Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero forcing, minimum rank, andapplications to control of quantum systems
Leslie Hogben
Iowa State University andAmerican Institute of Mathematics
Joint work with Daniel Burgarth, Domenico D’Alessandro, SimoneSeverini, Michael Young
SIAM Annual ConferenceMinneapolis, MN, July 12, 2012
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
IntroductionZero forcingMatrices and GraphsLie AlgebrasControl
Results on controllabilityMatrix resultsZero forcing results
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero Forcing Number
Color change ruleIf G is a graph with each vertex colored either white orblack, u is a black vertex of G , and exactly one neighbor vof u is white, then change the color of v to black.
Example
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
I A subset Z ⊆ VG defines a coloring by coloring allvertices in Z black and all the vertices not in Z white.
I Color change rule: If u is a black vertex of G , andexactly one neighbor v of u is white, then change thecolor of v to black.
I A zero forcing set for a graph G is a subset of verticesZ such that if initially the vertices in Z are coloredblack and the remaining vertices are colored white, thethe result of applying the color change rule until nomore changes result is that all vertices are black.
I The zero forcing number Z(G ) is the minimum of |Z |over all zero forcing sets Z ⊆ V (G ).
Zero forcing is also called graph infection.
Zero forcing is used in the study of minimum rank/maximumnullity problems and in control of quantum systems.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Matrices and Graphs
Hn(R) denotes the real vector space of real symmetricmatrices.
The graph of A ∈ Hn(R) is G(A) = (V ,E ) where
I V = {1, ..., n},I E = {ij : aij 6= 0 and i 6= j}.I Diagonal of A is ignored.
Example: G(A)
A =
2 −1 3 5−1 0 0 0
3 0 −3 05 0 0 0
1 2
34
G(AG ) = G and G(LG ) = G , where AG is the adjacencymatrix and LG is the Laplacian matrix.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Maximum nullity and minimum rank
The maximum nullity of graph G is
M(G ) = max{nullA : A ∈ Hn(R),G(A) = G}.
The minimum rank of graph G is
mr(G ) = min{rankA : A ∈ Hn(R),G(A) = G}.
I M(G ) + mr(G ) = |G |.
Theorem (AIM08)
For any graph G, M(G ) ≤ Z (G ).
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Lie algebras
For A1, . . . ,Ak ∈ Cn×n,
〈A1, . . . ,Ak〉[·,·]
is the real Lie algebra generated by A1, . . . ,Ak underaddition, real scalar multiplication, and the commutatoroperation.
gl(n,R) is the Lie algebra of all n × n real matrices(i.e., gl(n,R) = Rn×n).
u(n) is the Lie algebra of n × n skew-Hermitian matrices(over R).
su(n) is the Lie algebra of n × n skew-Hermitian matriceswith trace zero (over R).
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
For A1, . . . ,Ak ∈ Hn(R),
〈A1, . . . ,Ak〉[·,·] = gl(n,R) ⇐⇒ 〈iA1, . . . , iAk〉[·,·] = u(n).
For A ∈ Hn(R) and Z = {z1, . . . , zs} ⊂ Rn, the real Liealgebra generated by A and Z is
L(A,Z ) := 〈A, z1z1T , . . . , zszsT 〉[·,·].
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Linear control
A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s, B = [b1 . . . bs ] ∈ Rn×s .
Linear system
x = Ax +s∑
j=1
bjuj ,
Control matrix
C(A,B) := [B,AB, . . . ,An−1B]
Classical controllability conditionThe linear system is controllable if and only ifrank C(A,B) = n.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Walk matrix
A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s,B = [b1 . . . bs ] ∈ Rn×s , Z = {b1, . . . ,bs}.
Walk matrix
W (A,b1) := [b1,Ab1, . . . ,An−1b1],
Walk matrix (extended)
W (A,Z ) := [b1,Ab1, . . . ,An−1b1, . . . ,bs ,Abs , . . . ,A
n−1bs ]
W (A,B) := W (A,Z )
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
A ∈ Rn×n, bj ∈ Rn, j = 1, 2, . . . , s, B = [b1 . . . bs ] ∈ Rn×s .
x = Ax +s∑
j=1
bjuj
Walk matrix (extended)
W (A,B) := [b1,Ab1, . . . ,An−1b1, . . . ,bs ,Abs , . . . ,A
n−1bs ],
Control matrix
C(A,B) := [B,AB, . . . ,An−1B]
The linear system is controllable if and only ifrank W (A,B) = n.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Quantum control
For a finite dimensional closed quantum mechanical system:
Schrodinger equation
id
dt|ψ〉 = H(u)|ψ〉,
where |ψ〉 ∈ Cn is the quantum state, the Hamiltonianmatrix H = H(u) is Hermitian and depends on a controlu = u(t).
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
For a system linear in the state |ψ〉, the solution of theSchrodinger equation is |ψ(t)〉 = X (t)|ψ(0)〉 whereX = X (t) is the solution of
Schrodinger matrix equation
i X = H(u)X
with initial condition X (0) = In.
The solution X (t) is unitary at every time t.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Quantum control
Schrodinger matrix equation
i X = H(u)X
Lie algebra rank conditionThe Lie algebra generated by the matrices iH(u) is u(n) orsu(n) (as u varies in the set of admissible values for thecontrol).
Quantum controllability conditionThe system described by the Schrodinger matrix equation iscompletely controllable if and only if Lie algebra rankcondition is satisfied.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Matrix resultsQuantum control and walk matrices
Let A ∈ Hn(R), z ∈ Rn.
Theorem (Godsil Severini 2010)
If rankW (A, {z}) = n, then L(A, {z}) = gl(n,R).
TheoremIf L(A, {z}) = gl(n,R), then rankW (A, {z}) = n.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Quantum control and (extended) walk matrices
Let A ∈ Hn(R) such that G(A) is connected and all thenonzero off-diagonal elements of A have the same sign.Let S ⊆ {1, . . . , n} and Z = {ej : j ∈ S}.
Theoremrank W (A,Z ) = n if and only if L(A,Z ) = gl(n,R).
Corollary
rank W (A, {ej : j ∈ S}) = n, i.e., the linear systemx = Ax +
∑sj=1 ejuj is controllable, if and only if
〈iA, {iejej T : j ∈ S}〉[·,·] = u(n), i.e., the quantum system
associated with the Hamiltonians iA and iejeTj , j ∈ S, is
controllable.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
The hypothesis G(A) is connected is necessary.
Example Let A =
[A1 00 A2
]with Ai being ni × ni .
Let zi ∈ Rni , i = 1, 2 such that rank W (Ai , {zi}) = ni fori = 1, 2. Define z1 := [zT1 , 0
T ]T and z2 := [0T , zT1 ]T .
Then W (A, {z1, z2}) has rank n, but L(A, {z1, z2) containsonly block diagonal matrices.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
The hypothesis all the nonzero off-diagonal elements of Ahave the same sign is necessary.
Example Let A =
0 1 0 11 0 −1 00 −1 0 11 0 1 0
, and Z = {e1, e3}.
Then rank W (A, {e1, e3}) = 4. However,dimL(A, {e1, e3}) ≤ 8 (by exhibiting an 8-dimensional Liesubalgebra of gl(4,R) containing A, e1eT1 , e3e
T3 ).
Since dim gl(4,R) = 16, L(A, {e1, e3}) 6= gl(4,R).
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Zero forcing and quantum control
Let A ∈ Hn(R) such that G(A) is connected and all thenonzero off-diagonal entries of A have the same sign. LetV := {1, 2, . . . , n} be the set of vertices for G(A), andZ ⊆ V be a zero forcing set of G(A).
TheoremL(A, {ejej T : j ∈ Z}) = gl(n,R).
Corollary
〈iA, {iejej T : j ∈ Z}〉[·,·] = u(n)and the corresponding quantum system is controllable.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
The converse is false.
Example Consider the path on four vertices P4 with thevertices numbered in order. The set {e2} is not a zeroforcing set for P4. However,
W (AP4 , {e2}) =
0 1 0 21 0 2 00 1 0 30 0 1 0
and rank W (AP4 , {e2}) = 4, so L(AP4 , {e2}) = gl(n,R) byour previous results.
Zero forcing &quantum control
Leslie Hogben
Introduction
Zero forcing
Matrices and Graphs
Lie Algebras
Control
Results
Matrix results
Zero forcing results
Thank you!