In this chapter, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. In a two qubit system, the ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of unitary evolution, g , are decomposed into fast generators k (local Hamiltonians) and slow generators p (couplings) as a Cartan decomposition g = p ⊕ k . Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. The main contribution of the chapter is, we carry out a global analysis of time optimality.
- Kostant convexity
- spin dynamics
- Cartan decomposition
- Cartan subalgebra
- Weyl group
- time optimal control
A rich class of model control problems arise when one considers dynamics of two coupled spin . The dynamics of two coupled spins, forms the basis for the field of quantum information processing and computing  and is fundamental in multidimensional NMR spectroscopy [2, 3]. Numerous experiments in NMR spectroscopy, involve synthesizing unitary transformations [4, 5, 6] that require interaction between the spins (evolution of the coupling Hamiltonian). These experiments involve transferring, coherence and polarization from one spin to another and involve evolution of interaction Hamiltonians . Similarly, many protocols in quantum communication and information processing involve synthesizing entangled states starting from the separable states [1, 7, 8]. This again requires evolution of interaction Hamiltonians between the qubits.
A typical feature of many of these problems is that evolution of interaction Hamiltonians takes significantly longer than the time required to generate local unitary transformations (unitary transformations that effect individual spins only). In NMR spectroscopy [2, 3], local unitary transformations on spins are obtained by application of rf-pulses, whose strength may be orders of magnitude larger than the couplings between the spins. Given the Schróedinger equation for unitary evolution
where represents a coupling Hamiltonian, and are controls that can be switched on and off. What is the minimum time required to synthesize any unitary transformation in the coupled spin system, when the control generators are local Hamiltonians and are much stronger than the coupling between the spins (can be made large). Design of time optimal rf-pulse sequences is an important research subject in NMR spectroscopy and quantum information processing [4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], as minimizing the time to execute quantum operations can reduce relaxation losses, which are always present in an open quantum system [22, 23]. This problem has a special mathematical structure that helps to characterize all the time optimal trajectories . The special mathematical structure manifested in the coupled two spin system, motivates a broader study of control systems with the same properties.
The Hamiltonian of a spin can be written in terms of the generators of rotations on a two dimensional space and these are the Pauli matrices , where,
where is the matrix commutator and
The Hamiltonian for a system of two coupled spins takes the general form
where . The Hamiltonians and are termed local Hamiltonians and operate on one of the spins. The Hamiltonian
is the coupling or interaction Hamiltonian and operates on both the spins.
The following notation is therefore common place in the NMR literature.
The operators and commute and therefore
The unitary transformations of the kind
obtained by evolution of the local Hamiltonians are called local unitary transformations.
The coupling Hamiltonian can be written as
Written explicitly, some of these matrices take the form
for , form the basis for the Lie algebra , the , traceless skew-Hermitian matrices. For the coupled two spins, the generators and the evolution operator in Eq. (1) is an element of , the , unitary matrices of determinant .
The Lie algebra has a direct sum decomposition , where
Here is a subalgebra of made from local Hamiltonians and nonlocal Hamiltonians. In Eq. (1), we have and , It is easy to verify that
This special structure of Cartan decomposition arising in dynamics of two coupled spins in Eq. (1), motivates study of a broader class of time optimal control problems.
Consider the following canonical problems. Given the evolution
where , the special Unitary group (determinant , matrices such that , is conjugate transpose). Where , skew symmetric matrices and
We assume , the Lie algebra (and its matrix commutators) generated by generators is all of . We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls . Here again we have a Cartan decomposition on generators. Given , traceless skew-Hermitian matrices, generators of , we have , where , where is traceless symmetric and . As before, and . We want to find time optimal ways to steer this system. We call this problem. For , this system models the dynamics of two coupled nuclear spins in NMR spectroscopy.
In general, is in a compact Lie group (such as ), with in its real semisimple (no abelian ideals) Lie algebra and
Given the Cartan decomposition , where , and (product of exponentials of ) a closed subgroup of G, We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls . Since , any rotation (evolution) in subgroup can be synthesized with evolution of [25, 26]. Since there are no bounds on , this can be done in arbitrarily small time . We call this problem.
The special structure of this problem helps in complete description of the reachable set . The elements of the reachable set at time , takes the form
where , and all commute, and , . This reachable set is formed from evolution of and commuting Hamiltonians . Unbounded control suggests that can be synthesized in negligible time.
This reachable set can be understood as follows. The Cartan decomposition of the Lie algebra , in Eq. (13) leads to a decomposition of the Lie group . Inside is contained the largest abelian subalgebra, denoted as . Any is conjugate to an element of , i.e. for some .
Then, any arbitrary element of the group can be written as
The results in this chapter suggest that and can be synthesized by unbounded controls in negligible time. The time consuming part of the evolution is synthesized by evolution of Hamiltonian . Time optimal strategy suggests evolving and its conjugates where all commute.
Written as evolution
where take negligible time to synthesize using unbounded controls and time-optimality is characterized by synthesis of commuting Hamiltonians . This characterization of time optimality, involving commuting Hamiltonians is derived using convexity ideas [4, 28]. The remaining chapter develops these notions.
The chapter is organized as follows. In Section 2, we study the problem. In Section 3, we study the general problem. The main contribution of the chapter is, we carry out a global analysis of time optimality.
Given Lie algebra , we use killing form as an inner product on . When , we also use the inner product . We call this standard inner product.
2. Time optimal control for problem
Remark 1. Birkhoff’s convexity states, a real matrix is doubly stochastic (, for ) if it can be written as convex hull of permutation matrices (only one and everything else zero in every row and column). Given and , we have where is a column vector containing diagonal entries of and and hence , making a doubly stochastic matrix, which can be written as convex sum of permutations. Therefore , i.e. diagonal of a symmetric matrix , lies in convex hull of its eigenvalues and its permutations. This is called Schur convexity.
Remark 2. has a closed subgroup and a Cartan decomposition of its Lie algerbra as , for and where is traceless symmetric and is maximal abelian subalgebra of , such that where . KAK decomposition in Eq. (17) states for , where and
Remark 3. We now give a proof of the reachable set (16), for the problem. Let be a solution to the differential Eq. (14)
To understand the reachable set of this system we make a change of coordinates , where, . Then
If we understand reachable set of , then the reachable set in Eq. (14) is easily derived.
Theorem 1. Let be a solution to the differential equation
and and . The elements of the reachable set at time , take the form , where and (lies in convex hull of and its permutations), where .
Proof. As a first step, discretize the evolution of , as piecewise constant evolution, over steps of size . The total evolution is then
For , choose small step , such that , then .
By KAK, , where . To begin with, assume eigenvalues , where is an integer. When we take a small step of size , changes to as change to
where, , and . Let , which can be written as
We equate and to first order in . This gives,
Multiplying both sides with gives
where, and .
We evaluate , for .
such that is skew symmetric and is traceless symmetric matrix with . Note and onto , by appropriate choice of .
Given , we decompose it as
with denoting the projection onto (where .) w.r.t to standard inner product and to the orthogonal component. In Eq. (24), , we can solve for such that . This gives . Let and choose .
With this choice of and , and are matched to first order in and
Consider the case, when is degenerate. Let,
where is fold degenerate (modulo sign) described by block. WLOG, we arrange
Consider the decomposition
where denotes projection onto blocks in Eq. (25) and , the orthogonal complement.
where are blocks.
Then we write
Let be a rotation formed from block diagonal matrix
where is sub-block in . is chosen such that
is a diagonal matrix. Let , where is skew symmetric, such that
is sub-block in , related by (see 26)
Note lies in convex hull of eigenvalues of . This is true if we look at the diagonal of , it follows from Schur Convexity. The diagonal of is zero as its inner product
as has block diagonal form which is perpendicular to . Therefore diagonal of is same as diagonal of .
Now using , from 28, we have
where the above expression can be written as
where , , are chosen such that
Let and we relate the eigenvalues, of and . Given , as above, with , and a ordered set of eigenvalues of F, denote , there exists an ordering (correspondence) of eigenvalues of , such that .
Choose an ordering of call that minimizes .
and , where is diagonal with diagonal as , let ,
By Schur convexity,
where are permutations. Therefore .
is regulated by size of , which is bounded by , where is smallest non-zero difference. is chosen small enough such that .
For each point , we choose an open nghd , such that for . forms a cover of . We can choose a finite subcover centered at (see Figure 1A). Consider trajectory at points . Let be the point in intersection of and . Let and . We consider points as shown in Figure 1B.
Then we get the following recursive relations.
where and correspond to in Eq. (33) and lie in the convex hull of the eigenvalues .
Adding the above equations,
where in Eq. (38) is diagonal.
where and .
Note, . This implies that belongs to the compact set , else it has minimum distance from this compact set and by making and hence , we can make this arbitrarily small. In Eq. (18), as . Hence belongs to compact set . q.e.d.
Corollary 1. Let be a solution to the differential equation
where , the Lie algebra generated by , is and . The elements of reachable set at time , takes the form , where and , where and the set belongs to the closure of reachable set.
Proof. Let , where, . Then
From Theorem 1, we have . Therefore . Given
We can synthesize in negligible time, therefore , for any desired . Hence is in closure of reachable set. q.e.d.
Remark 4. We now show how Remark 2 and Theorem 1 can be mapped to results on decomposition and reachable set for coupled spins/qubits. Consider the transformation
The transformation maps the algebra to , four dimensional skew symmetric matrices, i.e., . The transformation maps to , where is traceless symmetric and maps to , space of diagonal matrices in , such that gets mapped to the four vector (the diagonal) .
Corollary 2. Canonical decomposition. Given the decomposition of SU(4) from Remark 2, we can write
where . We write above as
Multiplying both sides with gives
where local unitaries and we can rotate to .
Corollary 3. Digonalization. Given , there exists a local unitary such that
Note . Then choose such that and hence
where is a local unitary. We can rotate to ensure .
Corollary 4. Given the evolution of coupled qubits , we can diagonalize by local unitary , , which we write as triple . From this, there are 24 triples obtained by permuting and changing sign of any two by local unitary. Then where
Furthermore belongs to the closure of the reachable set. Alternate description of is
Proof. Let , where . Then
Consider the product
where and , where . Then,
Observe and . Then using results from Theorem 1, we have
Multiplying both sides with , we get
which we can write as
where using , we get,
Furthermore . Hence the proof. q.e.d.
3. Time optimal control for problem
Remark 5. Stabilizer: Let be Cartan decomposition of real semisimple Lie algebra and be its Cartan subalgebra. Let . is symmetric in basis orthonormal wrt to the killing form. We can diagonalize . Let be eigenvectors with nonzero (negative) eigenvalues . Let , .
are independent, as implies . Since are independent, are independent. Given , then , otherwise we can decompose it in eigenvectors of , i.e., , where are zero eigenvectors of . Since , which means . This is a contradiction. are orthogonal, implies are orthogonal, . Let satisfy . Then .
denote eigenvectors that have as non-zero integral multiples of . are related to . We now reserve for non-zero eigenvectors that are not integral multiples of .
where forms a basis of , forms a basis of . Let .
The range of in , is perpendicular to . Given such that . The norm of , such that part of satisfies
where is the smallest nonzero eigenvalue of such that is not an integral multiple of .
stabilizes and . If , is stabilized by , , i.e., . This means is an subalgebra, as the Lie bracket of for is stabilized by .
Let , be an integral manifold of . Let be the solution to or . is closed, . We show that is a manifold. Given element , where is closed, we have a nghd of , in ball nghd of , which is one to one. For , , implies,
then by one to one property of , we get and . Therefore is a nghd of .
Given a sequence converging to , for large enough . Then is in invariant manifold . Hence is closed and hence compact.
Let , then there exists a such that . We maximize the function , over the compact group , for regular element and is the killing form. At the maxima, we have at , .
if , then . The bracket is invariant and, hence, belongs to . We can choose so that gradient is not zero. Hence . For such that , we have .
as is invariant, hence . In above, we worked with killing form. For , we may use standard inner product.
Remark 6. Kostant’s convexity:  Given the decomposition , let and ,. Let such that are distinct, Weyl points. Then projection (w.r.t killing form) of on lies in convex hull of these Weyl points. The be the convex hull and let projection lie outside this Hull. Then there is a separating hyperplane , such that . W.L.O.G we can take to be a regular element. We minimize , with choice of and find that minimum happens when , i.e. is a Weyl point. Hence , for and . The result is true with a projection w.r.t inner product that satisfies , like standard inner product on .
Theorem 2 Given a compact Lie group and Lie algebra . Consider the Cartan decomposition of a real semisimple Lie algebra . Given the control system
where , the Cartan subalgebra and , a closed subgroup of . The end point
where and are Weyl points, and .
Proof. As in proof of Theorem 1, we define
and show that
where for ,
where is projection w.r.t killing form and , the centralizer in as defined in Remark 5, is a second order term that can be made small by choosing . .
To show Eq. (46), we show there exists such that
where and are constructed by a iterative procedure as described in the proof below.
Given and as matrices, considered elements of a matrix Lie algebra , we have,
We bound the largest element (absolute value) of , denoted as , given and , where , , .
where and .
Given decomposition of , with respect to the negative definite killing form . Furthermore there is decomposition of .
where , and , such that , which we just abbreviate as (we follow this convention below).
We describe an iterative procedure
where and , such that the limit
Note and are elements of and need not be contained in and .
Where, using bound in , which gives . Using the bound again, we obtain, . We can decompose, , into subspaces , where , and , where , where is Frobenius norm and . Let .
This gives, , and . This gives
For , we have, . Continuing and using .
Note, is a Cauchy sequences which converges to , where
The above exercise was illustrative. Now we use an iterative procedure as above to show Eq. (47).
where and , consider again the iterations
We refer to Remark 5, Eq. (45). Given such that . If , then (killing norm).
, is bounded , where as before converts between two different norms. Using bounds derived above , and , , we obtain.
which gives . For appropriate , we have
where is chosen small.
where , such that .
where, using bounds derived above , and , where using the bound , we obtain
which gives .
We can decompose, , into subspaces , where , and , where as before converts between two different norms.
For , we have, ,
Note, is a Cauchy sequences which converges to , where
The above iterative procedure generates and in Eq. (47), such that
where . By using a stabilizer , we can rotate them to such that
such that is in and is projection onto such that
This follows because the orthogonal part of to written as remains orthogonal of
(), remains orthogonal to . Therefore .
Lemma 1 Given , where . We can express
where is Weyl element of . Furthermore
Proof. Note, , commutes with . This implies
commutes with . This implies , i.e., , which implies that . Recall, from Remark 5,
This implies , implying . Therefore,
We have shown existence of such that , using as before,
Applying the theorem again to
Lemma 2 Given , we have , and , where . From above we can express
where and are first and second order increments to in the positive direction. The remaining notation is self-explanatory.
Using Lemma 1 and 2, we can express
Letting go to , we have
Hence the proof of theorem. q.e.d.
In this chapter, we studied some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. We saw how dynamics was decomposed into fast generators (local Hamiltonians) and slow generators (couplings) as a Cartan decomposition . Using this decomposition, we used some convexity ideas to completely characterize the reachable set and time optimal control for these problems.