## Abstract

We show that the series product, which serves as an algebraic rule for connecting state-based input–output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie–Trotter product formula.

## 1. Introduction

The aim of this paper is to make some striking connections between the rules for combining models in series in control system theory and the Weyl commutation relations. In the process, we develop a more intrinsic view of the unitary adapted processes of Hudson & Parthasarathy [1] as non-abelian versions of the Weyl unitaries—where the non-abelian nature arises from the presence of the initial space. Our starting point is a surprising connection between the theory of classical linear state-space models and the canonical commutation relations.

### (a) State-based input–output systems

Let and be finite-dimensional vector spaces over the reals. A controlled flow on the state space is given by the dynamical equations
where *u* is a -valued function of time called the input process. An output process *y* taking values in is given by some relation of the general form

The situation is sketched in figure 1, along with the case where we further decompose the value spaces into subspaces.

### (b) Linear systems

We consider a vector input *u*(⋅) leading to a vector output *y*(⋅) according to the model
1.1
or
Here, *x*(⋅) is the state vector, initialized at some value *x*_{0}, and **V** is referred to as the *model matrix* for the model. For *u*(⋅) integrable, the solution can be written immediately as : we also note that the input–output relation is described by the transfer function *T*(*s*)=*N*+*M*(*sI*−*K*)^{−1}*L*, which is determined from the model matrix. The situation is sketched in the top left picture in figure 1.

As the inputs and outputs are vector-valued, they may be further decomposed as say and . This is sketched on the right in figure 1. The model matrix is then
1.2
In each case, we have a port for each input–output. The lines external to the block represent an input or output, while the lines internal to the block correspond to a non-zero entry *N*_{ij} connect input port *j* to output port *i*. The picture on the bottom of figure 2 sketches the situation where *N*_{12}=*N*_{21}=0.

### (c) Concatenation

Suppose that we have a pair of such models with the same state space (with variable *x*) and model matrices , that is,

We may superimpose the two models to get the *concatenated* model
Writing *v*_{i}(*x*)=*K*_{i}*x*+*M*_{i}*u*_{i} for the separate state velocity fields (*i*=1,2), the concatenation rule effectively takes the combined velocity field
1.3
At the level of model matrices, this corresponds to the rule (figure 3)
1.4

The concatenation sum of two model matrices will result in the type of situation depicted in the picture in figure 3, that is, model (1.2) with *N*_{11}=*N*_{1}, *N*_{22}=*N*_{2}, *N*_{12}=0=*N*_{21}.

It is worth remarking that the addition rule (1.3) makes sense for stochastic flows, either in the Itō or Stratonovich form: here, we would have stochastic differential equations
where *U* is a semi-martingale with , formally. A concatenation would then take the form

### (d) Series product

Following this (assuming the dimensions match), we may then introduce feedback into the concatenated model (1.4) by setting the output *y*_{1}(⋅) of the first system equal to the input *u*_{2}(⋅) of the second. Setting *u*_{2}=*y*_{1}(=*L*_{1}*x*+*N*_{1}*u*_{1}) and eliminating these as internal signals in the concatenated system above, we reduce to a linear system
with model matrix
1.5

We refer to the binary operation * as the (general) *series product*, and this will recur in this paper under various guises (figure 4).

### (e) The Heisenberg group

The collection of square model matrices of a fixed dimension, and with lower block *N* invertible, forms a group with the series product as law. A straightforward representation *ρ* of these groups as a subgroup of higher dimensional upper block-triangular matrices (with the series product now replaced by ordinary matrix multiplication) is given by

We now make the observation that we have obtained (in the case *N*=*I*) the Heisenberg group associated with the canonical commutation relations: we refer to the situation *N*≠*I* as the extended Heisenberg group. For a single-input, single-output, single-variable system, we see that the Lie group is generated by
and we note the product table
so that the non-zero Lie brackets are , and .

### (f) Cascading

We should explain that the term ‘series’ is meant for driving fields acting on a given system in series and the use of the single state variable *x* allows for the possibility of variable sharing. The situation where two separate systems connected in series will be termed ‘cascading’, and we should emphasize that this is indeed a special case. Here, the joint state is the direct sum of the states *x*_{1} and *x*_{2} of the first and second system, respectively, and the cascaded system is then
which gives the correct matrix of coefficients for the systems
under the identification *u*_{2}=*y*_{1} (figure 5).

## 2. Quantum stochastic models

### (a) Second quantization

We recall the basic ideas of the (bosonic) second quantization over a separable Hilbert space . The Fock space over is , and a total set of vectors is provided by the exponential vectors defined, for test vector , by
The creation and annihilation operators with test vector *ϕ* are denoted as *a*^{†}(*ϕ*) and *a*(*ϕ*), respectively, and, along with the differential second quantization *dΓ*(*X*) of a self-adjoint operator *X*, they can be defined by
The closures of these operators then satisfy the canonical commutation relations (CCRs) [*a*(*f*),*a*^{†}(*g*)]=〈*f*|*g*〉.

### Definition 2.1

Let be a fixed separable Hilbert space. We denote by the group of unitary operators on with the strong operator topology. The Euclidean group over is the semi-direct product of with the translation group on and consists of pairs (*T*,*ϕ*) where and . The group law is (*T*_{2},*ϕ*_{2})°(*T*_{1},*ϕ*_{1})=(*T*_{2}*T*_{1},*ϕ*_{2}+*T*_{2}*ϕ*_{1}). The extended Heisenberg group over is defined to be
whose basic elements are triples (*T*,*ϕ*,*θ*) with the group law given by
2.1

For , we obtain the Weyl unitary *W*(*T*,*ϕ*) on defined on the domain of exponential vectors by
The special cases of a pure rotation *Γ*(*T*)=*W*(*T*,0), with *Γ*(e^{iX})=e^{i dΓ(X)}, and a pure translation lead to the second quantization and the Weyl displacement unitaries, respectively. The map however yields only a projective unitary representation of the Euclidean group since we have
which is the Weyl form of the CCR and the presence of the multiplier is equivalent to the original CCR.

### Proposition 2.2

*A unitary representation of* *in terms of unitaries on the Bose Fock space* *is then given by the modified Weyl operators* *W*(*T*,*ϕ*,*θ*) *with action*

The role of the ‘scalar phase’ *θ* here is of course to absorb the Weyl multiplier.

### (b) Non-abelian Weyl canonical commutation relation

We now turn to a question, first posed by Hudson & Parthasarathy [2], on how to obtain a non-abelian generalization of the Weyl CCR version wherein the role of *U*(1) phase is replaced by a (sub-)group of unitaries over a fixed separable Hilbert space . In this paper, we show that the appropriate non-abelian extensions are
where is the set of bounded self-adjoint operators on . The corresponding law replacing (2.1) is the series product.

### Definition 2.3

Let and be a fixed separable Hilbert spaces. The extended Heisenberg group is defined to be the set of triples , with group law given by the (special) series product 2.2

Unlike the general situation in quantum groups, the product does in fact lead to a group law! It originated in the work of one of the authors in relation to a systems theoretic approach to ‘cascaded’ quantum stochastic models [3,4].

The original answer provided by Hudson & Parthasarathy involved the quantum Itō calculus [1,5] with initial space and multiplicity space , see below, in which a triple (*S*,*L*,*H*) encoded the information on the coefficients of a quantum stochastic evolution. Apart from a restriction to quantum Itō diffusions (*S*=*I*), they also considered only the operator product of the unitary quantum evolutions, which forced the introduction of time dependence—effectively the coefficients (*S*_{1},*L*_{1},*H*_{1}) will be evolved by the unitary process generated by the second set (*S*_{2},*L*_{2},*H*_{2}). The *S*≠*I* case is readily handled with the aid of quantum stochastic calculus employing the gauge process.

We shall show that the natural Lie–Trotter product formula for a pair of quantum stochastic evolutions leads naturally to the series product (2.2), which from the above is the generalization of the Weyl canonical commutations relations to the non-abelian setting.

### (c) Quantum stochastic evolutions

We recall the quantum stochastic calculus of Hudson & Parthasarathy [1]. The Hilbert space for the system and noise is , where is a fixed separable Hilbert space called the initial space (modelling a quantum mechanical system), and we have the Fock space over the space of square-integrable -valued functions on . Note that . For transparency of presentation, we restrict to the case where is , however, the general case of a separable Hilbert space presents no difficulties. Let be a basis of (the multiplicity space) and define the operators
where 1_{[0,t]} is the characteristic function of the interval [0,*t*] and *χ*_{[0,t]} is the operator on corresponding to multiplication by 1_{[0,t]}. Hudson & Parthasarathy [1] have developed a quantum Itō calculus where integrals of adapted processes with respect to the fundamental processes *Λ*^{αβ}. The Itō table is then
where is the Evans–Hudson [6] delta defined to be unity if *α*=*β*∈{1,…,*n*} and zero otherwise. This may be written as

In particular, we have the following theorem [1].

### Theorem 2.4

*There exists a unique solution V (⋅,⋅) to the quantum stochastic integro-differential equation*
2.3
*(t≥s≥0), where*
*with* *. (We adopt the convention that we sum repeated Greek indices over the range 0,1,…,n.)*

We refer to as the *coefficient matrix*, and *V* as the left process generated by **G**. With respect to the decomposition , we may write
where and . In the situation where is , we have *G*_{00}=*K*, *L* is the column vector [*G*_{i0}], *M* is the row vector [*G*_{0j}] and *N*_{ij}=*G*_{ij}.

Adopting the convention that repeated Latin indices are summed over the range 1,…,*n*, we may write in more familiar notation [1]
For emphasis, we shall often write *V* _{G}(⋅,⋅) when we wish to emphasize the dependence on the coefficients **G**. We remark that the process satisfies the following properties.

— Flow law:

*V*_{G}(*t*,*r*)*V*_{G}(*r*,*s*)=*V*_{G}(*t*,*s*) whenever*t*≥*r*≥*s*.— Stationarity:

*Γ*(*θ*_{τ})*V*_{G}(*t*,*s*)*Γ*(*θ*_{τ})=*V*_{G}(*t*+*τ*,*s*+*τ*) where*θ*_{τ}is the shift map on .— Localization: with respect to the decomposition ,

*V*_{G}(*t*,*s*) acts trivially on the factors and .

It is convenient to introduce the projection matrix (the Hudson–Evans delta) The key result from Hudson & Parthasarathy [1] is the following concerning unitary evolutions.

### Theorem 2.5

*Necessary and sufficient conditions on* **G** *to generate a unitary family are that it satisfies the identities
*
*and this is equivalent to* **G** *taking the form
*
2.4
*where S is a unitary and H is self-adjoint. We then refer to the triple (S,L,H) as Hudson–Parthasarathy coefficients.*

We shall refer to a coefficient matrix as being a *unitary Itō generator matrix* if it leads to a unitary process. We may likewise consider right processes, defined as the solution to , and denote these as *U*_{G}. We find that . It turns out that it is technically easier to establish existence of right processes, especially when the *G*_{αβ} are unbounded.

### (d) The general series product

### Definition 2.6

The (general) series product of two coefficient matrices is defined to be 2.5 With respect to the standard decomposition above, this corresponds to 2.6

The series product is not commutative, however, it is readily seen to be associative. Let us define the *model matrix* **V** associated with a coefficient matrix **G** to be

### Remark 2.7

The series product for two coefficient matrices implies the corresponding law **V**_{2}^{*}**V**_{1} for the associated model matrices given by
Note that this is the natural generalization to the rule (1.5) already seen for classical linear state based models in series!

### Remark 2.8

For Itō generating matrices for unitary process, we have which again leads to a unitary process. Therefore, the general series product defined in (2.6) implies the special series product (2.2).

### Lemma 2.9

*The increment* d*G* *associated with* *is related to the increments* d*G*_{i} *associated with* *V* _{Gi} *through the identity*
2.7
*and this is equivalent to the algebraic relation (2.5) or (2.6)*.

This follows from a straightforward application of the quantum Itō calculus.

### (e) The group of coefficient matrices

### Definition 2.10

Denote by the subset of consisting of operators of the form
with respect to the decomposition of and where *N* is required to be invertible. GL becomes a group under the general series product given in (2.6).

We note that the zero operator is the group identity, and that the series product inverse of is . The extended Heisenberg group is then a subgroup of GL inheriting the series product as law.

The set was introduced in Gough & James [3] as the collection of all Itō generator matrices (2.4) and was shown to be a group under the series product (2.2), though not identified as a Heisenberg group.

### Remark 2.11

The isometry and co-isometry conditions in theorem (2.5) imply that a two-sided inverse of for the series product is given by **G**^{†}∼ (*S*^{†},−*S*^{†}*L*,−*H*). The inverse being of course unique.

### Lemma 2.12

*The mapping*: given *by*
*is an injective group homomorphism*.

One readily checks that and .

This representation is the basis for Belavkin’s formalism of quantum stochastic calculus [7,8]. The Lie algebra of (in the Belavkin representation) consists of matrices
where now the entries *κ*,*λ*,*μ*,*ν* are operators and the exponential map is then with the entries *K*,*L*,*M*,*N* given by
2.8
where we encounter the ‘decapitated exponential’ functions which are the entire analytic functions *e*_{1}(*z*)=(e^{z}−1)/*z*, *e*_{2}(*z*)=(e^{z}−1−*z*)/*z*^{2}.

With an abuse of notation, we shall take the Lie algebra of to be the vector space of operators with entries matched with the representation element above and Lie bracket With this convention, the exponential map from to takes to with entries given by (2.8), and this corresponds to

The Lie algebra for the subgroup will have elements *κ*=−*iη* and *ν*=−*iσ* with and , while is arbitrary but with *μ*=−*λ*^{†}. The exponential map then leads to the element with Hudson–Parthasarathy parameters

## 3. Lie–Trotter formulae

We set , with each element (*t*,*s*)∈Δ^{2} determining an associated interval [*s*,*t*) in . Let be a Hausdorff topological semi-group.

### Definition 3.1

Given an -valued function *V* (⋅,⋅) on Δ^{2}, we set
3.1
where is a partition of the interval [*s*,*t*]. The grid size is and we say that the limit
exists if converges in the topology to a fixed element *a* of independently of the sequence of partitions used, that is, for every open neighbourhood *U* of *a* there exists a *δ*>0 such that if . If the limit is well defined for all *t*>*s*≥0, then we shall write the corresponding two-parameter function as .

### (a) Examples

#### (i) Trivial

If we start with a quantum stochastic exponential *V* =*V* _{G}, the flow property implies that we trivially have for any partition .

#### (ii) Quantum stochastic exponentials

In the setting of quantum stochastic calculus, we let *G*(*t*)=*G*_{αβ}⊗*Λ*^{αβ}(*t*), with *G*_{αβ} bounded, and set (*I*+Δ*G*)(*t*,*s*)=*I*+*G*(*t*)−*G*(*s*), then

#### (iii) Holevo’s time-ordered exponentials

In the same setting, we let *H*(*t*)=*H*_{αβ}⊗*Λ*^{αβ}(*t*) and set e^{ΔH}(*t*,*s*)=e^{H(t)−H(s)}, then the limit is the Holevo time-ordered exponential [9]
often written as . Holevo established strong convergence for such limits, including an extension to the situation where with *H*_{αβ}(⋅) strongly continuous -valued functions with the *H*_{i0}(⋅) and *H*_{0j}(⋅) square integrable, and the *H*_{ij}(⋅) integrable.

We should think of the **H**=[*H*_{αβ}] of the Holevo time-ordered exponential as an element of the Lie algebra . In particular, we have the following result.

### Lemma 3.2

*The Holevo time-ordered exponential* *Y* _{H} *is equivalent to the quantum stochastic exponential* *V* _{G} *where* .

### Proof.

We observe that the integro-differential equation (2.3) can be given the infinitesimal form while for the time-ordered exponential, we have For the two to be equal, we need the coefficients of to coincide, but from the Itō table, this implies .

### (b) The quantum stochastic Lie–Trotter formula

### Definition 3.3

Given -valued functions *V* _{1}(⋅,⋅) and *V* _{2}(⋅,⋅) on Δ^{2}, we define their product *V* _{2}⋅*V*_{1} *interval-wise*, that is
3.2

Note that the product *V*_{2}⋅*V*_{1} will not generally satisfy the flow property even when *V*_{1} and *V*_{2} do, with the result that the limit may now not be trivial.

As an example, take the algebra of *n*×*n* matrices and define *U*_{A}(*t*,*s*)=e^{(t−s)A}, then the Lie product formula can be recast in the form
The extension to the algebra of operators over a Hilbert space with strong operator topology was subsequently given by Trotter. For instance, if *A*=−*iH*_{1} and *B*=−i*H*_{2}, where *H*_{1} and *H*_{2} are self-adjoint, with *H*_{1}+*H*_{2} essentially self-adjoint, on the overlap of their domains, then the strong limit exists (theorem VIII.31 [10]). The case of strongly continuous contractive semigroups on Banach spaces is given as theorem X.5.1 in Reed & Simon [11].

We are now able to formulate our main result.

### Theorem 3.4

*Let* **G**_{1} *and* **G**_{2} *be a pair of bounded coefficient matrices on the same Hudson–Parthasarathy space, then in the strong operator topology,
*
3.3

Similarly, we find , where the interval-wise multiple products are defined in the obvious way.

### Proof.

To see where this comes from, we note from the infinitesimal form that should satisfy the analogous equation
where d*G*=d*G*_{1}+d*G*_{2}+d*G*_{2} d*G*_{1}, but by (2.7), we recognize this as just the infinitesimal generator of . In contrast to the traditional Lie–Trotter formulae, the above limit depends on the order of *V*_{G2}⋅*V*_{G1} and is therefore asymmetric under interchange of *V*_{G2} and *V*_{G1}.

### (c) Special cases

#### (i) Lie–Trotter formulae

The special case recovers the usual Lie–Trotter formulae.

#### (ii) Separate channels

Let be coefficient matrices with common initial space but different multiplicity spaces . We combine the multiplicity space into a single space and ampliate both coefficient matrices as follows:
then
The right-hand side is taken as the definition of the concatenation of the two separate coefficient matrices: this is consistent with the definition of concatenation introduced earlier for model matrices. Theorem (3.4) then implies that
This is equivalent to the result derived by Lindsay & Sinha [12]. We should also mention that the recent work of Das *et al.* [13] indicates that the Trotter formula should also hold at the level of flows.

## Acknowledgements

The authors are grateful to Prof. Kalyan Sinha for presenting his work on quantum stochastic Lie–Trotter formula [12] during a visit to Aberystwyth within the framework of the UK–India Education and Research Initiative.

## Footnotes

One contribution of 15 to a Theo Murphy Meeting Issue ‘Principles and applications of quantum control engineering’.

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