_k^{†}\right]=0$. The Hamiltonian operator can be written as where the sum over the latin indices represents the coordinate degrees of freedom of the system. If we take a particular coordinate, the creation and annihilation operators may be written:

Calculating commutator $\left[\widehat{g},\widehat{h}\right]=\widehat{g}\widehat{h}-\widehat{h}\widehat{g}$ for these operators we find:

To calculate $\left[{\widehat{p}}_x,\widehat{x}\right]$ use ${\widehat{p}}_x=-i{\displaystyle \frac{\partial }{\partial x}}$, we find the standard commutation relations: which means that we can write $\left[{\widehat{p}}_x,\widehat{x}\right]=-i$. Therefore the commutator of the creation and annihilation operators is given by $\left[{\widehat{a}}_x,{\widehat{a}}_x^{†}\right]=1$. The Hamiltonian operator for the spherical isotropic oscillator is:

Consider the operator ${\widehat{a}}_x^{†}{\widehat{a}}_x$. Writing this out in terms of the momentum and position operators

Rearranging, we therefore have $\omega \left({\widehat{a}}_x^{†}{\widehat{a}}_x+{\displaystyle \frac{1}{2}}\right)={\displaystyle \frac{1}{2}}\left({\omega }^2{x}^2+{\widehat{p}}_x^2\right)$. Summing over the other components, we find: as required. We have the formula for the shift operators given as ${\widehat{G}}_{ij}={\displaystyle \frac{1}{2}}({\widehat{a}}_i^{†}{\widehat{a}}_j+{\widehat{a}}_j{\widehat{a}}_i^{†})$. Take the component ${\widehat{G}}_{xx}$, the other components will follow similarly.

We have already evaluated the commutator, i.e. $\left[{\widehat{a}}_x,{\widehat{a}}_x^{†}\right]={\widehat{a}}_x{\widehat{a}}_x^{†}-{\widehat{a}}_x^{†}{\widehat{a}}_x=1$. Rearranging, ${\widehat{a}}_x{\widehat{a}}_x^{†}=1+{\widehat{a}}_x^{†}{\widehat{a}}_x$. Substituting this into the expression for ${\widehat{G}}_{xx}$ we find and similar for ${\widehat{G}}_{yy}$ and ${\widehat{G}}_{zz}$, hence the expression for the Hamiltonian operator is given by

From this perspective, we can see how the isotropic oscillator problem is related to the tensor defined through the creation and annihilation operators. We shall now discuss some basic properties of these matrix operators as related to SU(3).