Zero forcing, linear and quantum controllability for systems evolving on networks
We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.
This paper deals with several concepts from different fields such as linear algebra, graph theory and quantum and classical (linear) control theory. In the context of dynamics and control of systems on networks, it establishes a connection between a notion of graph theory (zero forcing) and concepts in control theory (quantum and classical controllability). We review these different concepts before we introduce the technical content of the paper and give physical motivation for our study.
For a dynamical system with a control input, the property of controllability describes to what extent one can go from one state to another with the evolution corresponding to an appropriate choice of the control. If all the possible state transfers can be obtained within a natural set (the phase space), then the system is said to be controllable.
For several classes of systems, controllability has been described in detail and controllability tests are known. In particular, for linear systems
, , , where both the state and the control functions enter the right hand side linearly, several equivalent conditions of controllability are known. The classical Kalman controllability condition (see, e.g., ) says the system (1) is controllable if and only if the matrix
has full rank , where . In this case, for any prescribed state transfer and interval , there exists a control such that the corresponding solution of (1) satisfies and . For quantum mechanical systems which are closed (i.e., not interacting with the environment) and finite dimensional, one considers the Schrödinger equation
where is the quantum state and the Hamiltonian matrix is Hermitian and depends on a control which in some cases can be assumed to be a switch between different Hamiltonians. If (2) is a system linear in the state , the solution of (2) is where is the solution of the Schrödinger matrix equation
with initial condition equal to the identity matrix .
Since is Hermitian for every value of and therefore
is skew-Hermitian, the solution of (3) is forced
to be unitary at every time . In this context, the system is
called completely controllable if for any unitary matrix in
At the beginning of the development of the theory of quantum
control, it was realized (see e.g., ) that system
(3) has a structure familiar in geometric control theory
 and therefore controllability conditions developed
there can be directly applied. In particular, the Lie algebra rank
condition  says that a necessary and sufficient condition for
complete controllability of system (3) is that the Lie
algebra generated by the matrices (as varies in
the set of admissible values for the control) is or .
In recent years there has been considerable interest in the study of control systems, both classical and quantum, which are naturally modeled on networks. Often one tries to relate the controllability of these systems to topological or graph theoretic properties of the network. For quantum systems, the nodes of the network may represent energy levels or particles which are interacting with each other. For these systems, the application of the Lie algebra rank condition to determine controllability can become cumbersome and subject to errors when the dimension of the system becomes large. It is preferable to have criteria based on graph theoretic properties of the network not only because they are typically checked more efficiently but also because they give more insight in the dynamics of the system. Work in this direction has been done in , , . In this context, a relevant property of a graph and a subset of its vertices is the capability of this set to ‘infect’ all the vertices of the graph, as explained in the next paragraph.
Every graph discussed is simple (no loops or multiple edges), undirected, and has a finite nonempty vertex set. Consider a graph and color each of its vertices black or white. A vertex is said to infect, or force a vertex if is black, is white, is a neighbor of , and is the only white neighbor of . In the case where infection of has occurred, we change the color of to black and continue the iterative procedure. The set is called a zero forcing set if this procedure, starting from a graph where only the vertices in are black, leads to a graph where all vertices are black. An example of a zero forcing (infection) process is shown in Figure 1, indicated by arrows; the set of black vertices is a zero forcing set.
For a real symmetric matrix , the graph of , denoted , is the graph with vertices and edges . Observe that , where and denote the adjacency matrix of and the Laplacian matrix of , respectively (here is the diagonal matrix of degrees). Zero forcing has been studied in detail in the context of linear algebra. This is because the size of the minimum zero forcing set of a given graph , which is called the zero forcing number , is an upper bound to the maximum nullity (or maximum co-rank) over any field of ; the maximum nullity is taken over all symmetric matrices such that (see  for background on the problem of determining maximum nullity).
Zero forcing appears then to be a valuable concept in the study of graph-theoretic properties that are captured by generalized adjacency matrices. Indeed, there are important classical parameters introduced with this purpose, e.g., the Colin de Verdiére number, the Haemers bound, etc. It has to be remarked that questions about the maximum nullity of a graph are generally difficult problems and the zero forcing number does not constitute an exception: it was shown in  that there is no poly-logarithmic approximation algorithm for the zero forcing number.
1.2 Contribution of the paper and physical motivation
In this paper, we consider the dynamics of a system defined on a
network and relate the above notions and criteria of
controllability with the graph theoretic concept of zero forcing. Abstractly,
we consider a graph and a subset of its vertices . The
dynamics are that of a quantum system (3) where the
Hamiltonian is allowed to take the values . Here is the adjacency matrix of , Laplacian matrix of , or more generally a real symmetric matrix such that with all nonzero off-diagonal entries of having the same sign (which is the
typical situation in transport models). The vectors are the
On physical grounds, our motivation for considering a Hamiltonian specified by a matrix with the given graph comes from the study of continuous time quantum walks which model transport phenomena in many physical and biological systems . A recent review is given in . Most of the studies consider this sole Hamiltonian and concern the statistical (diffusion) properties of the dynamics. We add here the Hamiltonians where is the characteristic vector of a given node of the network and study the nature of the states that the resulting dynamics can achieve, in particular whether an arbitrary (unitary) state transfer can be achieved between the states of the quantum system. The Hamiltonians model a prescribed energy difference between the corresponding node and all the other nodes of the network which are assumed to be at the same energy level. Therefore the dynamics is the alternating of a diffusion process (modeled by the Hamiltonian ) and a rearrangement of the energies of the various states by selecting one of the states as high energy state and all the other at the same (lower) energy.
Theoretical research in network theory has focused on a number of discrete time, deterministic diffusion processes on graphs. While zero forcing has not been studied in this context, there are two directions of research that are closely related: as it was already noted in , the threshold model introduced for studying influence in social networks shares with zero forcing certain issues underlying its computational complexity ; the model of complex networks controllability recently proposed in  also makes a natural use of the Kalman rank condition and it singles out certain combinatorial properties to determine when the condition is satisfied. Determining whether zero forcing has a place in the metrology of complex networks is a point worth further interest.
The paper is organized as follows. In Section 2 we introduce notation and give background and basic results concerning Lie algebras that will be used in the following sections. The connection between quantum (Lie algebraic) controllability and the Kalman criterion for linear systems is established in Section 3. There we also prove the converse of the main result of . The relation with the zero forcing property is established in Section 4, and Section 5 contains concluding remarks.
2 Lie algebra terminology and preliminary results
Standard material on Lie algebras can be found in . For , denotes the real Lie algebra generated by under addition, real scalar multiplication, and the commutator operation. Let denote the real vector space of symmetric matrices. For , the notation means for the and entries of are both . Observe that can be expressed as
The following proposition is well known (a proof appears in ). It provides a link between an appropriate Lie algebra of real matrices and the Lie algebra rank condition of quantum control theory, thereby allowing us to work with real matrices only. Recall that the Lie algebra consisting of all real matrices is denoted by , denotes the Lie algebra of real matrices with zero trace, denotes the Lie algebra of all skew-Hermitian (complex) matrices, and denotes the Lie algebra of all skew-Hermitian (complex) matrices with zero trace. All these Lie algebras are considered as vector spaces over the field of real numbers.
The next lemma is used in the proof of Theorem 3.7 in the next section.
Let , with . Define and let denote the smallest ideal of that contains . If and for some , then .
For the result is clear, so assume , , and for some . Observe that is spanned by and . Since , we have
It is known that , because is a nonzero ideal in and is a simple Lie algebra. Since and , . Thus . ∎
The next lemma is used in the proof of Theorem 3.1 in the next section. Let be a Lie algebra, , and let be a subspace of . Recall that the operation is defined as , and the normalizer of is
It follows from the Jacobi identity that is a subalgebra of [12, p. 7].
Let . Assume and define
where and are nonnegative integers. Then .
First note that because we have assumed that and generate . Clearly . Since is a subalgebra of and and generate , . Thus is an ideal of . Notice that since is skew-Hermitian with zero trace and and are skew-Hermitian. Since is an ideal of , is an ideal of , and . Since is a simple Lie algebra, by definition it has only the trivial ideals and . Therefore . ∎
For and , the real Lie algebra generated by and is defined as
3 Controllability and walk matrices
For and , the extended walk matrix of and is the real matrix
A special case is when (with denoting the -th standard basis vector) for some subset for a graph and is the adjacency matrix of the graph. In this case, the relevant walk matrix is .
For the connection between the walk matrix in (6) and the Lie algebra in (5) was studied in . It was shown [10, Lemma 1] that implies , or equivalently, (cf. Proposition 2.1). The next theorem states that the converse of this result is also true.
Consider a matrix in and a vector . Then, (or equivalently ) implies that .
The equivalence of the hypotheses is justified by Proposition 2.1. The result is clear if , so assume . We use a contradiction argument. Assume the rank of the walk matrix is less than but , where . There exists a vector such that . Consider the rank matrix . We claim that commutes with every matrix in , where is as in (4). To see this, notice that from (4), all elements in are linear combinations of monomials of the form , for some , , and appearing at least once with exponent greater than zero. When multiplying with , with on the left, write as for some matrix , so we have
which follows immediately from the condition for , and by using the Cayley-Hamilton theorem for . Analogously, when multiplying on the right of , we write as , for some matrix , and we have
since also implies . Therefore commutes with all elements of .
Observe that since is simple, is an irreducible representation of . Therefore, since commutes with all elements of , it follows from Schur’s Lemma that must be a scalar multiple of the identity [12, p. 26]. However this is not possible since has rank . This gives the desired contradiction and thus completes the proof. ∎
We study the generalization of this result to multiple vectors () but for matrices and vectors related to a connected graph . In particular, , all nonzero off-diagonal entries of have the same sign, and will be the characteristic vectors associated to a subset of the vertices. In the next section we will relate this to the zero forcing property of the set . In the context of graphs, it is important to consider multiple vectors because if is a graph and , then for any one vector . On the other hand we will see that if is a zero forcing set for and , then (see Theorem 4.1 below).
For and , define
For and , the associative algebra generated by is equal to , because
if and only if .
Clearly if and only if . First assume . For any matrix with , there exist vectors , , such that . Since , each is expressible as a linear combination of the columns of , i.e., as a linear combination of vectors of the form , and similarly for . Thus each , and hence , is expressible as a linear combination of . Thus the matrices of the form span .
For the converse, observe that if is a basis for , then
If , then , so the matrices in cannot span . ∎
The distance between two distinct vertices and of a connected graph , denoted by is the minimum number of edges in a path from to .
Let such that is connected and all nonzero off-diagonal entries of have the same sign. If and , then .
Let . The entry is a sum of terms which are each the product of nonzero entries of . Since is the distance between and , only off-diagonal entries can appear in this product. Thus every term has the same sign and . ∎
Let be such that is connected and all nonzero off-diagonal entries of have the same sign. Let and be the subset of standard basis vectors. Then .
The proof of Lemma 1 in  shows that for any real symmetric matrix and vector , for all . Applying this, we obtain that for all . The result will follow if we are able to show that for all , with different from .
Consider the distance between the nodes and in , which is because is connected. From the fact that both and are in , we have in ,
It follows from Lemma 3.5 that , and so .
So, . Similarly, . Finally,
So, . ∎
The following theorem establishes the connection between quantum Lie algebraic controllability and the rank condition for an extended walk matrix modeled on a graph.
Let such that is connected and all the nonzero off-diagonal elements of have the same sign. Let and be a subset of standard basis vectors. Then if and only if .
Let such that is connected and all the nonzero off-diagonal elements of have the same sign, and let . Then if and only if , i.e., the quantum system associated with the Hamiltonians and , , is controllable.
The result of  for the case showing that implies , (and the converse proved in Theorem 3.1 in this paper) were proved in reference to systems on graphs. The proofs however go through for an arbitrary symmetric matrix and vector . It is natural to ask whether the conditions on the matrix that we have used in Theorem 3.7 are really necessary. To this purpose, we can observe that the result is not true if we give up either of the hypotheses that 1) is connected or 2) the off-diagonal entries of have the same sign, as shown in the next two examples.
To see the necessity of assuming that is connected, consider a block diagonal matrix with and symmetric matrices of dimensions and , respectively, with , and and two vectors that have zeros in the last or first entries, respectively, and such that the corresponding matrices and have ranks and , respectively. In this case, the walk matrix has rank , but the Lie algebra generated by , , and contains only block diagonal matrices.
To see the necessity of assuming that all nonzero off-diagonal entries of have the same sign, consider , and . It is straightforward to verify that the walk matrix has rank . However, , as can be seen as follows. Let
Since for all , is a Lie subalgebra of . Clearly and .
4 Zero forcing and controllability
The neighborhood of is
Let such that is connected and all the nonzero off-diagonal entries of have the same sign. Let be the set of vertices for , and be a zero forcing set of . Then
After a (possibly empty) sequence of forces, denote by the set of currently black vertices, and assume that for all , . The hypotheses of Lemma 3.6 are satisfied for , so for all , .
If , then there is a vertex that has a unique neighbor outside . For that we have
where if and if . For all such that , , so . Thus . Since
so . Since is a zero forcing set, we obtain for all , and thus we conclude that ∎
Applying Proposition 2.1 we obtain the next corollary.
If is a connected graph, , all the nonzero off-diagonal entries of have the same sign, and is a zero forcing set of , then and the corresponding quantum system is controllable.
Note that the converse of Theorem 4.1 is false.
Consider the path on four vertices with the vertices numbered in order. The set is not a zero forcing set for . However,
so by Theorem 3.7.
Motivated by the control and dynamics of systems modeled on networks both classical and quantum, we have established a connection between various tests of controllability and the notion of zero forcing in graph theory. Lie algebraic quantum controllability is necessary and sufficient for linear (Kalman-like) controllability of an associated system and both notions are implied by the zero forcing property of the associated set of vertices. Linear systems have a very well developed theory  and it is an open question to investigate to what extent this analogy can be further used to discover properties of quantum systems and systems on networks.
Acknowledgements. The authors would like to thank Mark Hunacek for productive discussions concerning Lie algebras. This work has been done while Daniel Burgarth was supported by EPSRC grant EP/F043678/1. Domenico D’Alessandro is supported by NSF under Grant No. ECCS0824085. Simone Severini is a Newton International Fellow. Michael Young’s postdoctoral fellowship is supported by NSF through DMS 0946431.
- Following standard notation, is the special unitary group, i.e., the matrix group of unitary matrices having determinant 1.
- Following standard notation, is the Lie algebra of skew-Hermitian matrices and is the Lie algebra of skew-Hermitian matrices with zero trace.
- The vector has the th entry equal to one and every other entry equal to zero and is also called the th standard basis vector.
- As a vector space, is the same as . We use the latter notation when we want to stress the Lie algebra structure on .
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