Open access peer-reviewed chapter

Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics

By Navin Khaneja

Submitted: March 7th 2018Reviewed: July 28th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.80567

Downloaded: 179

Abstract

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.

Keywords

  • Kostant convexity
  • spin dynamics
  • Cartan decomposition
  • Cartan subalgebra
  • Weyl group
  • time optimal control

1. Introduction

A rich class of model control problems arise when one considers dynamics of two coupled spin 12. The dynamics of two coupled spins, forms the basis for the field of quantum information processing and computing [1] 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 [2]. 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

U̇=iHc+j=1nujHjU,U0=I,E1

where Hcrepresents a coupling Hamiltonian, and ujare 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 Hjare local Hamiltonians and are much stronger than the coupling between the spins (ujcan 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 [4]. 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 12can be written in terms of the generators of rotations on a two dimensional space and these are the Pauli matrices iσx,iσy,iσz, where,

σz=121001;σy=120ii0;σx=120110.E2

Note

σxσy=iσz,σyσz=iσx,σzσx=iσy,E3

where AB=ABBAis the matrix commutator and

σx2=σy2=σz2=14,E4

The Hamiltonian for a system of two coupled spins takes the general form

H0= aασα1+ bβ1σβ+ Jαβσασβ,E5

where α,βxyz. The Hamiltonians σα1and 1σβare termed local Hamiltonians and operate on one of the spins. The Hamiltonian

Hc= Jαβσασβ,E6

is the coupling or interaction Hamiltonian and operates on both the spins.

The following notation is therefore common place in the NMR literature.

Iα=σα1;Sβ=1σβ.E7

The operators Iαand Sβcommute and therefore expiαaαIα+βbβSβ=

expiαaαIαexpiβbβSβ=expiαaασα1(1expiβbβσβ,E8

The unitary transformations of the kind

expiαaασαexpiβbβσβ,

obtained by evolution of the local Hamiltonians are called local unitary transformations.

The coupling Hamiltonian can be written as

Hc=JαβIαSβ.E9

Written explicitly, some of these matrices take the form

Iz=σz1=121000010000100001.E10

and

IzSz=σzσz=141000010000100001.E11

The 15operators,

iIαSβIαSβ,

for α,βxyz, form the basis for the Lie algebra g=su4, the 4×4, traceless skew-Hermitian matrices. For the coupled two spins, the generators iHc,iHjsu4and the evolution operator Utin Eq. (1) is an element of SU4, the 4×4, unitary matrices of determinant 1.

The Lie algebra g=su4has a direct sum decomposition g=pk, where

k=iIαSβ,p=iIαSβ.E12

Here kis a subalgebra of gmade from local Hamiltonians and pnonlocal Hamiltonians. In Eq. (1), we have iHjkand iHcp, It is easy to verify that

kkk,kpp,ppp.E13

This decomposition of a real semi-simple Lie algebra g=pksatisfying (13) is called the Cartan decomposition of the Lie algebra g[24].

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

U̇=Xd+jujtXjU,U0=1,E14

where USUn, the special Unitary group (determinant 1, n×nmatrices Usuch that UU=1, 'is conjugate transpose). Where Xjk=son, skew symmetric matrices and

Xd=iλ1000λ2000λn,λi=0.

We assume XjLA, the Lie algebra (Xjand its matrix commutators) generated by generators Xjis all of son. We want to find the minimum time to steer this system between points of interest, assuming no bounds on our controls ujt. Here again we have a Cartan decomposition on generators. Given g=sun, traceless skew-Hermitian matrices, generators of SUn, we have g=pk, where p=iA, where Ais traceless symmetric and k=son. As before, Xdpand Xjk. We want to find time optimal ways to steer this system. We call this SUnSOnproblem. For n=4, this system models the dynamics of two coupled nuclear spins in NMR spectroscopy.

In general, Uis in a compact Lie group G(such as SUn), with Xd,Xjin its real semisimple (no abelian ideals) Lie algebra gand

U̇=Xd+jujtXjU,U0=1.E15

Given the Cartan decomposition g=pk, where Xdp, XjLA=kand K=expk(product of exponentials of k) 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 ujt. Since XjLA=k, any rotation (evolution) in subgroup Kcan be synthesized with evolution of Xj[25, 26]. Since there are no bounds on ujt, this can be done in arbitrarily small time [4]. We call this GKproblem.

The special structure of this problem helps in complete description of the reachable set [27]. The elements of the reachable set at time T, takes the form UT

S=K1expTkαkWkXdWk1K2,E16

where K1,K2,Wkexpk, and WkXdWk1all commute, and αk>0, αk=1. This reachable set is formed from evolution of K1,K2and commuting Hamiltonians WkXdWk1. Unbounded control suggests that K1,K2,Wkcan be synthesized in negligible time.

This reachable set can be understood as follows. The Cartan decomposition of the Lie algebra g, in Eq. (13) leads to a decomposition of the Lie group G[24]. Inside pis contained the largest abelian subalgebra, denoted as a. Any Xpis AdKconjugate to an element of a, i.e. X=Ka1K1for some a1a.

Then, any arbitrary element of the group Gcan be written as

G=K0expX=K0expAdKa1=K1expa1K2,E17

for some Xpwhere KiKand a1a. The first equation is a fact about geodesics in G/Kspace [24], where K=expkis a closed subgroup of G. Eq. (17) is called the KAK decomposition [24].

The results in this chapter suggest that K1and K2can be synthesized by unbounded controls Xiin negligible time. The time consuming part of the evolution expa1is synthesized by evolution of Hamiltonian Xd. Time optimal strategy suggests evolving Xdand its conjugates WkXdWk1where WkXdWk1all commute.

Written as evolution

G=K1kexptkWkXdWk1K2=K1kWkexptkXdWk1K2.

where K1,K2,Wktake negligible time to synthesize using unbounded controls uiand time-optimality is characterized by synthesis of commuting Hamiltonians WkXdWk1. 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 SUnSOnproblem. In Section 3, we study the general GKproblem. The main contribution of the chapter is, we carry out a global analysis of time optimality.

Given Lie algebra g, we use killing form xy=tradxadyas an inner product on g. When g=sun, we also use the inner product xy=trxy. We call this standard inner product.

2. Time optimal control for SUn/SOnproblem

Remark 1. Birkhoff’s convexity states, a real n×nmatrix Ais doubly stochastic (i Aij=j Aij=1, for Aij0) if it can be written as convex hull of permutation matrices Pi(only one 1and everything else zero in every row and column). Given ΘSOnand X=λ1000λ2000λn, we have diagΘXΘT=BdiagXwhere diagXis a column vector containing diagonal entries of Xand Bij=Θij2and hence i Bij=j Bij=1, making Ba doubly stochastic matrix, which can be written as convex sum of permutations. Therefore BdiagX=iαiPidiagX, i.e. diagonal of a symmetric matrix ΘXΘT, lies in convex hull of its eigenvalues and its permutations. This is called Schur convexity.

Remark 2. G=SUnhas a closed subgroup K=SOnand a Cartan decomposition of its Lie algerbra g=sunas g=pk, for k=sonand p=iAwhere Ais traceless symmetric and ais maximal abelian subalgebra of p, such that a=iλ100000λn,where iλi=0. KAK decomposition in Eq. (17) states for USUn, U=Θ1expΩΘ2where Θ1,Θ2SOnand

Ω=iλ100000λn,

where iλi=0.

Remark 3. We now give a proof of the reachable set (16), for the SUnSOnproblem. Let UtSUnbe a solution to the differential Eq. (14)

U̇=Xd+iuiXiU,U0=I.

To understand the reachable set of this system we make a change of coordinates Pt=KtUt, where, K̇=iuiXiK. Then

Ṗt=AdKtXdPt,AdKXd=KXK1.

If we understand reachable set of Pt, then the reachable set in Eq. (14) is easily derived.

Theorem 1. Let PtSUnbe a solution to the differential equation

Ṗ=AdKtXdP,

and KtSOnand Xd=iλ1000λ2000λn. The elements of the reachable set at time T, take the form K1expiμTK2, where K1,K2SOnand μλ(μlies in convex hull of λand its permutations), where λ=λ1λn.

Proof. As a first step, discretize the evolution of Pt, as piecewise constant evolution, over steps of size τ. The total evolution is then

Pn=iexpAdkiXdτ,E18

For tn1τ, choose small step Δ, such that t+Δ<, then Pt+Δ=expAdKXdΔPt.

By KAK, Pt=K1expiϕ10000expiϕ200000000expiϕnAK2, where K1,K2SOn. To begin with, assume eigenvalues ϕjϕk, where nis an integer. When we take a small step of size Δ, Ptchanges to Pt+Δas K1,K2,Achange to

K1t+Δ=expΩ1ΔK1,K2t+Δ=expΩ2ΔK2,At+Δ=expaΔA,

where, Ω1, Ω2kand aa. Let Qt+Δ=K1t+ΔAt+ΔK2t+Δ, which can be written as

Qt+Δ=expΩ1ΔK1expaΔAexpΩ2ΔK2.E19
Qt+Δ=expΩ1ΔexpK1aK1ΔexpK1AΩ2AK1ΔPt.E20

Observe

Pt+Δ=expAdKXdΔPt.E21

We equate Pt+Δand Qt+Δto first order in Δ. This gives,

AdKXd=Ω1+K1aK1+K1AΩ2AK1.E22

Multiplying both sides with K1K1gives

AdK¯Xd=Ω1+a+AΩ2A.E23

where, K¯=K1Kand Ω1=KΩK.

We evaluate AΩ2A, for Ω2son.

AΩ2Akl=expiϕkϕlΩ2kl=cosϕkϕlΩ2klSkl+isinϕkϕlΩ2klRkl.E24

such that Sis skew symmetric and Ris traceless symmetric matrix with iRp. Note iRaand onto a, by appropriate choice of Ω2.

Given AdK¯Xdp, we decompose it as

AdK¯Xd=PAdK¯Xd+AdK¯Xd=Ω1+a+AΩ2A,

with Pdenoting the projection onto a(a=iλ100000λn,where iλi=0.) w.r.t to standard inner product and AdK¯Xdto the orthogonal component. In Eq. (24), ϕkϕl0,π, we can solve for Ω2klsuch that iR=AdK¯Xd. This gives Ω2. Let a=PAdK¯Xdand choose Ω1=AdK¯XdAΩ2A=Sk.

With this choice of Ω1,Ω2and a, Pt+Δand Qt+Δare matched to first order in Δand

Pt+ΔQt+Δ=oΔ2.

Consider the case, when Ais degenerate. Let,

A=A1000A2000An,E25

where Akis nkfold degenerate (modulo sign) described by nk×nkblock. WLOG, we arrange

Ak=expiϕkIr×r00Is×s.E26

Consider the decomposition

AdK¯Xd=PAdK¯Xd+AdK¯Xd,

where Pdenotes projection onto nk×nkblocks in Eq. (25) and AdKXd, the orthogonal complement.

PX11X12X1nX21X22X2nXn1Xn2Xnn=X11000X22000Xnn,E27

where Xijare blocks.

Then we write

Qt+Δ=expΩ1ΔK1expPAdK¯XdΔAexpΩ2ΔK2.E28

where in Eq. (24) we can solve for Ω2klsuch that iR=AdK¯Xd. This gives Ω2. Choose, AdK¯XdAΩ2A=Ω1k, this gives Ω1=K1Ω1K1. Again Pt+ΔQt+Δ=oΔ2.We write Eq. (28) slightly differently.

Let H1be a rotation formed from block diagonal matrix

H1=Θ1000Θ2000Θn,E29

where Θkis nk×nksub-block in SOnk. H1=exph1is chosen such that

H1PAdK¯XdH1=a

is a diagonal matrix. Let H2=exp(A1h1Ah2, where h2is skew symmetric, such that

h1=θ1000θ2000θn,h2=θ̂1000θ̂2000θ̂n,E30

where

θk,θ̂kis nk×nksub-block in sonk, related by (see 26)

θ̂k=AkθkAk,θk=θ11r×rθ12θ12θ22s×s,θ̂k=θ11θ12θ12θ22E31

Note H1PAdkXdH1=alies in convex hull of eigenvalues of Xd. This is true if we look at the diagonal of H1AdKXdH1, it follows from Schur Convexity. The diagonal of H1AdkXdH1is zero as its inner product

tra1H1AdkXdH1=trH1a1H1AdkXd=0.

as H1a1H1has block diagonal form which is perpendicular to AdkXd. Therefore diagonal of H1PAdkXdH1is same as diagonal of H1AdKXdH1.

Now using H1AH2=A, from 28, we have

Qt+Δ=expΩ1ΔK1expPAdK¯XdΔH1AH2expΩ2ΔK2.E32
Qt+Δ=expΩ1ΔK1H1expaΔAH2expΩ2ΔK2.E33

where the above expression can be written as

Qt+Δ=expΩ1ΔexpK1H1aH1K1ΔexpK1AΩ2AK1ΔPt.

where Ω1, H1,a,Ω2, are chosen such that

Ω1+K1H1aH1K1+K1AΩ2AK1=AdKXd.
Ω1+H1aH1+AΩ2A=AdK¯Xd.
Qt+ΔPt+Δ=oΔ2Pt.
Qt+Δ=I+oΔ2Pt+Δ.
Qt+ΔQt+ΔT=I+oΔ2Pt+ΔPTt+ΔI+oΔ2=Pt+ΔPTt+ΔI+oΔ2.
Pt+ΔPTt+Δ=K1expi2ϕ1000expi2ϕ2000expi2ϕnK1T.

Let F=Pt+ΔPTt+Δand G=Qt+ΔQTT+Δwe relate the eigenvalues, of Fand G. Given F,G, as above, with FGε, and a ordered set of eigenvalues of F, denote λF=expi2ϕ1expi2ϕ2expi2ϕn, there exists an ordering (correspondence) of eigenvalues of G, such that λFλG<ε.

Choose an ordering of λGcall μthat minimizes λFλG.

F=U1DλU1and G=U2DμU2, where Dλis diagonal with diagonal as λ, let U=U1U2,

FG2=DλUDμU2=λ2+μ2trDλUDμU+UDμUDλ,

By Schur convexity,

trDλUDμU+UDμUDλ=iαiλPiμ+Piμλ,

where Piare permutations. Therefore FG2>λμ2.

Therefore,

λQQTt+Δ=λPPTt+Δ+oΔ2.

The difference

oΔ2=expΩ1+K1H1aH1K1+K1AΩ2AK1ΔexpAdKXdΔexpΩ1ΔexpK1H1aH1K1ΔexpK1AΩ2AK1Δ,

is regulated by size of Ω2, which is bounded by Ω2Xdsinϕiϕj, where sinϕiϕjis smallest non-zero difference. Δis chosen small enough such that oΔ2<εΔ.

For each point t0T, we choose an open nghd Nt=tNtt+Nt, such that otΔ2<εΔfor ΔNt. Ntforms a cover of 0T. We can choose a finite subcover centered at t1,,tn(see Figure 1A). Consider trajectory at points Pt1,,Ptn. Let ti,i+1be the point in intersection of Ntiand Nti+1. Let Δi+=ti,i+1tiand Δi+1=ti+1ti,i+1. We consider points Pti,Pti+1,Pti,i+1,Qti+Δi+Qi+,Qti+1Δi+1Qi+1as shown in Figure 1B.

Figure 1.

A. Collection of overlapping neighborhoods forming the finite subcover. B. Depiction of P i , P i + 1 , Q i + , Q i − , P i , i + 1 as in proof of Theorem 1.

Then we get the following recursive relations.

λQi+Qi+T=exp2ai+Δi+λPiPiTE34
λPi,i+1Pi,i+1T=λQi+Qi+T+oΔi+2E35
λQi+1Qi+1T=λPi,i+1Pi,i+1T+oΔi+12E36
exp2ai+1Δi+1λPi+1Pi+1T=λQi+1Qi+1TE37

where ai+and ai+1correspond to ain Eq. (33) and lie in the convex hull of the eigenvalues Xd.

Adding the above equations,

λPi+1Pi+1T=expoΔ2exp(2ai+Δi++ai+1Δi+1λPiPi.E38
λPnPnT=exp(oΔ2εTexp2iai+Δi++ai+1Δi+1λP1P1T.E39

where oΔ2in Eq. (38) is diagonal.

λPnPnT=exp(oΔ2εTexp2TkαkPkλλP1P1T=exp(oΔ2εTexp2μTλP1P1T,E40

where μλand P1=I.

Pn=K1exp12oΔ2εTexpμTK2.E41

Note, PnK1expμTK2=oε. This implies that Pnbelongs to the compact set K1expμTK2, else it has minimum distance from this compact set and by making Δ0and hence ε0, we can make this arbitrarily small. In Eq. (18), PnPTas τ0. Hence PTbelongs to compact set K1expμTK2. q.e.d.

Corollary 1. Let UtSUnbe a solution to the differential equation

U̇=Xd+iuiXiU,

where XiLA, the Lie algebra generated by Xi, is sonand Xd=iλ1000λ2000λn. The elements of reachable set at time T, takes the form UTK1expiμTK2, where K1,K2SOnand μλ, where λ=λ1λnand the set S=K1expiμTK2belongs to the closure of reachable set.

Proof. Let Vt=KtUt, where, K̇=iuiXiK. Then

V̇t=AdKtXdVt.

From Theorem 1, we have VTK1expiμTK2. Therefore UTK1expiμTK2. Given

U=K1expiμTK2=K1expijαjPjλTK2=K1jexpitjXdKj,tj=T.

We can synthesize Kjin negligible time, therefore UTU<ε, for any desired ε. Hence Uis 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

W=expIySyexpiπ2Iz

The transformation maps the algebra k=su2×su2=IαSαto k1=so4, four dimensional skew symmetric matrices, i.e., AdWk=k1. The transformation maps p=IαSβto p1=iA, where Ais traceless symmetric and maps a=iIxSxIySyIzSzto a1=iSz2Iz2IzSz, space of diagonal matrices in p1, such that axIxSx+ayIySy+azIzSzgets mapped to the four vector (the diagonal) λ1λ2λ3λ4=ay+azaxax+ayazax+ay+azax+azay.

Corollary 2. Canonical decomposition. Given the decomposition of SU(4) from Remark 2, we can write

U=expΩ1expiλ100000λ4expΩ2,

where Ω1,Ω2so4. We write above as

U=expΩ1expiax2Sz+ay2Iz+azIzSzexpΩ2,

Multiplying both sides with W.Wgives

WUW=K1expiaxIxSx+ayIySy+azIzSzK2,

where K1,K2SU2×SU2local unitaries and we can rotate to axayaz.

Corollary 3. Digonalization. Given iHc=iαβJαβIαSβ, there exists a local unitary Ksuch that

KiHcK=iaxIxSx+ayIySy+azIzSz,axayaz.

Note WiHcWp1. Then choose ΘSOnsuch that ΘWiHcWΘ=iax2Sz+ay2Iz+azIzSzand hence

WexpΩWiHcWexpΩW=iaxIxSx+ayIySy+azIzSz.

where K=WexpΩWis a local unitary. We can rotate to ensure axayaz.

Corollary 4. Given the evolution of coupled qubits U̇=iHc+jujHjU, we can diagonalize Hc=αβJαβIαSβby local unitary Xd=KHcK=axIxSx+ayIySy+azIzSz, axayaz, which we write as triple axayaz. From this, there are 24 triples obtained by permuting and changing sign of any two by local unitary. Then UTSwhere

S=K1expTiαiaibiciK2,αi>0iαi=1.

Furthermore Sbelongs to the closure of the reachable set. Alternate description of Sis

U=K1expiαIxSx+βIySy+γIzSzK2,αβγ,

αaxTand α+β±γax+ay±azT.

Proof. Let Vt=KtUt, where K̇=ijujXjK. Then

V̇t=AdKtiXdVt.

Consider the product

V=iexpAdKiiXdΔt

where KiSU2SU2and Xd=axIxSx+ayIySy+azIzSz, where axayaz. Then,

WVW=iexpAdWKiWiWXdWΔt

Observe WKiWSO4and WXdW=diagλ1λ2λ4. Then using results from Theorem 1, we have

WVW=J1expJ2=J1expijαjPjλJ2,J1,J2SO4,μλT

Multiplying both sides with WW, we get

V=K1expTiαiaibiciK2,αi>0iαi=1.

which we can write as

V=K1expiαIxSx+βIySy+γIzSzK2,αβγ,

where using μλT, we get,

α+βγax+ayazTE42
αaxTE43
α+β+γax+ay+azT.E44

Furthermore U=KV. Hence the proof. q.e.d.

3. Time optimal control for G/Kproblem

Remark 5. Stabilizer: Let g=pkbe Cartan decomposition of real semisimple Lie algebra gand apbe its Cartan subalgebra. Let aa. ada2:ppis symmetric in basis orthonormal wrt to the killing form. We can diagonalize ada2. Let Yibe eigenvectors with nonzero (negative) eigenvalues λi2. Let Xi=aYiλi, λi>0.

adaYi=λiXi,adaXi=λiYi.

Xiare independent, as αiXi=0implies αiλiYi=0. Since Yiare independent, Xiare independent. Given XXi, then aX=0, otherwise we can decompose it in eigenvectors of ada2, i.e., aX=iαiai+jβjYj, where aiare zero eigenvectors of ada2. Since 0=X[aaX=aX2, which means aX=0. This is a contradiction. Yiare orthogonal, implies Xiare orthogonal, aYiaYj=[a,aYiYj=λi2YiYj=0. Let k0ksatisfy ak0=0. Then k0=Xi.

Y˜idenote eigenvectors that have λias non-zero integral multiples of π. X˜iare adarelated to Y˜i. We now reserve Yifor non-zero eigenvectors that are not integral multiples of π.

Let

f=aiY˜i,h=k0X˜i,

X˜i,Xl,kjwhere kjforms a basis of k0, forms a basis of k. Let A=expa.

AkA1=AiαiXi+lαlX˜l+jαjkjA, where kk

AkA1=iαicosλiXisinλiYi+l±αlX˜l+jαjkj

The range of AA1in p, is perpendicular to f. Given Ypsuch that Yf. The norm Xof Xk, such that ppart of AXA1p=Ysatisfies

XYsinλs.E45

where λs2is the smallest nonzero eigenvalue of ada2such that λsis not an integral multiple of π.

A2kA2stabilizes hkand fp. If kk, is stabilized by A2A2, λi=, i.e., kh. This means his an subalgebra, as the Lie bracket of yzkfor y,zhis stabilized by A2A2.

Let H=exph, be an integral manifold of h. Let H˜Kbe the solution to A2H˜A2=H˜or A2H˜H˜A2=0. H˜is closed, HH˜. We show that H˜is a manifold. Given element H0H˜K, where Kis closed, we have a expBδknghd of H0, in expBδball nghd of H0, which is one to one. For xBδk, A2expxA2=expx, implies,

A2expiαiXi+lβlX˜l+jγjkjH0A2=expiαicos2λiXisin2λiYi+lβlX˜l+jγjkjH0=expiαiXi+lβlX˜l+jγjkjH0,

then by one to one property of expBδ, we get αi=0and xh. Therefore expBδhH0is a nghd of H0.

Given a sequence Hiexphconverging to H0, for nlarge enough HnexpBδhH0. Then H0is in invariant manifold exph. Hence exphis closed and hence compact.

Let yf, then there exists a h0hsuch that exph0yexph0a. We maximize the function arexphyexph, over the compact group exph, for regular element araand ..is the killing form. At the maxima, we have at t=0, ddtarexph1texph0yexph0exph1t=0.

arh1exph0yexph0=h1arexph0yexph0,

if exph0yexph0a, then arexph0yexph0k. The bracket arexph0yexph0is AdA2invariant and, hence, belongs to h. We can choose h1so that gradient is not zero. Hence exph0yexph0a. For zpsuch that zf, we have exph0zexph0a.

aexph0zexph0=exph0aexph0z=0,

as exph0aexph0is AdA2invariant, hence exph0aexph0f. In above, we worked with killing form. For g=sun, we may use standard inner product.

Remark 6. Kostant’s convexity: [28] Given the decomposition g=pk, let apand Xa,. Let Wiexpksuch that WiXWiaare distinct, Weyl points. Then projection (w.r.t killing form) of AdKXon alies in convex hull of these Weyl points. The Cbe the convex hull and let projection PAdKXlie outside this Hull. Then there is a separating hyperplane a, such that AdKXa<Ca. W.L.O.G we can take ato be a regular element. We minimize AdKXa, with choice of Kand find that minimum happens when AdKXa=0, i.e. AdKXis a Weyl point. Hence PAdKXiαiWiXWi1, for αi>0and iαi=1. The result is true with a projection w.r.t inner product that satisfies xyz=xyz, like standard inner product on g=sun.

Theorem 2 Given a compact Lie group Gand Lie algebra g. Consider the Cartan decomposition of a real semisimple Lie algebra g=pk. Given the control system

Ẋ=AdKtXdX,P0=1

where Xda, the Cartan subalgebra apand Ktexpk, a closed subgroup of G. The end point

PT=K1expTiαiWiXdK2,

where K1,K2expkand WiXdaare Weyl points, αi>0and iαi=1.

Proof. As in proof of Theorem 1, we define

Pt+Δ=expAdKXdΔPt=expAdKXdΔK1expaK2

and show that

expAdKXdΔK1AK2=Kaexpa0Δ+CΔ2AKb=Kaexpa+a0Δ+CΔ2Kb,E46

where for K¯=K11K,

AdK¯Xd=PAdK¯Xda0+AdK¯Xd.

where Pis projection w.r.t killing form and a0f, the centralizer in pas defined in Remark 5, CΔ2fis a second order term that can be made small by choosing Δ. Ka,Kbexpk.

To show Eq. (46), we show there exists K1,K2Ksuch that

expk1K1expAdK¯XdΔexpAk2A1K2=expa0Δ+CΔ2,E47

where K1and K2are constructed by a iterative procedure as described in the proof below.

Given Xand Yas N×Nmatrices, considered elements of a matrix Lie algebra g, we have,

logeXeYX+Y=n>01n1n1inXr1Ys1XrnYsni=1nri+sir1!s1!rn!sn!,E48

where ri+si>0.

We bound the largest element (absolute value) of logeXeYX+Y, denoted as logeXeYX+Y0, given X0<Δand Y0<b0Δk, where k1, Δ<1, b0Δ<1.

logeXeYX+Y0n=1Nb0eΔk+1+n>11n2Ne2nb0Δn+k1nE49
Nb0eΔk+1+Ne22b0Δk+11+2Ne2Δ+E50
Nb0eΔk+1+Ne22b0Δk+112Ne2ΔM˜b0Δk+1E51

where 2NΔ<1and M˜Δ<1.

Given decomposition of g=pk, pkwith respect to the negative definite killing form BXY=tradXadY. Furthermore there is decomposition of p=aa.

Given

U0=expa0Δ+b0Δ+c0Δ,

where a0a, b0aand c0k, such that a00+b00+c00<1, which we just abbreviate as a0+b0+c0<1(we follow this convention below).

We describe an iterative procedure

Un=Πk=1nexpckΔU0Πk=0nexpbkΔ,E52

where ckkand bka, such that the limit

nUn=expa0Δ+CΔ2,E53

where a0,Ca.

U1=expc0Δexpa0Δ+b0Δ+c0Δexpb0Δ=expa0Δ+b0Δ+c0Δ2expb0Δ=expa0Δ+b0Δ2+c0Δ2=expa1+b1+c1Δ

Note b0and c0are elements of gand need not be contained in aand k.

Where, using bound in c0M˜c0, which gives a0+b0+c0Δa0+b0+c0. Using the bound again, we obtain, b0M˜b0. We can decompose, b0+c0Δ, into subspaces a0+b1+c1, where a0Mb0+c0Δ, b1Mb0+c0Δand c1Mb0+c0Δ, where BXXλmaxX2, where Xis Frobenius norm and BXXλminX2. Let M=Nλmaxλmin.

This gives, a0Mb0+c0Δ, b1Mb0+c0Δand c1Mb0+c0Δ. This gives

a1a0+M˜Mb0+c0Δ b1M˜Mb0+c0Δ c1M˜Mb0+c0Δ

For 4M˜MΔ<1, we have, a1+b1+c1a0+b0+c0. Continuing and using bk+ck2M˜MΔbk1+ck12M˜MΔkb0+c0.

Similarly,

akak102M˜MΔkb0+c0

Note, akbkckis a Cauchy sequences which converges to a,0,0, where

aa00b0+c0k=12M˜MΔk2MM˜Δb0+c012M˜MΔCΔ,

where C=4M˜Mb0+c0.

The above exercise was illustrative. Now we use an iterative procedure as above to show Eq. (47).

Writing

AdK¯Xd=PAdK¯Xda0+AdK¯Xdb0,

where a0fand b0f, consider again the iterations

U0=expc¯0Δexpa0Δ+b0Δexpb0Δ+c¯0Δ=expc¯0Δexpa0Δ+c¯0Δ+b0Δ2=expa0Δ+b0Δ2+c0Δ2=expa1Δ+b1Δ+c1Δ

We refer to Remark 5, Eq. (45). Given b0Δpsuch that b0Δf. If AkA=b0Δ+c¯0Δ, then khb0Δ(killing norm).

c¯0k, is bounded c¯0Mhb0, where Mas before converts between two different norms. Using bounds derived above b0M˜Mh+1b0, and c0M˜Mhb0, 2M˜Mh+1Δ<1, we obtain.

which gives a0+b0Δ+c¯0a0+b0M˜Mh+1Δ+Mh1. For appropriate M, we have

a1a0+M3b0+c0Δb1M3b0+c0Δc1M3b0+c0Δ

we obtain

a1+b1+c1a0+Mb0+c0Δa0+b0+c0

where Δis chosen small.

U1=expc1+c¯1Δexpa1Δ+b1Δ+c1Δexpb1Δ+c¯1Δ=expc1+c¯1Δexpa1Δ+c1+c¯1Δ+b1Δ2=expa1Δ+b1Δ2+c1Δ2=expa2Δ+b2Δ+c2Δ

where c¯1k, such that c¯1Mhb1.

where, using bounds derived above b1M˜Mh+1b1, and c1M˜Mhb1+c1, where using the bound 2M˜Mh+1Δ<1, we obtain

which gives a1+b1Δ+c1+c¯1a1+1+Mhb1+c1a0+b0+c0.

We can decompose, b1+c1Δ2, into subspaces a1+b2+c2Δ, where a1Mb1+c1Δ, b2Mb1+c1Δand c2Mb1+c1Δ, where Mas before converts between two different norms.

This gives

a2a1+4M˜M2hb1+c1Δ b24M˜M2hb1+c1Δ c24M˜M2hb1+c1Δ

For x=8M˜M2hΔ<23, we have, a2+b2+c2a1+b1+c1a0+b0+c0,

Using bk+ckxbk1+ck1xkb0+c0.

Similarly,

akak10xkb0+c0

Note, akbkckis a Cauchy sequences which converges to a,0,0, where

aa00xb0+c0k=0xkxb0+c01xCΔ,

where C=16M˜M2hb0+c0.

The above iterative procedure generates k1and k2in Eq. (47), such that

expK1AdKXdK1Δ=expk1expa0Δ+CΔ2expAk2A.

where a0Δ+CΔ2f. By using a stabilizer H1,H2, we can rotate them to asuch that

expAdKXdΔK1AK2=KaH1expa0Δ+CΔ2AH2Kb

such that H11a0Δ+CΔ2H1=a0Δ+CΔ2is in aand a0=PH11a0H1is projection onto asuch that

PH11a0H1=kαkWkXd.

This follows because the orthogonal part of AdK¯Xdto fwritten as AdK¯Xdremains orthogonal of f

H1AdKXdHa=AdKXdHaH1=AdKXda'=0

(af), remains orthogonal to a. Therefore PH11a0H1=PH11AdK¯XdH1=kαkWkXd.

expAdKXdΔK1AK2=Kaexpa+a0Δ+CΔ2Kb.

Lemma 1 Given P=K1expa+a1ΔA1K2=K3expbb1ΔA2K4, where a,b,a1,b1a. We can express

expb=Kaexpa+a1Δ+Wb1ΔKb,

where Wb1is Weyl element of b1. Furthermore

expb+b2Δ=Kaexpa+a1Δ+Wb1Δ+Wb2ΔKb.

Proof. Note, A2=K31PK41, commutes with b1. This implies

A2=K˜expa+a1ΔKcommutes with b1. This implies A2b1A21=b1, i.e., K˜expa+a1ΔAdKb1expa+a1ΔK˜'=b1, which implies that AdKb1f. Recall, from Remark 5,

expa+a1ΔAdKb1expa+a1Δ=kckYkcosλk+Xksinλk.

This implies kcksinλkXk=0, implying λk=. Therefore,

exp2a+a1ΔAdKb1exp2a+a1Δ=AdKb1.

We have shown existence of H1such that H1AdKb1H11a, using H1,H2as before,

K˜expa+a1ΔKexpb1Δ=K˜H2expa+a1ΔH1expAdKb1ΔK=Kaexpa+a1Δ+Wb1ΔKb.

Applying the theorem again to

Kaexpa+a1Δ+Wb1ΔKbexpb2Δ=Kaexpa+a1Δ+Wb1Δ+Wb2ΔKb.

Lemma 2 Given Pi=K1iAiK2i=K1iexpaiK2i, we have Pi,i+1=expHi+Δi+Pi, and Pi,i+1=expHi+1Δi+1Pi+1, where Hi+=AdKiXd. From above we can express

Pi,i+1=Kai+expai+a1i+Δ+i+a2i+Δ+i2Kbi+.

where a1i+and a2i+are first and second order increments to aiin the positive direction. The remaining notation is self-explanatory.

Pi,i+1=Kai+1expai+1a1i+1Δi+1a2i+1Δi+12Kbi+1.
expai+1=K1expai+a1i+Δ+i+a2i+Δ+i2+Wa1i+1Δi+1+a2i+1Δi+12K2.
Wa1i+1Δi+1+a2i+1Δi+12=PWa1i+1Δi+1+PWa2i+1Δi+12=kαkWkXdΔi+1+oΔi+12

where, ai,a1i,a2ia.

Using Lemma 1 and 2, we can express

PnT=K1expanexpK2=K1expiWai+Δi++Wai+1Δi+1expoΔ2εTK2

Letting εgo to 0, we have

PnT=K1expTiαiWiXdK2.

Hence the proof of theorem. q.e.d.

4. Conclusion

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 k(local Hamiltonians) and slow generators p(couplings) as a Cartan decomposition g=pk. Using this decomposition, we used some convexity ideas to completely characterize the reachable set and time optimal control for these problems.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Navin Khaneja (November 5th 2018). Convexity, Majorization and Time Optimal Control of Coupled Spin Dynamics, Applied Modern Control, Le Anh Tuan, IntechOpen, DOI: 10.5772/intechopen.80567. Available from:

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