Time optimal control of coupled spin dynamics: A global analysis

Automatica 111 (2020) 108639

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Time optimal control of coupled spin dynamics: A global analysis✩ Navin Khaneja Systems and Control Engineering, IIT Bombay - 400076, India

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Article history: Received 10 May 2018 Received in revised form 17 August 2019 Accepted 29 September 2019 Available online xxxx

a b s t r a c t In this paper, 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 paper is that we carry out a second order analysis of time optimality. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction A rich class of model control problems arise when one considers dynamics of two coupled spin 21 . The dynamics of two coupled spins forms the basis for the field of quantum information processing and computing (Nielsen & Chuang, 2000), and is fundamental in multidimensional NMR spectroscopy (Cavanagh, Fairbrother, Palmer, & Skelton, 1996; Ernst, Bodenhausen, & Wokaun, 1987). Numerous experiments in NMR spectroscopy involve synthesizing unitary transformations (D’Alessandro, 2001; Khaneja, Brockett, & Glaser, 2001; Khaneja & Glaser, 2001) 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 (Ernst et al., 1987). Similarly, many protocols in quantum communication and information processing involve synthesizing entangled states, starting from the separable states (Bennett et al., 2002; Kraus & Cirac, 2001; Nielsen & Chuang, 2000). 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 (Cavanagh et al., 1996; Ernst et al., 1987), local unitary ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor James Lam under the direction of Editor Ian R Petersen. E-mail address: [email protected]. https://doi.org/10.1016/j.automatica.2019.108639 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

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 Schródinger equation for unitary evolution U˙ = −i[Hc +

n ∑

uj Hj ]U , U(0) = I ,

(1)

j=1

where Hc represents a coupling Hamiltonian, and uj 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 Hj are local Hamiltonians and are much stronger than the coupling between the spins (uj can be made large). Design of time optimal rf-pulse sequences is an important research subject in NMR spectroscopy and quantum information processing (Brockett, 1972; D’Alessandro, 2001; Jurdjevic & Sussmann, 1972; Khaneja et al., 2001; Khaneja & Glaser, 2001, 2002; Khaneja, Glaser, & Brockett, 2002; Khaneja et al., 2007; Reiss, Khaneja, & Glaser, 2003; Vidal, Hammerer, & Cirac, 2002; Yuan & Khaneja, 0000, 2005, 2006; Yuan, Zeier, & Khaneja, 2011; Zeier, Yuan, & Khaneja, 2008), as minimizing the time to execute quantum operations can reduce relaxation losses which are always present in an open quantum system (Lindblad, 1976; Redfield, 1965). This problem has a special mathematical structure that helps to characterize all the time optimal trajectories (Khaneja et al., 2001). 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 21 can be written in terms of the generators of rotations on a two dimensional space and these are

2

N. Khaneja / Automatica 111 (2020) 108639

the Pauli matrices −iσx , −iσy , −iσz , where,

σz =

[

0

[

−1 ] 1 .

1 1 2 0

1 0 σx = 2 1

]

[

1 0 2 i

; σy =

−i

The Lie algebra g = su(4) has a direct sum decomposition g = p ⊕ k, where

] (2)

;

0

[σx , σy ] = iσz , [σy , σz ] = iσx , [σz , σx ] = iσy ,

(3)

where [A, B] = AB − BA is the matrix commutator and 1

σ =σ =σ =

. (4) 4 The Hamiltonian for a system of two coupled spins takes the general form H0 =

2 z

2 y

∑

aα σα ⊗ 1 +

∑

bβ 1 ⊗ σβ +

∑

Jαβ σα ⊗ σβ ,

(5)

where α, β ∈ {x, y, z }. The Hamiltonians σα ⊗ 1 and 1 ⊗ σβ are termed local Hamiltonians and operate on one of the spins. The Hamiltonian Hc =

∑

Jαβ σα ⊗ σβ

(6)

is the coupling or interaction Hamiltonian, and operates on both the spins. The following notation is therefore common place in the NMR literature. Iα = σα ⊗ 1 ; Sβ = 1 ⊗ σβ .

(7)

The ∑ operators Iα and Sβ commute and therefore exp(−i β bβ Sβ ) = exp(−i

∑

aα Iα ) exp(−i

∑

α

(exp(−i

∑

bβ S β ) =

α

(8)

aα σα ) ⊗ 1 )(1 ⊗ exp(−i

∑

α

bβ σβ )).

β

The unitary transformations of the kind exp(−i

∑

aα σα ) ⊗ exp(−i

∑

α

bβ σβ )

β

obtained by evolution of the local Hamiltonians are called local unitary transformations. The coupling Hamiltonian can be written as Hc =

∑

Jαβ Iα Sβ .

[k, k] ⊂ k,

[k, p] ⊂ p, [p, p] ⊂ p.

U˙ = (Xd +

∑

0 1 0 0

0 0 −1 0

where U ∈ SU(n), the special Unitary group (determinant 1, n × n matrices U such that UU ′ = 1 , ′ is conjugate transpose). Where Xj ∈ k = so(n), skew symmetric matrices and

⎡

λ1

(10)

and

⎡

1 1 ⎢0 Iz S z = σ z ⊗ σ z = ⎣ 4 0 0

0

−1 0 0

0 0 −1 0

⎤

0 0⎥ . 0⎦ 1

... ... .. .

0

⎢0 Xd = −i ⎢ ⎣ .. .

λ2 .. .

0

0

...

0 0⎥

⎤

.. ⎥ ⎦, . λn

∑

λi = 0.

We assume {Xj }LA , the Lie algebra (Xj and its matrix commutators) generated by generators Xj is all of so(n). We want to find the minimum time to steer this system between points of interest assuming no bounds on our controls uj (t). Here again we have a Cartan decomposition on generators. Given g = su(n), traceless skew Hermitian matrices , generators of SU(n), we have g = p ⊕ k, where p = −iA where A is traceless symmetric and k = so(n). As before, Xd ∈ p and Xj ∈ k. We want to find time optimal ways SU(n) to steer this system. We call this SO(n) problem. For n = 4, this system models the dynamics of two coupled nuclear spins in NMR spectroscopy. Consider another problem evolving on SU(2n). U˙ = (Xd +

∑

uj (t)Xj )U , U(0) = 1 .

(15)

j

⎡

⎤

0 0 ⎥ . 0 ⎦ −1

(14)

Here

Written explicitly, some of these matrices take the form 1 1 ⎢0 Iz = σz ⊗ 1 = ⎣ 2 0 0

uj (t)Xj )U , U(0) = 1 ,

j

(9)

⎡

(13)

This decomposition of a real semi-simple Lie algebra g = p ⊕ k, satisfying (13), is called the Cartan decomposition of the Lie algebra g (Helgason, 1978). 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,

aα I α +

β

∑

(12)

Here k is a sub-algebra of g, made from local Hamiltonians, and p represents nonlocal Hamiltonians. In Eq. (1) , we have −iHj ∈ k and −iHc ∈ p. It is easy to verify that

0

Note

2 x

k = −i{Iα , Sβ }, p = −i{Iα Sβ }.

(11)

The 15 operators,

−i{Iα , Sβ , Iα Sβ }, for α, β ∈ {x, y, z }, form the basis for the Lie algebra g = su(4), the 4 × 4, traceless skew Hermitian matrices. For the coupled two spins, the generators −iHc , −iHj ∈ su(4) and the evolution operator U(t) in Eq. (1) is an element of SU(4), the 4 × 4, unitary matrices of determinant 1.

0

⎢ .. ⎢ . ⎢ ⎢ 0 Xd = ⎢ ⎢−λ1 ⎢ . ⎣ . .

... .. . ... ... .. .

0

.. .

0 0

λ1 .. .

... .. .

0

... .. . ...

...

0

0

⎤

.. ⎥ .⎥ ⎥ λn ⎥ ⎥, 0⎥ .. ⎥ ⎦ .

.. .. . . 0 . . . −λn 0 0 [ ] A 0 and Xj ∈ k = , space of block diagonal traceless skew 0

B

Hermitian matrices. We assume {Xj }LA is all of k. We want to find the minimum time to steer this system between points of interest SU(2n), assuming no bounds on our controls uj (t). Here again, we have [a Cartan ]decomposition, of g = su(2n) as g = p ⊕ k and 0 Z SU(2n) , Xd ∈ p and Xj ∈ k. We call this SU(n)×SU(n)×U(1) p = −Z ′ 0 problem. For n = 2, this system models the dynamics of coupled electron-nuclear spin system in EPR (Zeier et al., 2008).

N. Khaneja / Automatica 111 (2020) 108639

In general, U is in a compact Lie group G (such as SU(n)), with Xd , Xj in its real semisimple (no abelian ideals) Lie algebra g and U˙ = (Xd +

∑

uj (t)Xj )U , U(0) = 1 .

(16)

j

Given the Cartan decomposition g = p ⊕ k, where Xd ∈ p, {Xj }LA = k and K = exp(k) (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 uj (t). Since {Xj }LA = k, any rotation (evolution) in subgroup K can be synthesized with evolution of Xj (Brockett, 1972; Jurdjevic & Sussmann, 1972). Since there are no bounds on uj (t), this can be done in arbitrarily small time (Khaneja et al., 2001). We call this G problem. K The special structure of this problem helps in complete description of the reachable set (Jurdjevic, 1997). The elements of the reachable set at time T , take the form U(T ) ∈ S = K1 exp(T

∑

αk Wk Xd Wk−1 )K2 ,

(17)

k

−1 where K∑ all commute, and 1 , K2 , Wk ∈ exp(k), and Wk Xd Wk αk > 0, αk = 1. This reachable set is formed from evolution of K1 , K2 and commuting Hamiltonians Wk Xd Wk−1 . Unbounded control suggests that K1 , K2 , Wk can 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 (Helgason, 1978). Inside p is contained the largest abelian sub-algebra, denoted as a. Any X ∈ p is AdK conjugate to an element of a, i.e. X = Ka1 K −1 for some a1 ∈ a. Then, any arbitrary element of the group G can be written as

G = K0 exp(X ) = K0 exp(AdK (a1 )) = K1 exp(a1 )K2 ,

(18)

for some X ∈ p where Ki ∈ K and a1 ∈ a. The first equation is a fact about geodesics in G/K space (Helgason, 1978), where K = exp(k) is a closed subgroup of G. Eq. (18) is called the KAK decomposition (Helgason, 1978). The results in this paper suggest that K1 and K2 can be synthesized by unbounded controls Xi in negligible time. The time consuming part of the evolution exp(a1 ) is synthesized by evolution of Hamiltonian Xd . Time optimal strategy suggests evolving Xd and its conjugates Wk Xd Wk−1 where Wk Xd Wk−1 all commute. Written as evolution G = K1

∏

= K1

∏

exp(tk Wk Xd Wk−1 ) K2

k

−1

Wk exp(tk Xd )Wk

K2 .

k

where K1 , K2 , Wk take negligible time to synthesize using unbounded controls ui and time-optimality is characterized by synthesis of commuting Hamiltonians Wk Xd Wk−1 . This characterization of time optimality, involving commuting Hamiltonians, is derived using convexity ideas (Khaneja et al., 2001; Kostant, 1973). The remaining paper develops these notions. The paper is organized as follows. In Section 2, we study the SU(n) problem. In Section 3, we study the general KG problem. SO(n) The main contribution of the paper is, we carry out a second order analysis of time optimality. We conclude in Section 4 , with facts about roots and reflections, with application to dynamics of coupled spins. Given Lie algebra g, we use killing form ⟨x, y⟩ = tr(adx ady ) as an inner product on g. When g = su(n), we also use the inner product ⟨x, y⟩ = tr(x′ y). We call this standard inner product.

3

2. Time optimal control for SU (n)/SO(n) problem Result 1. Birkhoff convexity states, a real n × n matrix A is doubly ∑ ∑ stochastic ( i Aij = A = 1, for Aij ≥ 0) iff it can be written ij j as convex hull of permutation matrices Pi (only one 1 and everything ⎡ else zero in every row ⎤ and column). Given Θ ∈ SO(n), and λ1 0 . . . 0 ⎢ 0 λ2 . . . 0 ⎥ T X = ⎢ . . .⎥ ⎣. ⎦, we have diag(Θ X Θ ) = B diag(X ),

..

..

..

.. λn

0 0 ... where diag(X ) is a column vector ∑containing ∑ diagonal entries of X , and Bij = (Θij )2 , and hence B = ij i j Bij = 1, making B a doubly stochastic matrix, which can be∑ written as convex sum of permutations. Therefore B diag(X ) = i αi Pi diag(X ), i.e., diagonal of a symmetric matrix Θ X Θ T lies in convex hull of its eigenvalues and its permutations. This is called Schur convexity. Result 2. G = SU(n) has a closed subgroup K = SO(n), and a Cartan decomposition of its Lie algebra g = su(n) as g = p ⊕ k, for k = so(n) and p = −iA where A is traceless symmetric abelian subalgebra of p, such that ⎡ and a is maximal ⎤ λ1 . . . 0

a = −i ⎣ 0

..

0

0

⎢

.

0 ⎦, where

⎥

∑

i

λi = 0. KAK decomposition

λn

in Eq. (18) states for U ∈ SU(n), U = Θ1 exp(Ω )Θ2 , where Θ1 , Θ2 ∈ SO(n), and

⎡ λ1 ⎢ Ω = −i ⎣ 0

... .. .

0 where

∑

i

0

0

⎤

0 ⎦,

⎥

λn

λi = 0.

Remark 1. We now give a proof of the reachable set (17) for the SU(n) problem. Let U(t) ∈ SU(n) be a solution to the differential SO(n) equation (14) U˙ = (Xd +

∑

ui Xi )U , U(0) = I .

i

To understand the reachable set of this system, ∑we make a change of coordinates P(t) = K ′ (t)U(t), where K˙ = ( i ui Xi )K . Then

˙ = AdK ′ (t) (Xd )P(t), AdK (Xd ) = KXK −1 . P(t) If we understand reachable set of P(t), then the reachable set in Eq. (14) is easily derived. Theorem 1. equation

Let P(t) ∈ SU(n) be a solution to the differential

P˙ = AdK (t) (Xd )P ,

⎡

λ1

⎢0 and K (t) ∈ SO(n) and Xd = −i ⎢ ⎣ ..

0

λ2 .. .

... ... .. .

0 0⎥ .⎥ ⎦. The elements

⎤

.. 0 0 . . . λn of the reachable set at time T take the form K1 exp(−iµT )K2 , where K1 , K2 ∈ SO(n), and µ ≺ λ (µ lies in convex hull of λ and its permutations), where λ = (λ1 , . . . , λn )′ . .

Proof. As a first step, discretize the evolution of P(t) as piecewise constant evolution, over steps of size τ . The total evolution after T = N τ is then PN =

∏ j

exp(Adkj (Xd )τ ),

(19)

4

N. Khaneja / Automatica 111 (2020) 108639

For t ∈ [(n − 1)τ , nτ ], choose small step ∆, such that t + ∆ < nτ , then P(t + ∆) = exp(Ad ⎡K (Xd )∆)P(t). ⎤ exp(iφ1 ) 0 0 0 exp(iφ2 ) 0 0 ⎢ 0 ⎥ ⎥ K2 , By KAK, P(t) = K1 ⎢ . ⎣ ⎦ . . 0 0 0 0 0 0 exp(iφn )







A

where K1 , K2 ∈ SO(n). To begin with, assume eigenvalues φj − φk ̸= nπ , where n is an integer. When we take a small step of size ∆, P(t) changes to P(t + ∆), as K1 , K2 , A change to K1 (t + ∆) = exp(Ω1 ∆)K1 , K2 (t + ∆) = exp(Ω2 ∆)K2 , A(t + ∆) = exp(a∆)A, where, Ω1 , Ω2 ∈ k and a ∈ a. Let Q (t + ∆) = K1 (t + ∆)A(t + ∆)K2 (t + ∆), which can be written as Q (t + ∆) = exp(Ω1 ∆)K1 exp(a∆)A exp(Ω2 ∆)K2 .

(20)

exp(Ω1 ∆) exp(K1 aK1 ∆) exp(K1 AΩ2 A K1 ∆)P(t). ′

AdK¯ (Xd ) = P(AdK¯ (Xd )) + AdK¯ (Xd )⊥ , where P denotes projection onto nk × nk blocks in Eq. (26), and AdK (Xd )⊥ the orthogonal complement. X11 ⎢X21

′

(21)

Observe P(t + ∆) = exp(AdK (Xd )∆)P(t).

P(⎢ ⎣ ..

.

.. .

Xn1

Xn2

(22)

AdK (Xd ) = Ω1 + K1 aK1 + K1 AΩ2 A K1 . ′

′

(23)

′

Multiplying both sides with K1 (·)K1 gives ′

(24)

where, K¯ = K1′ K and Ω1′ = K ′ Ω K . We evaluate AΩ2 A† , for Ω2 ∈ so(n). (25)

Rkl

⎡ λ1 ⎢ with P denoting the projection onto a ( a = −i ⎣ 0 0

... .. . 0

0

⎤

0 ⎦,

⎥

λn

⊥ where i λi = 0.) w.r.t to standard inner product and AdK¯ (Xd ) to the orthogonal component. In Eq. (25), φk − φl ̸ = 0, π , we can solve for (Ω2 )kl such that iR = AdK¯ (Xd )⊥ . This gives Ω2 . Let a = P(AdK¯ (Xd )) and choose Ω1′ = AdK¯ (Xd )⊥ − AΩ2 A† = −S ∈ k. With this choice of Ω1 , Ω2 and a, P(t + ∆) and Q (t + ∆) are matched to first order in ∆ and

∑

A=⎢ ⎣ ..

.

.. .

0

0

...

0 0⎥

⎤

.. ⎥ ⎦, .

(26)

An

[

Ir × r 0

Xnn

0

0

...

.. ⎥ ⎦, .

(28)

Xnn

Q (t + ∆) = exp(Ω1 ∆)K1 exp(P(AdK¯ (Xd )∆))A exp(Ω2 ∆)K2 .

(29)

where in Eq. (25) we can solve for (Ω2 )kl such that iR = AdK¯ (Xd )⊥ . This gives Ω2 . Choose, AdK¯ (Xd )⊥ − AΩ2 A† = Ω1′ ∈ k. This gives Ω1 = K1 Ω1′ K1′ . Again P(t + ∆) − Q (t + ∆) = o(∆2 ). We write

⎡ Θ1 ⎢0 H1 = ⎢ ⎣ .. .

... ... .. .

0

Θ2 .. .

0

...

0

0 0⎥

⎤

.. ⎥ ⎦, . Θn

(30)

where Θk is nk × nk sub-block in SO(nk ). H1 = exp(h1 ) is chosen such that

is a diagonal matrix. Let H2 = exp(A−1 h1 A), where h2 is skew

⎡ θ1 ⎢0 h1 = ⎢ ⎣ .. .

θ2 .. .

0

0

0

... ... .. . ...

⎡ θˆ1 ⎢0 ⎢ .. ⎥ . ⎦ , h2 = ⎢ ⎣ .. . θn 0 0 0⎥

⎤

0

θˆ2 .. . 0

... ... .. . ...

⎤

0 0⎥ ⎥

.. ⎥ , .⎦ θˆn

(31)

where

θk , θˆk is nk × nk sub-block in so(nk ), related by (see (27)) θˆk = A′k θk Ak , ⎤ ⎡ r ×r  [ ] ⎢ θ11 −θ12 θ12 ⎥ ⎥ , θˆk = θ11 (32) θk = ⎢ † † ⎣−θ12 θ22 ⎦ θ12 θ22  s×s

′

Note H1 P(Adk (Xd ))H1 = a lies in convex hull of eigenvalues of Xd . This is true if we look at the diagonal of H1′ AdK (Xd )H1 , it follows from Schur Convexity. The diagonal of H1′ Adk (Xd )⊥ H1 is zero as its inner product tr(a1 H1′ Adk (Xd )⊥ H1 ) = tr(H1 a1 H1′ Adk (Xd )⊥ )

†

K1 exp(P(AdK¯ (Xd )∆))H1 AH2 exp(Ω2 ∆)K2 . Q (t + ∆) =

where Ak is nk fold degenerate ( modulo sign) described by nk × nk block. WLOG, we arrange Ak = exp(iφk )

.. .

⎤

to Adk (Xd )⊥ . Therefore diagonal of H1′ P(Adk (Xd ))H1 is same as diagonal of H1′ AdK (Xd )H1 . † Now using H1 AH2 = A, from (29), we have Q (t + ∆) = exp(Ω1 ∆)×

Consider the case, when A is degenerate. Let,

... ... .. .

.

0 0 ⎥

= 0 as H1 a1 H1′ has block diagonal form which is perpendicular

P(t + ∆) − Q (t + ∆) = o(∆2 ).

0 A2

.

... ... .. .

h2

P(AdK¯ (Xd )) + AdK¯ (Xd )⊥ = Ω1′ + a + AΩ2 A′ ,

A1 ⎢0

0 X22

symmetric, such that

such that S is skew symmetric and R is traceless symmetric matrix with iR ∈ p. Note iR ⊥ a and onto a⊥ , by appropriate choice of Ω2 . Given AdK¯ (Xd ) ∈ p, we decompose it as AdK¯ (Xd ) =

⎡

...

⎡

  

{AΩ2 A† }kl = exp{i(φk − φl )}(Ω2 )kl = cos(φk − φl )(Ω2 )kl +i sin(φk − φl )(Ω2 )kl .       Skl

⎤

H1′ P(AdK¯ (Xd ))H1 = a

AdK¯ (Xd ) = Ω1 + a + AΩ2 A . ′

X1n X11 X2n ⎥ ⎢0 ⎢ .. ⎥ ⎦) = ⎣ ..

where Xij are blocks. Then we write

We equate P(t + ∆) and Q (t + ∆) to first order in ∆. This gives, ′

... ... .. .

X12 X22

⎡

Eq. (29) slightly differently. Let H1 be a rotation formed from block diagonal matrix

Q (t + ∆) = ′

Consider the decomposition

(33)

where the above expression can be written as

]

0 . −Is×s

†

exp(Ω1 ∆)K1 H1 exp(a∆)AH2 exp(Ω2 ∆)K2 ,

(27)

Q (t + ∆) = exp(Ω1 ∆) exp(K1 H1 aH1′ K1′ ∆) exp(K1 AΩ2 A′ K1′ ∆)P(t).

N. Khaneja / Automatica 111 (2020) 108639

5

where Ω1 , H1 , a, Ω2 , are chosen such that (Ω1 + K1 H1 aH1′ K1′ + K1 AΩ2 A′ K1′ ) = AdK (Xd ). (Ω1′ + H1 aH1′ + AΩ2 A′ ) = AdK¯ (Xd ). Q (t + ∆) − P(t + ∆) = o(∆2 )P(t). Q (t + ∆) = (I + o(∆2 ))P(t + ∆). Q (t + ∆)Q (t + ∆)T = (I + o(∆2 ))P(t + ∆)P T (t + ∆)(I + o(∆2 ))

= P(t + ∆)P T (t + ∆)[I + o(∆2 )].

Fig. 1. Figure A shows collection of overlapping neighborhoods forming the finite subcover. Figure B depicts Pi , Pi+1 , Qi+ , Qi− , Pi,i+1 as in proof of Theorem 1.

P(t + ∆)P T (t + ∆) exp(i2φ1 ) 0 ⎢

⎡ = K1 ⎢ ⎣

.. .

0 exp(i2φ2 )

.. .

0

... ... .. .

0 0

.. .

...

0

⎤ Then we get the following recursive relations.

⎥ T ⎥K . ⎦ 1

+ T λ(Qi+ QiT+ ) = exp(2a+ i ∆i ) λ(Pi Pi )

exp(i2φn )

2 λ(Pi,i+1 PiT,i+1 ) = λ(Qi+ QiT+ ) + o((∆+ i ) )

Let F = P(t +∆)P T (t +∆) and G = Q (t +∆)Q T (T +∆), we relate the eigenvalues of F and G. Given F , G as above, with⎡|F − G| ≤⎤ ϵ, exp(i2φ1 ) ⎢exp(i2φ2 )⎥ ⎥. and a ordered set of eigenvalues of F, denote λ(F ) = ⎢ . ⎣ ⎦

..

exp(i2φn ) There exists an ordering (correspondence) of eigenvalues of G such that |λ(F ) − λ(G)| < ϵ . Choose an ordering of λ(G) call µ that minimizes |λ(F ) − λ(G)|. F = U1 D(λ)U1′ and G = U2 D(µ)U2′ , where D(λ) is diagonal with diagonal as λ, let U = U1′ U2 ,

2 λ(Q(i+1)− Q(iT+1)− ) = λ(Pi,i+1 PiT,i+1 ) + o((∆− i+1 ) )

− T T exp(−2a− i+1 ∆i+1 ) λ(Pi+1 Pi+1 ) = λ(Q(i+1)− Q(i+1)− ) − where a+ i and ai+1 correspond to a in Eq. (33) and lie in the convex hull of the eigenvalues Xd . Adding the above equations, † + − − λ(Pi+1 PiT+1 ) = exp(o(∆2 )) exp(2(a+ i ∆i + ai+1 ∆i+1 )) λ(Pi Pi ). (34)

λ(Pn PnT ) = exp(

∑

o(∆2 )) exp(2





≤ϵ T

∑

+ − − T a+ i ∆i + ai+1 ∆i+1 ) λ(P1 P1 ).

i



(35)

′ 2

|F − G|2 = |D(λ) − UD(µ)U | = |λ|2 + |µ|2 − tr(D(λ)′ UD(µ)U ′ + (UD(µ)U)′ D(λ)).

where o(∆ ) in Eq. (34) is diagonal. λ 2

By Schur convexity, tr(D(λ) UD(µ)U + (UD(µ)U ) D(λ))



∑

∑

′

=

exp(

∑

′

′ ′

αi (λ′ Pi (µ) + Pi (µ)′ λ)

= exp( 

i

where Pi are permutations. Therefore |F − G|2 > |λ − µ|2 . Therefore, T

2

≤ϵ T

Pn = K1 exp(

exp((Ω1 + K1 H1 aH1′ K1′ + K1 AΩ2 A′ K1′ )∆) −





exp(Ω1 ∆) exp(K1 H1 aH1′ K1′ ∆) exp(K1 AΩ2 A′ K1′ ∆), is regulated by size of Ω2 , which is bounded by |Ω2 | ≤

∥Xd ∥ , sin(φi −φj )

where sin(φi − φj ) is smallest non-zero difference. ∆ is chosen small enough such that |o(∆2 )| < ϵ ∆. For each point t ∈ [0, T ], we choose a open nghd N(t) = (t − Nt , t + Nt ), such that ot (∆2 ) < ϵ ∆ for ∆ ∈ N(t). N(t) forms a cover of [0, T ]. We can choose a finite sub-cover centered at t1 , . . . , tn (see Fig. 1A). Consider trajectory at points P(t1 ), . . . , . . . P(tn ). Let ti,i+1 be the point in intersection of N(ti ) and N(ti+1 ). Let ∆+ = ti,i+1 − ti and ∆− i i+1 = ti+1 − ti,i+1 . We − consider points P(ti ), P(ti+1 ), P(ti,i+1 ), Q (ti + ∆+ i ), Q (ti+1 − ∆i+1 )

 as shown in Fig. 1B.



Qi+

 



Q(i+1)−

k

o(∆2 )) exp(2µT ) λ(P1 P1T ),  

The difference o(∆ ) =

exp(AdK (Xd )∆)

αk Pk (λ)) λ

=

(P1 P1T )

≤ϵ T

2



∑

where µ ≺ λ and P1 = I.

λ(QQ (t + ∆)) = λ(PP (t + ∆)) + o(∆ ). T

o(∆ )) exp(2T   2

(Pn PnT )



1∑ 2

o(∆2 )) exp(µT ) K2 .  

(36)

≤ϵ T

Note, |Pn − K1 exp(µT )K2 | = o(ϵ ). This implies that Pn belongs to the compact set K1 exp(µT )K2 , else it has minimum distance from this compact set and by making ∆ → 0 and hence ϵ → 0, we can make this arbitrarily small. In Eq. (19), Pn → P(T ) as τ → 0. Hence P(T ) belongs to compact set K1 exp(µT )K2 . □ Corollary 1. equation U˙ = (Xd +

Let U(t) ∈ SU(n) be a solution to the differential

∑

ui Xi )U ,

i

where ⎡ {Xi }LA , λ1 0 ⎢ 0 λ2 −i ⎢ . ⎣.

..

..

the Lie algebra generated by Xi , is so(n) and Xd = ⎤ ... 0 ... 0 ⎥ .⎥ . ⎦. The elements of reachable set at time T

..

.. 0 0 . . . λn take the form U(T ) ∈ K1 exp(−iµT )K2 , where K1 , K2 ∈ SO(n) and

6

N. Khaneja / Automatica 111 (2020) 108639

µ ≺ λ, where λ = (λ1 , . . . , λn )′ and the set S = K1 exp(−iµT )K2 belongs to the closure of reachable set. Proof. Let V (t) = K ′ (t)U(t), where, K˙ = (

i ui Xi )K . Then

∑

αi (ai , bi , ci ))K2 , αi > 0

i

From Theorem 1, we have V (T ) ∈ K1 exp(−iµT )K2 . Therefore U(T ) ∈ K1 exp(−iµT )K2 . Given U = K1 exp(−iµT )K2 = K1 exp(−i

∑

S = K1 exp(T

∑

V˙ (t) = AdK ′ (t) (Xd )V (t).

∑

by permuting and changing sign of any two by local unitary. Then U(T ) ∈ S where

αj Pj (λ)T )K2

αi = 1.

i

Furthermore S belongs to the closure of the reachable set. Alternate description of S is U = K1 exp(−i(α Ix Sx + β Iy Sy + γ Iz Sz ))K2 ,

α ≥ β ≥ |γ |, α ≤ ax T and α + β ± γ ≤ (ax + ay ± az )T .

j

= K1

∏

∑

exp(−itj Xd )Kj ,

Proof. Let V (t) = K ′ (t)U(t), where, K˙ = (−i

tj = T .

j

We can synthesize Kj in negligible time, therefore |U(T ) − U | < ϵ , for any desired ϵ . Hence U is in closure of reachable set. □

W = exp(−iπ Iy Sy ) exp(−i

π 2

Iz ) .

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

⎡

λ1

U = exp(Ω1 ) exp(−i ⎣ 0

⎢

0

... .. .

0

U = exp(Ω1 ) exp(−i(−

2

Sz +

exp(AdKi (−iXd )∆t)

where Ki ∈ SU(2) ⊗ SU(2) and Xd = ax Ix Sx + ay Iy Sy + az Iz Sz , where ax ≥ ay ≥ |az |. Then,

⎥

2

exp(AdWKi W ′ (−iWXd W ′ )∆t).

i

Observe WKi W ′ ∈ SO(4) and WXd W ′ = diag(λ1 , λ2 , . . . , λ4 ). Then using results from Theorem 1, we have WVW ′ = J1 exp(−iµ)J2 = J1 exp(−i

∑

αj Pj (λ))J2 ,

j

J1 , J2 ∈ SO(4), µ ≺ λT . Multiplying both sides with W ′ (·)W , V = K1 exp(T

∑

αi (ai , bi , ci ))K2 , αi > 0,

i

∑

αi = 1.

which can be written as,

λ4 ay

∏

i

V = K1 exp(−i(α Ix Sx + β Iy Sy + γ Iz Sz ))K2 ,

where Ω1 , Ω2 ∈ so(4). We write above as ax

∏ i

⎤

0 ⎦) exp(Ω2 ),

0

uj Xj )K . Then

Consider the product

WVW ′ =

The transformation maps the algebra k = su(2) × su(2) = {Iα , Sα } to k1 = so(4), four dimensional skew symmetric matrices, i.e., AdW (k) = k1 . The transformation maps p = {Iα Sβ } to p1 = −iA, where A is traceless symmetric and maps a = −i{Ix Sx , Iy Sy , Iz Sz } to a1 = −i{− S2z , I2z , Iz Sz }, space of diagonal matrices in p1 , such that ax Ix Sx + ay Iy Sy + az Iz Sz gets mapped to the four vector (the diagonal) (λ1 , λ2 , λ3 , λ4 ) = (ay + az − ax , ax + ay − az , −(ax + ay + az ), ax + az − ay ).

j

V˙ (t) = AdK ′ (t) (−iXd )V (t).

V = Remark 2. We now show how Result 2 and Theorem 1 can be mapped to results on decomposition and reachable set for coupled spins/qubits. Consider the transformation

∑

α ≥ β ≥ |γ |, where using µ ≺ λT , we get,

Iz + az Iz Sz )) exp(Ω2 ),

Multiplying both sides with W (.)W gives ′

W ′ UW = K1 exp(−iax Ix Sx + ay Iy Sy + az Iz Sz )K2 , where K1 , K2 ∈ SU(2) × SU(2) local unitaries and we can rotate to ax ≥ ay ≥ |az |. Corollary 3 (Digonalization). Given −iHc = −i exists a local unitary K such that

∑

αβ Jαβ Iα Sβ ,

there

K (−iHc )K ′ = −i(ax Ix Sx + ay Iy Sy + az Iz Sz ), ax ≥ ay ≥ |az |. Note W (−iHc )W ′ ∈ p1 . Then choose Θ ∈ SO(n) such that a Θ W (−iHc )W ′ Θ ′ = −i(− a2x Sz + 2y Iz + az Iz Sz ) and hence (W ′ exp(Ω )W )(−iHc )(W exp(Ω )W ′ )′ = −i(ax Ix Sx +ay Iy Sy +az Iz Sz ). Where K = W ′ exp(Ω )W is a local unitary. We can rotate to ensure ax ≥ ay ≥ |az |. Corollary 4. Given the evolution of ∑ coupled qubits U˙ = −i(Hc + ∑ j uj Hj )U, we can diagonalize Hc = αβ Jαβ Iα Sβ by local unitary Xd = K ′ Hc K = ax Ix Sx + ay Iy Sy + az Iz Sz , ax ≥ ay ≥ |az |, which we write as triple (ax , ay , az ). From this, there are 24 triples obtained

α + β − γ ≤ (ax + ay − az )T ,

(37)

α ≤ ax T , α + β + γ ≤ (ax + ay + az )T .

(38)

Furthermore U = KV .

(39)

□

3. Time optimal control for G /K problem 3.1. Stabilizer Let g = p ⊕ k be Cartan decomposition of real semisimple Lie algebra g, and a ∈ p be its Cartan subalgebra. Let a ∈ a. ad2a : p → p is symmetric in basis orthonormal w.r.t to the killing form. We can diagonalize ad2a . Let Yi be eigenvectors with nonzero [a , Y ] (negative) eigenvalues −λ2i . Let Xi = λ i , λi > 0. i

ada (Yi ) = λi Xi , ada (Xi ) = −λi Yi . Xi are independent, as αi Xi = 0, implies − αi λi Yi = 0. Since Yi are independent, Xi are independent. Given X ⊥ Xi , then [a, X ] = 0, otherwise we ∑ can decompose it in eigenvectors of ad2a , ∑ i.e., [a, X ] = α a + i i i j βj Yj , where ai are zero eigenvectors of ad2a . Since 0 = ⟨X [a[a, X ]⟩ = −∥[a, X ]∥2 , which means [a, X ] = 0. This is a contradiction. Yi are orthogonal, implies Xi

∑

∑

N. Khaneja / Automatica 111 (2020) 108639

are orthogonal, ⟨[a, Yi ][a, Yj ]⟩ = ⟨[a, [a, Yi ]Yj ⟩ = λ2i ⟨Yi Yj ⟩ = 0. Let k0 ∈ k satisfy [a, k0 ] = 0. Then k0 = {Xi }⊥ . Y˜i denote eigenvectors that have λi as non-zero integral multiples of π . X˜ i are ada related to Y˜i . We now reserve Yi for non zero eigenvectors that are not integral multiples of π . Let

f = {ai } ⊕ Y˜i ,

h = k0 ⊕ X˜ i ,

AkA−1 = A(

∑

αi Xi +

i

where k ∈ k. AkA

∑

αl X˜ l +

∑

l

−1

αj kj )A− ,

j

=

αi [cos(λi )Xi − sin(λi )Yi ] +

∑

i

±αl X˜ l +

l

∑

αj kj .

j

The range of A(·)A−1 in p, is perpendicular to f. Given Y ∈ p such that Y ∈ f⊥ . The norm ∥X ∥ of X ∈ k, such that p part of AXA−1 |p = Y satisfies

∥Y ∥ . sin λs

∥X ∥ ≤

where λ is the smallest nonzero eigenvalue of − such that λs is not an integral multiple of π . A2 kA−2 stabilizes h ∈ k and f ∈ p. If k ∈ k is stabilized by 2 A (·)A−2 , λi = nπ , i.e., k ∈ h. This means h is an sub-algebra, as the Lie bracket of [y, z ] ∈ k for y, z ∈ h is stabilized by A2 (·)A−2 . ˜ ∈ K be the Let H = exp(h) be an integral manifold of h. Let H ˜ −2 = H˜ or A2 H˜ − HA ˜ −2 = 0. H˜ is closed, H ∈ H. ˜ solution to A2 HA ˜ is a manifold. Given element H0 ∈ H˜ ∈ K , where We show that H K is closed, we have a exp(Bkδ ) nghd of H0 , in exp(Bδ ) ball nghd of H0 , which is one to one. For x ∈ Bkδ , A2 exp(x)A−2 = exp(x), implies,

∑

ad2a

α i Xi +

i

= exp(

∑

⟨a, exp(h0 )z exp(−h0 )⟩ = ⟨exp(−h0 )a exp(h0 ), z ⟩ = 0, as exp(−h0 )a exp(h0 ) is AdA2 invariant, hence exp(−h0 )a exp(h0 ) ∈ f. In above, we worked with killing form. For g = su(n), we may use standard inner product.

Result 3 (Kostant Convexity (Kostant, 1973)). Given the decomposition g = p ⊕ k, let a ⊂ p and X ∈ a. Let Wi ∈ exp(k) such that Wi X Wi ∈ a are distinct Weyl points. Then projection (w.r.t killing form) of AdK (X ) on a lies in convex hull of these Weyl points. The C be the convex hull, and let projection P(AdK (X )) lie outside this Hull. Then there is a separating hyperplane a, such that ⟨AdK (X ), a⟩ < ⟨C , a⟩. W.L.O.G, we can take a to be a regular element. We minimize ⟨AdK (X ), a⟩, with choice of K , and find that minimum happens when [Ad ∑K (X ), a] =−10, i.e. AdK (X ) is a Weyl point. Hence P(Ad (X )) ∈ K i αi Wi X Wi , for αi > 0 and ∑ i αi = 1. The result is true with a projection w.r.t inner product that satisfies ⟨x, [y, z ]⟩ = ⟨[x, y], z ]⟩, like standard inner product on g = su(n).

(40)

2 s

A2 exp(

exp(h0 )z exp(−h0 ) ∈ a⊥ .

3.2. Kostant convexity

X˜i , Xl , kj where kj forms a basis of k0 , forms a basis of k. Let A = exp(a).

∑

7

∑

βl X˜ l +

l

∑

γj kj )H0 A−2

j

αi cos(2λi )Xi − sin(2λi )Yi

Theorem 2. Given a compact Lie group G and its Lie algebra g, consider the Cartan decomposition of a real semisimple Lie algebra g = p ⊕ k. Given the control system X˙ = AdK (t) (Xd )X , P(0) = 1 , where Xd ∈ a, the Cartan subalgebra a ∈ p and K (t) ∈ exp k, a closed subgroup of G. The end point P(T ) = K1 exp(T

∑

αi Wi (Xd ))K2 ,

i

where K1 , K2 ∈ exp(k) and Wi (Xd ) ∈ a are Weyl points, αi > 0 and ∑ i αi = 1. Proof. As in proof of Theorem 1, we define P(t + ∆) = exp(AdK (Xd )∆)P(t) = exp(AdK (Xd )∆)K1 exp(a)K2 , and show that

i

+

∑

βl X˜ l +

l

= exp(

∑

exp(AdK (Xd )∆)K1 AK2 = Ka exp(a0 ∆ + C ∆2 )AKb

γj kj )H0

= Ka exp(a + a0 ∆ + C ∆2 )Kb ,

j

∑ i

αi Xi +

∑ l

βl X˜ l +

∑

γj kj )H0 ,

j

then by one to one property of exp(Bδ ), we get αi = 0 and x ∈ h. h Therefore exp(Bδ )H0 is a nghd of H0 . Given a sequence Hi ∈ exp(h) converging to H0 , for n large h enough, Hn ∈ exp(Bδ )H0 . Then H0 is in invariant manifold exp(h). Hence exp(h) is closed and hence compact. Let y ∈ f, then there exists a h0 ∈ h such that exp(h0 )y exp (−h0 ) ∈ a. We maximize the function ⟨ar , exp(h)y exp(h)⟩ over the compact group exp(h), for regular element ar ∈ a and ⟨., .⟩ is the killing form. At the maxima, we have at t = 0, d ⟨ar , exp(h1 t)(exp(h0 )y exp(−h0 )) exp(−h1 t)⟩ = 0. dt

⟨ar , [h1 exp(h0 )y exp(−h0 )]⟩ = −⟨h1 , [ar exp(h0 )y exp(−h0 )]⟩, if exp(h0 )y exp(−h0 ) ̸ = a, then [ar , exp(h0 )y exp(−h0 )] ∈ k. The bracket [ar , exp(h0 )y exp(−h0 )] is AdA2 invariant and hence belongs to h. We can choose h1 so that gradient is not zero. Hence exp(h0 )y exp(−h0 ) ∈ a. For z ∈ p such that z ∈ f⊥ , we have

(41)

where for K¯ = K1−1 K , AdK¯ (Xd ) = P(AdK¯ (Xd )) +AdK¯ (Xd )⊥ .







a0

where P is projection w.r.t killing form and a0 ∈ f, the centralizer in p as defined in remark 3.1, C ∆2 ∈ f is a second order term that can be made small by choosing ∆. Ka , Kb ∈ exp(k). To show Eq. (41), we show there exist K1′′ , K2′′ ∈ K such that exp(k′′1 ) exp(AdK¯ (Xd )∆) exp(Ak′′2 A−1 )

  



K1′′

 K2′′



= exp(a0 ∆ + C ∆2 ),

(42)

where K1′′ and K2′′ are constructed by an iterative procedure as described in the proof below. Given X and Y as N × N matrices, considered elements of a matrix Lie algebra g, we have log(eX eY ) − (X + Y ) =

∑ (−1)n−1 ∑ n>0

n

[X r1 Y s1 . . . X rn Y sn ] , i=1 (ri + si )r1 !s1 ! . . . rn !sn !

∑n 1≤i≤n

(43)

8

N. Khaneja / Automatica 111 (2020) 108639

where ri + si > 0. Denote by |X |0 , largest element (absolute value) of X . We bound |log(eX eY ) − (X + Y )|0 . Given |X |0 < ∆ and |Y |0 < b0 ∆k , where k ≥ 1, ∆ < 1, b0 ∆ < 1, 2N ∆ < 1, |log(eX eY ) − (X + Y )|0 ≤

∑

Nb0 e∆

n=1

k+1

+

∑ 1 (2Ne2 )n b0 ∆n+k−1

n>1 2 2

n

n

≤ Nb0 e∆k+1 + (Ne ) b0 ∆k+1 (1 + 2Ne2 ∆ + · · ·) (Ne2 )2 b0 ∆k+1 ˜ 0 ∆k+1 , ≤ Mb ≤ Nb0 e∆k+1 + 1 − 2Ne2 ∆ ˜ ∆ < 1. where by choosing small ∆, we have M Given decomposition of g = p ⊕ k, p ⊥ k with respect to the negative definite killing form B(X , Y ) = tr(adX adY ). Furthermore there is decomposition of p = a ⊕ a⊥ . Given U0 = exp(a0 ∆ + b0 ∆ + c0 ∆), where a0 ∈ a, b0 ∈ a⊥ and c0 ∈ k, such that |a0 |0 +|b0 |0 +|c0 |0 < 1, which we just abbreviate as a0 + b0 + c0 < 1 (we follow this convention below) We describe an iterative procedure Uk = exp(−ck ∆) Uk−1 exp(−bk ∆),

(44)

where ck ∈ k and bk ∈ a⊥ , such that the limit n → ∞ Un = exp(a0 ∆ + C ∆2 ),

(45)

AdK¯ (Xd ) = P(AdK¯ (Xd )) + AdK¯ (Xd )⊥ ,



 a0



   b0

⊥

where a0 ∈ f and b0 ∈ f , consider again the iterations, U0 = exp(−¯c0 ∆) exp(a0 ∆ + b0 ∆) exp(−b0 ∆ + c¯0 ∆) = exp(−¯c0 ∆) exp(a0 ∆ + c¯0 ∆ + b′0 ∆2 ) = exp(a0 ∆ + b′0 ∆2 + c0′ ∆2 ) = exp(a1 ∆ + b1 ∆ + c1 ∆) We refer to Eq. (40). Given b0 ∆ ∈ p such that b0 ∆ ∈ f⊥ . If Ak′ A′ = −b0 ∆ + c¯0 ∆, then ∥k′ ∥ ≤ h∥b0 ∆∥ (killing norm). c¯0 ∈ k, is bounded c¯0 ≤ Mhb0 , where M as before converts between two different norms. Using bounds derived above b′0 ≤ ˜ ˜ ˜ M(Mh + 1)b0 , and c0′ ≤ MMhb 0 , 2M(Mh + 1)∆ < 1, we obtain ˜ a0 + b′0 ∆ + c¯0 ≤ a0 + b0 (M(Mh + 1)∆ + Mh) ≤ 1. For appropriate M ′ , we have a1 ≤ a0 + b1 ≤

M

M′ 3

(b0 + c0 )∆

′

3 M′

(b0 + c0 )∆

(b0 + c0 )∆ 3 we obtain c1 ≤

a1 + b1 + c1 ≤ a0 + M ′ (b0 + c0 )∆ ≤ a0 + b0 + c0 where ∆ is chosen small. U1 =

where a0 , C ∈ a. U1 =

exp(−(c1 + c¯1 )∆) exp(a1 ∆ + b1 ∆ + c1 ∆)

exp(−c0 ∆) exp(a0 ∆ + b0 ∆ + c0 ∆) exp(−b0 ∆)

exp(−b1 ∆ + c¯1 ∆) =

= exp(a0 ∆ + b0 ∆ + c0′ ∆2 ) exp(−b0 ∆) = exp(a0 ∆ + b′0 ∆2 + c0′ ∆2 ) = exp((a1 + b1 + c1 )∆)

exp(−(c1 + c¯1 )∆) exp(a1 ∆ + (c1 + c¯1 )∆ + b′1 ∆2 )

Note b′0 and c0′ are elements of g and need not be contained in a⊥ and k. ˜ 0 , which gives a0 + b0 + c ′ ∆ ≤ Where, using bound in c0′ ≤ Mc 0 ˜ 0 . We a0 + b0 + c0 . Using the bound again, we obtain, b′0 ≤ Mb can decompose, (b′0 + c0′ )∆, into subspaces a′′0 + b1 + c1 , where a′′0 ≤ M(b′0 + c0′ )∆, b1 ≤ M(b′0 + c0′ )∆ and c1 ≤ M(b′0 + c0′ )∆, where −B(X , X ) ≤ λmax |X |2 , where |X | is Frobenius norm and −B(X , X ) ≥ λmin |X |2 . Let M = Nλλmax . min

This gives, a′′0 ≤ M(b′0 + c0′ )∆, b1 ≤ M(b′0 + c0′ )∆ and c1 ≤ M(b′0 + c0′ )∆. This gives

˜ a1 ≤ a0 + MM(b 0 + c0 )∆

= exp(a1 ∆ + b′1 ∆2 + c1′ ∆2 ) = exp(a2 ∆ + b2 ∆ + c2 ∆) where c¯1 ∈ k, such that c¯1 ≤ Mhb1 . ˜ Where, using bounds derived above b′1 ≤ M(Mh + 1)b1 , and ′ ˜ ˜ c1 ≤ M(Mhb + c ). Using the bound 2 M(Mh + 1) ∆ < 1, we 1 1 obtain a1 + b′1 ∆ + (c1 + c¯1 ) ≤ a1 + ((1 + Mh)b1 + c1 ) ≤ a0 + b0 + c0 . We can decompose, (b′1 + c1′ )∆2 , into subspaces (a′′1 + b2 + c2 )∆, where a′′1 ≤ M(b′1 + c1′ )∆, b2 ≤ M(b′1 + c1′ )∆ and c2 ≤ M(b′1 + c1′ )∆, where M as before converts between two different norms. This gives

˜ 2 h(b1 + c1 )∆ a2 ≤ a1 + 4MM ˜ 2 h(b1 + c1 )∆ b2 ≤ 4MM ˜ 2 h(b1 + c1 )∆ c2 ≤ 4MM

˜ b1 ≤ MM(b 0 + c0 )∆ ˜ c1 ≤ MM(b 0 + c0 )∆ ˜ ∆ < 1, we have, a1 + b1 + c1 ≤ a0 + b0 + c0 . Continuing For 4MM ˜ ∆(bk−1 + ck−1 ) ≤ (2MM ˜ ∆)k (b0 + c0 ). and using (bk + ck ) ≤ 2MM Similarly, ˜ ∆) (b0 + c0 ). |ak − ak−1 |0 ≤ (2MM k

Note, (ak , bk , ck ) is a Cauchy sequence which converges to (a∞ , 0, 0), where

|a∞ − a0 |0 ≤ (b0 + c0 )

Writing

∞ ∑

˜ ∆(b0 + c0 ) 2M M

k=1

˜ ∆ 1 − 2MM

˜ ∆)k ≤ (2MM

≤ C ∆,

˜ where C = 4MM(b 0 + c0 ). The above exercise was illustrative. Now we use an iterative procedure as above to show Eq. (42).

˜ 2 h∆ < 2 , we have, a2 + b2 + c2 ≤ a1 + (b1 + c1 ) ≤ For x = 8MM 3 a0 + b0 + c0 , Using (bk + ck ) ≤ x(bk−1 + ck−1 ) ≤ xk (b0 + c0 ). Similarly, |ak − ak−1 |0 ≤ xk (b0 + c0 ). Note, (ak , bk , ck ) is a Cauchy sequence which converges to (a∞ , 0, 0), where

|a∞ − a0 |0 ≤ x(b0 + c0 )

∞ ∑ k=0

xk ≤

x(b0 + c0 ) 1−x

≤ C ∆,

˜ 2 h(b0 + c0 ). where C = 16MM The above iterative procedure generates k′1 and k′′2 in Eq. (42), such that exp((K1′ AdK (Xd )K1 )∆) = exp(−k′′1 ) exp(a0 ∆ + C ∆2 ) exp(−Ak′′2 A′ ).

N. Khaneja / Automatica 111 (2020) 108639

where a0 ∆ + C ∆2 ∈ f. By using a stabilizer H1 , H2 , we can rotate them to a such that exp(AdK (Xd )∆)K1 AK2 = Ka H1 exp(a′0 ∆ + C ′ ∆2 )AH2 Kb ,

P(H1−1 a0 H1 ) =

2

∑

′

′

= P (W (a(i1 +1)− ))∆i−+1 + P (W (a2(i+1)− ))(∆i−+1 )2 ∑ = αk Wk (Xd )∆i−+1 + o((∆i−+1 )2 ), k

such that H1 (a0 ∆ + C ∆ )H1 = a0 ∆ + C ∆ is in a and a0 = P(H1−1 a0 H1 ) is projection onto a such that −1

2

′

αk Wk (Xd ).

where, ai , ai1 , ai2 ∈ a. Using Lemmas 1 and 2 , we can express Pn (T ) = K1 exp(an ) exp K2 =

k

This follows because the orthogonal part of AdK¯ (Xd ) to f written as AdK¯ (Xd )⊥ remains orthogonal of f. ⟨H −1 AdK (Xd )⊥ H , a⟩ =

⟨AdK (Xd )⊥ , H aH −1 ⟩ = ⟨AdK (Xd )⊥ , a′′ ⟩ = 0

Ka exp(a + a′0 ∆ + C ′ ∆2 )Kb . Given P = K1 exp(a + a1 ∆) K2 = K3 exp(b − b1 ∆) K4 ,





A1



where a, b, a1 , b1 ∈ a. We can express





A2

exp(b) = Ka exp(a + a1 ∆ + W (b1 )∆)Kb , where W (b1 ) is Weyl element of b1 . Furthermore exp(b + b2 ∆) = Ka′′ exp(a + a1 ∆ + W (b1 )∆ + W (b2 )∆)Kb′′ . Proof. Note, A2 = K3−1 PK4−1 , commutes with b1 . This implies A2 = K˜ exp(a + a1 ∆)K commutes with b1 . This implies 1 ˜ ˜′ A2 b1 A− 2 = b1 , i.e., K exp(a + a1 ∆)AdK (b1 ) exp(−(a + a1 ∆))K = b1 , which implies that AdK (b1 ) ∈ f. Recall, from Section 3.1, exp(a + a1 ∆)AdK (b1 ) exp(−(a + a1 ∆))

∑

∑

W (ai )∆i + W (ai+1 )∆i+1 ) exp( +

+

−

−

i

∑

o(∆2 ))K2 .





≤ϵ T



Pn (T ) = K1 exp(T

∑

αi Wi (Xd ))K2 .

i

Hence the proof of Theorem 2. □



=

K1 exp(

Letting ϵ go to 0, we have

(a′′ ∈ f), remains ∑ orthogonal to a. Therefore P(H1−1 a0 H1 ) = −1 P(H1 AdK¯ (Xd )H1 ) = k αk Wk (Xd ). exp(AdK (Xd )∆)K1 AK2 =

Lemma 1.

9

ck (Yk cos(λk ) + Xk sin(λk )).

Corollary 5. Given U in compact Lie group G, with Xd , Xj in its Lie algebra g. Given the Cartan decomposition g = p ⊕ k, where Xd ∈ a ⊂ p U˙ = (Xd +

∑

uj (t)Xj )U , U(0) = 1 ,

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j

and {Xj }LA = k. The elements of the reachable set at time T , take the form U(T ) ∈ S = K1 exp(T

∑

αk Wk Xd Wk−1 )K2 ,

k

where Wk are Weyl elements and K1 , K2 , Wk ∈ exp(k). S belongs to the closure of reachable set.

exp(2(a + a1 ∆))AdK (b1 ) exp(−2(a + a1 ∆)) = AdK (b1 ).

Theorem 3 (Co-ordinate Theorem). Let g = p ⊕ k be a Cartan decomposition with Cartan subalgebra a ∈ p. Let a ∈ a be a regular element such that f = a. Given ad2a : p → p symmetric. Let Yi be the [a , Y ] eigenvectors of ad2a that are orthogonal to a = {Zj }. Let Xi = λ i ,

We have shown existence of H1 such that H1 AdK (b1 )H1−1 ∈ a, using H1 , H2 as before,

λi > 0, where −λ2i is an eigenvalue of ad2a . Then k = {Xi }+k0 = {Xk }, where [a, k0 ] = 0 and Xi ⊥ k0 .

k

This implies

∑

k ck

sin(λk )Xk = 0, implying λk = nπ . Therefore,

i

K˜ exp(a + a1 ∆)K exp(b1 ∆)

ada (Yi ) = λi Xi , ada (Xi ) = −λi Yi

= K˜ H2 exp(a + a1 ∆)H1 exp(AdK (b1 )∆)K = Ka exp(a + a1 ∆ + W (b1 )∆)Kb .

AXi A−1 = cos(λi )Xi − sin(λi )Yi ,

Applying the theorem again to

where A = exp(a). Given U = K1 exp(a)K2 , consider the map U(ai , bj , ck ) =

Ka exp(a + a1 ∆ + W (b1 )∆)Kb exp(b2 ∆)

= Ka′′ exp(a + a1 ∆ + W (b1 )∆ + W (b2 )∆)Kb′′ . K1i Ai K2i

K1i

i

exp(

)K2i ,

Lemma 2. Given Pi = = exp(a we have Pi,i+1 = − − + exp(Hi+ ∆+ )P , and P = exp( − H ∆ )P = i i , i + 1 i + 1 , where Hi i i+1 i+1 AdKi (Xd ). From above we can express

∑

ck Xk ) K1 exp(

k

∑

bj Zj ) ×

j

∑

exp(a) exp(

ai Xi )K2 ,

i

∂U | = ∂ ai (0,0,0)

Pi,i+1 = Kai+ exp(ai + ai1+ ∆i+ + ai2+ (∆i+ )2 )Kbi+ .

such that U(0, 0, 0) = U.

where ai1+ and ai2+ are first and second order increments to ai in the positive direction. The remaining notation is self explanatory.

(cos(λi )AdK1 (Xi ) − sin(λi )AdK1 (Yi )) U ∂U |(0,0,0) = AdK1 (Zj ) U ∂ bj ∂U |(0,0,0) = Xk U . ∂ ck

(i+1)−

Pi,i+1 = Ka(i+1)− exp(ai+1 − a1

(i+1)−

(∆i−+1 )2 )Kb

(i+1)−

∆i−+1

∆i−+1 − a2

(i+1)−

exp(ai+1 ) = K1 exp(ai + ai1+ ∆i+ + ai2+ (∆i+ )2 + W (a1

+ a(i2 +1)− (∆i−+1 )2 ))K2 . (i+1)−

W (a1

(i+1)−

∆i−+1 + a2

(∆i−+1 )2 )

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.

Yi , Zj span p, AdK1 (Yi ), AdK1 (Zj ), span p. AdK1 (Yi ), AdK1 (Zj ), Xk span p ⊕ k. cos(λi )AdK1 (Xi ) − sin(λi )AdK1 (Yi ), AdK1 (Zj ) and Xk , span p ⊕ k. By inverse function theorem U(ai , bj , ck ) is a nghd of U, any curve U(t) passing through U, at t = 0, for t ∈ (−δ, δ ) can be written

10

N. Khaneja / Automatica 111 (2020) 108639

Theorem 4 (Reflection). g = p ⊕ k, Let Yα ± iXα , where that Yα ∈ p and Xα ∈ k, are the roots, such that,

as U(t) = exp(

∑

ck (t)Xk )K1 exp(

∑

k

exp(a) exp(

∑

bj (t)Zj )

[a, Yα ] = α (a)Xα ; [a, Xα ] = −α (a)Yα

j

ai (t)Xi )K2 = K1 (t)A(t)K2 (t).

[a, Yα + iXα ] = −iα (a)(Yα + iXα )

i

(ai , bj , ck ) are coordinates of nghd of U. Given U(0) = K1 exp(a)K2 such that a is regular (f = a, see ˙ remark 3.1), we can represent a curve U(t) = AdK (t) (Xd )U(t) passing through U(0) as U(t) = K1 (t)A(t)K2 (t), where K˙1 = Ω1 (t)K1 , ˙ K˙2 = Ω2 (t)K2 and A(t) = Ω (t)A(t) where Ω1 (t), Ω2 (t) ∈ k and Ω (t) ∈ a. Differentiating, we get

Note, [Xα , Yα ] ∈ a. Note, [Xα , Yα ] ∈ p, let a0 ∈ a be regular. ada0 ([Xα , Yα ]) =

[ada0 (Xα ), Yα ] + [Xα , ada0 (Yα )] = 0. Observe, exp(sXα )[Xα , Yα ] exp(−sXα ) =

β s2

which gives for K¯ = K1−1 K , and Ω1′ = K1−1 Ω1 K1 ∈ k,

[Xα , Yα ] + · · · 2 where β < 0. The above expression can be written as, exp(sXα ) [Xα , Yα ] exp(−sXα ) =

AdK¯ (Xd ) = Ω1′ + Ω + AΩ2 A−1 .

cos( |β|s)[Xα , Yα ] −

AdK (t) (Xd )U(t) = (Ω1 + K1 Ω K1−1 + K1 AΩ2 A−1 K1−1 )U(t),

[Xα , Yα ] + β sYα +

√

Using AΩ2 A−1 ⊥ a, we obtain Ω = P(AdK¯ (Xd )), projection of AdK¯ (Xd ) on a. A(t) evolves as this projection, which lies in convex hull of Weyl points of Xd by Kostant Convexity theorem.

By choosing, s =

√ √ |β| sin( |β|s)Yα .

√π , |β|

we have U [Xα , Yα ]U −1 =

exp(sXα )[Xα , Yα ] exp(−sXα ) = −[Xα , Yα ]. Given Z = c [Xα , Yα ] +

4. Roots and reflections

∑

αk Zk ,

k

Roots: Let g be real, compact, semisimple Lie algebra, with negative definite killing form ⟨., .⟩. Let Ei be basis of g, orthonormal, w.r.t to the killing form. adX is skew symmetric matrix, w.r.t to this basis. ⟨Ei , adX (Ej )⟩ = tr(adEi ad[X ,Ej ] ) = tr(adEi [adX adEj ]) = −⟨Ej , adX (Ei )⟩. where, we use, ad[X ,Y ] = [adX , adY ], which follows from Jacobi identity, [[x, y], z ] = [x[y, z ]] − [y[x, z ]], ad[X ,Y ] = [adX , adY ]. Let a ∈ a. Eigenvalues of A = ada , are imaginary (A is skew symmetric). Given, A(x + iy) = iλ(x + iy), let, x = xp + xk and y = yp + yk , be direct decomposition in p + k parts. A(xp + xk + i(yp + yk )) = iλ(xp + xk + i(yp + yk ));

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Axp = −λyk ; Ayk = λxp ;

(49)

Axk = −λyp ; Ayp = λxk .

(50)

A(xp + iyk ) = iλ(xp + iyk ); A(yp − ixk ) = iλ(yp − ixk )

(51)

Eigenvectors of A, have the form xp ± iyk , with conjugate eigenvalues. Choose a basis for a as ai , with Ai = adai . Since Ai , commute, we can simultaneously diagonalize them. The eigenvectors take the form xp + iyk , which we abbreviate as p + ik and call them roots. We also use the notation k + ip for roots, obtained by multiplication by i. p + ik and p − ik have conjugate eigenvalues. p + ik and p′ + ik′ corresponding to distinct eigenvalue (for any of Ai ) are orthogonal. There are also trivial roots. In p they are just basis of a and in k, we call them k0 . Given a root vector e = p + ik, (with p, k normalized to killing norm 1) its value α defined as [a, p + ik] = −iα (a)(p + ik), can be read by taking inner product with vector [p, k] ∈ a.

⟨a, [p, k]⟩ = ⟨[a, p], k⟩ = α. We represent the root by its representative vector e = [p, k] ∈ a. Choose a basis for the roots ek . We can express all roots in terms of ek as coefficients (c1 , . . . , ck , . . . , cn ). The ones with positive leading non-zero entry are called positive and vice versa.

where ⟨[Xα , Yα ], Zk ⟩ = 0. This implies that [Zk , Xα ] = 0, else [Zk , Xα ] = α (Zk )Yα . Since

⟨Yα , [Zk , Xα ]⟩ = ⟨Zk , [Xα , Yα ]⟩ = 0, implying α (Zk ) = 0. This implies for U = exp( √π|β| Xα ) UZU −1 = −c [Xα , Yα ] +

∑

αk Zk .

k

This is reflection in the plane given by α (.) = ⟨[Xα , Yα ], .⟩ = 0. In orthonormal basis Ei (for a), Z , [Xα , Yα ] and Zk , take ∑the form of coordinates, z, m, zk , respectively, where, z = c m + k zk , and zk ⊥ m, the reflection formula takes the form Rm (z) = z − 2

∑ ⟨m, z⟩ m = −c m + zk . ⟨m, m⟩ k

Remark 3. When a is one-dimensional in Theorem 2, we can choose U as in Theorem 4, such that UXd U −1 = −Xd . Let X (T ) ∈ KUF , belong to coset of UF , where X˙ = AdK (Xd )X . Let the length L(X (t)) = β T , where β = |AdK (Xd )|. Form of geodesics say that we have for l ≤ β T such that exp(Yl) ∈ KUF , where |Y | = 1. Therefore, exp((β Y ) βl ) ∈ KUF . Let AdK (Xd ) = β Y , by appropriate choice of K . This is achieved by Maximization of ⟨AdK (Xd ), Y ⟩, w.r.t K , which yields [AdK (Xd ), Y ] = 0. This, gives AdK (Xd ) = ±β Y . We can choose either, by the choice of U. Therefore, X (T ) = K1 exp(tXd )K2 , where t = βl ≤ T . For t < T , we can use U, to insure UF = K1 exp(T (α Xd + (1 − α )UXd U −1 ))K2 . We get the form of the reachable set in Theorem 2, by a geodesic argument.

Result 4. Let e+ = kj + ipj , be positive roots. The roots dij vide a, into connected regions called Weyl chambers defined by sign(e+ j (x)) = ±1, where the signs do not change over a connected region. On the boundary of a Weyl chamber, some of + e+ j (x) = 0. Principal Weyl chamber c is defined as ej (x) > 0. There exists a basis B for the a such that all positive roots can ∑ be expressed as f = αj ej where ej ∈ B and αj > 0 are integers. We call B fundamental roots. They make obtuse angle among themselves. Principal Weyl chamber c is then defined as ei > 0, for ei ∈ B. By a sequence of reflections sj , around roots e+ j , we can map c to any Weyl chamber transitively, such that if

N. Khaneja / Automatica 111 (2020) 108639

x is an interior point, then after reflections it stays an interior point. Boundary points go to boundary points. For y ∈ c, if W (y) ∈ c, then W is identity, i.e. Weyl rotations act simple on Weyl chambers (Helgason, 1978). Simple action entails that any W can be written as product of finite reflections. Theorem 5. Let g = p ⊕ k be simple algebra (no ideals). Given any a1 ∈ a, We show a is spanned by Weyl points, Wi (a1 ) = Adki (a1 ). Consider the reflection around the root e1 , where e1 is independent of a1 , this gives, a1 → a1 − 2⟨a1 , e1 ⟩e1 = a2 . a2 is independent of a1 . Let ek be independent of the generated vectors a1 , . . . , ak , and not perpendicular to these, then reflecting these around ek produces ak+1 , which is independent of these. If no such ek can be found beyond k − 1, chain terminates. Then we can divide the root vectors into two categories, R1 = {e1 , . . . , eq } ∈ span{a1 , . . . , ak } = a1 and R2 = {eq+1 , . . . , eN } ⊥ span{a1 , . . . , ak } = a2 . Given e ∈ R1 and f ∈ R2 , then e ⊥ f . [e, f ], if not zero, is e + f , has non-vanishing inner product with e and f . Then [e, f ] is root that is neither parallel or perpendicular to span{ak }, therefore [e, f ] = 0. This divides nontrivial roots into two commuting sets R1 and R2 . Let k1 , and k2 be the k part of the roots k + ip comprising R1 and R2 . Similarly p1 and p2 . [a1 , k1 ] ∈ p1 , [a1 , p1 ] ∈ k1 , [a1 , k2 ] = 0 and [a1 , p2 ] = 0. Similarly, for a2 . [k1 , p1 ] ∈ p1 ⊕ a1 , [k2 , p2 ] ∈ p2 ⊕ a2 . Follows from a2 and p2 commute with k1 and p1 and vice versa. Then for k1 ∈ k1 and k2 ∈ k2 , [k1 , k2 ] = 0. This follows from [k1 + ip1 , k2 ± ip2 ] = 0. Similarly [p1 , p2 ] = 0 and [p1 , k2 ] = 0, [k1 , p2 ] = 0, [k1 , k1 ] ⊥ k2 , [k2 , k2 ] ⊥ k1 , [p1 , p1 ] ⊥ k2 , [p2 , p2 ] ⊥ k1 . Let k0 be trivial roots in k, i.e., [a, k0 ] = 0. [k0 , k1 ] ∈ k1 , [k0 , p1 ] ∈ p1 . Similarly for k2 , p2 . Let B1 ∈ k0 be generated by k1 orthogonal part of [k1 , k1 ] and [p1 , p1 ]. Let B2 ∈ k0 be generated by k2 orthogonal part of [k2 , k2 ] and [p2 , p2 ]. [B1 , a1 ] = 0, [B1 , k1 ] ∈ k1 and [B1 , p1 ] ∈ p1 , [B1 , B1 ] ∈ k˜1 , where, k˜1 = k1 ⊕ B1 and k˜2 = k2 ⊕ B2 . Let B3 be part of k0 , orthogonal to B1 and B2 . Note, [k˜i , k˜i ] ∈ k˜i . k˜1 ⊥ k˜2 . [B3 , k˜i ] ∈ k˜i . Then I1 = a1 ⊕ p1 ⊕ k˜1 and I2 = a2 ⊕ p2 ⊕ k˜2 , are non-trivial ideals. Therefore k = n, i.e., span{a1 , . . . , an } = a. We show positive span of {a1 , . . . , an } = a. Consider the convex hull of C = Wi a1 , where Wi = Adki . Suppose origin is not in the convex hull. By Hahn Banach theorem we can find a separating ∑ Hyperplane such that ⟨c , x⟩ = ci xi > 0 for all Weyl points of a1 . We can write ∑ the hyperplane ∑ in terms of n independent root vectors as bj sj . Let y be chosen such j bj ⟨sj , x⟩, c = that sign⟨sj , y⟩ = −sign(bj ). By reflecting around plane sj , if sign(⟨sj , x⟩) ̸ = sign(⟨sj , y⟩), we decrease the distance between a1 and y (this∑ is same idea as in Result 4). In finite steps ∑ bj ⟨sj , x⟩ ≤ 0. Therefore j bj ⟨sj , x⟩ ≤ 0. Therefore 0 ∈ C , i.e., j αj Adkj a1 = 0, i.e., −a1 =

∑

a1 we can produce i αi W1 (a1 ) = −a1 leaving ai , i ̸ = 1 invariant. We can synthesize a convex combination that synthesizes ±a1 . Using the construction detailed before in Theorem 5, we can synthesize V1 . Similarly we can synthesize all Vj , and hence any V . Let ki and pi be the subspace formed from the k and p parts of the roots in Vi . Then [ki , pj ] = 0, [ki , aj ] = 0, [ki , pj ] = 0, where i ̸ = j. We have [ki , pi ] ⊥ pj , [ki , pi ] ⊥ aj , for i ̸ = j. This implies [ki , p∑ i ] ∈ pi ⊕ ai . We have [k0 , a] = 0 and [k0 , pi ] ∈ pi . This implies m p˜ = i=1 ai ⊕ pi , where m < s, is invariant under adk and Adk . Given Xd ∈ p˜ , the solution to the differential equation

⟨[kl , pl ][km , pm ]⟩ = ⟨kl [pl , [km , pm ]]⟩ = 0 where we use Jacobi identity. Let ai be the subspace spanned by root vectors Vi , a direct decomposition of a, into root spaces,

a = a1 ⊕ a2 . . . ⊕ as . Given an element a ∈ a, we can decompose, a = a1 + a2 + · · · + as Reflecting in root vectors in Vi , only reflects root vector ai . Reflecting

∑

X˙ = (Xd +

ui ki )X , ki ∈ k

i

˜ = exp({˜p, k}). K = exp(k) is is confined to the invariant, manifold G ˜ = exp(p˜ )K . Given Y ∈ p˜ , we closed subgroup, we can decompose G can rotate it to Cartan subalgebra ∈ a. Since Y ∈ p˜ ∑m a, i.e., AdK (Y ) ∑ m ˜ is AdK invariant, AdK (Y ) ∈ i=1 bi )K2 , where i=1 ai . G = K1 exp( bi ∈ ai . We can synthesize

˜ = K1 exp( G

∑m

j αj Adkj Xd )K2 .

i=1

bi =

∑

j

αj Adkj Xd as detailed before.

∑

Theorem 7. Given g = p ⊕ k, Let a ∈ a. The number of Weyl points W aW −1 ∈ a is finite. Proof. For j = 1, . . . , m, we choose as basis of g, kj and pj (normalized to killing norm 1) where kj + ipj are nontrivial roots. The remaining basis can be chosen as basis of k0 and a. We can organize the basis as the first m vectors being pj followed by next m elements as kj respectively. In this basis ada takes the block form

[

]

A 0

0 , 0

where

[ A=

⎡ λ1 Λ ⎢ , Λ = ⎣ ... ]

0

−Λ

0

0

... .. . 0

⎤ ... .. ⎥ = i2σ ⊗ Λ. y . ⎦ λm

By performing a rotation by S = exp(−iσx ⊗ Im ), we have iΛ SAS = i2σz ⊗ Λ = 0

[

′

We define, S˜ =

[

S 0

]

0 . −iΛ

]

0 , The adjoint representation of AdK (a), In

takes the form

[ Θ1 S

˜′

αj Adkj (a1 ). Hence the proof.

Theorem 6. Let Vi be mutually commuting root vectors, such that no further subdivision in commuting sets is possible. Given roots el = kl + ipl ∈ Vl and em = km + ipm ∈ Vm , we have [kl ± ipl , km ± ipm ] = 0 which implies [kl , km ] = [kl , pm ] = [km , pl ] = [pl , pm ] = 0 The associated root vectors are [kl , pl ] and [km , pm ]. Then

11

∑

iΛ 0 0

0 −iΛ 0

0 0 S˜ Θ1 , 0

]

where Θ1 is matrix representation of AdK (·) over the chosen basis. It is orthonormal, as it preserves the killing norm. When AdK is ˜ Ad (a) S˜ ′ = an automorphism of a, we have Sad K iΛ 0 0

[ S˜ Θ1 S˜ ′

⎡

˜ iΛ = ⎣0 0

0 ˜ −iΛ 0

0 −iΛ 0

0 0 S˜ Θ1′ S˜ ′ 0

]

⎤

0 0⎦ . 0

Since eigenvalues are preserved transformation, [ by similarity ] ˜ iΛ 0 there are only finite possibilities ˜ , which means there 0 −iΛ are finite possibilities for adAdK (a) and AdK (a). Hence Weyl points

12

N. Khaneja / Automatica 111 (2020) 108639

are finite and therefore number of AdK automorphisms of a is finite. Example 1 (Coupled Nuclear Spins). Let

g = −i{Iα , Sβ , Iα Sβ } = −iIα Sβ ⊕ −iIα , Sβ ,

   p

   k

a = −i{Iα Sα }. Given α Ix Sx +β Iy Sy +γ Iz Sz , the roots are −i{Iy Sz ± γ ∓β Iz Sy + i 21 (Ix ± Sx )}, with value 2 , −i{Iz Sx ± Ix Sz ∓ i 21 (Iy ± Sy )} with γ ∓α

β∓α

value 2 and −i{Ix Sy ± Iy Sx + i 21 (Iz ± Sz )}, with value 2 . Regular β±α γ −β element is |α| ̸ = |β| ̸ = |γ |. The fundamental roots are 2 , 2 . Example 2 (Coupled Electron Nuclear Spin). Let g = −i{Iα , Sβ , Iα S β } =

−i{Ix , Iy , Ix Sα , Iy Sβ } ⊕ {−iIz , Sα , Iz Sβ },       p

k

a = −i{Ix , Ix Sz }. Given α Ix + β 2Ix Sz =

(α−β ) Ix ( 12 2

(α+β ) Ix ( 12 2

+ Sz ) +

− Sz ), the roots are −i{Iy ( 12 + Sz ) + iIz ( 12 + Sz )}, with value α+β , −i{Iy ( 12 − 2 α−β 1 Sz ) + iIz ( 2 − Sz )} with value 2 , −i{2Ix Sx + iSy } with value β , −i{2Iy Sx + i2Iz Sx } with value, α , −i{2Iy Sy + i2Iz Sy } with value α −i{2Ix Sy − iSx }, with value β . Regular element is |α| ̸= |β| ̸= 0. The fundamental roots are β and and β .

α−β 2

, with double root at α

5. 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 = p ⊕ k. Using this decomposition, we used some convexity ideas to completely characterize the reachable set and time optimal control for these problems. References Bennett, C. H., Cirac, J. I., Leifer, M. S., Leung, D. W., Linden, N., Popescu, S., & Vidal, G. (2002). Physical Review A, 66, 012305. Brockett, R. W. (1972). System theory on group manifolds and coset spaces. SIAM Journal of Control, 10, 265–284. Cavanagh, J., Fairbrother, W. J., Palmer, A. G., & Skelton, N. J. (1996). Protein NMR spectroscopy, principles and practice. Academic Press. D’Alessandro, D. (2001). Constructive controllability of one and two spin 1/2 particles. In Proceedings 2001 American control conference. Arlington, Virginia.

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Navin Khaneja received his B.Tech in Electrical Engineering from IIT Kanpur in 1994, followed by M.S. and M.A. in Electrical Engineering and Mathematics from Washington University, St. Louis, in 1997. He earned his Ph.D. from Harvard University in Applied Mathematics in 2000. He is recipient of the NSF career award, Sloan fellowship and Bessel Prize of the Humboldt Foundation. His research interests are in areas of control theory, NMR spectroscopy and quantum control.