Realization Theory for Linear Dynamical Quantum Systems

Abstract

This chapter presents a realization/network synthesis theory for linear quantum systems. This theory addresses the question of how one can go from an abstract description of a linear quantum system to a concrete realization of the system using quantum optical devices. Two distinct types of realization problems are introduced and treated: the strict (or hard) realization problem and the transfer function (or soft) realization problem. The system to be realized is decomposed into simpler subsystems, and how these subsystems can be realized, at least approximately, in the quantum optical setting is developed. In particular, it is shown that simpler realizations can be obtained for completely passive linear quantum systems.

The Introduction, Sect. 3.1 and the associated appendices contain materials adapted from [3] Copyright $\copyright$ 2009 Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved.

Section 3.2 and the associated appendices contain some materials reprinted, with permission, from [13] $\copyright$ 2010 IEEE.

Section 3.3 contain materials reprinted, with permission, from [16] $\copyright$ 2010 IEEE.

Appendices

Appendices

Appendix A: Adiabatic Elimination of Coupled Cavity Modes

In this section, we shall derive formulas for two coupled cavity modes in which one of the cavity has very fast dynamics compared to the other and can be adiabatically eliminated, leaving only the dynamics of the slow cavity mode. The cavities are each coupled to separate bosonic fields and are interacting with one another in a classically pumped nonlinear crystal. A mathematically rigorous theory for the type of adiabatic elimination/singular perturbation that we are interested in here was developed in [12].

The two cavity modes will be denoted by a and b, each defined on two distinct copies of the Hilbert space $\ell ^2$ of square-integrable sequences ($\mathbb {Z}_+$ denotes the set of all nonnegative integers). Thus, the composite Hilbert space for the two cavity modes is $\mathcal {H}=\ell ^2(\mathbb {Z}_+) \otimes l^2(\mathbb {Z}_+)$. The interaction in a nonlinear crystal is given, in some rotating frame, by an interaction Hamiltonian $H_{ab}$ of the form $H_{ab}=\alpha a^*b+\beta a^*b^*+ \alpha ^* ab^*+\beta ^* ab$, for some complex constants $\alpha$ and $\beta$. The mode a is coupled to a bosonic field $\mathcal {A}_1$, while b is coupled to the bosonic field $\mathcal {A}_2$, both fields in the vacuum state. The fields $\mathcal {A}_1$ and $\mathcal {A}_2$ live the boson Fock space $\mathcal {F}_2=\mathcal {F}_1 \otimes \mathcal {F}_1$. We take a to be the slow mode to be retained and b to be the fast mode to be eliminated.

We consider a sequence of linear quantum systems $G_k=(I,\tilde{L}^{(k)},H_{ab}^{(k)})$ with $\tilde{L}^{(k)}=(\sqrt{\gamma _1} a,k\sqrt{\gamma _2}b)^{\top }$ and $H_{ab}^{(k)}=\Delta _1 a^*a + k^2\Delta _2 b^*b+ k(\alpha a^*b+\beta a^*b^*+ \alpha ^* ab^*+\beta ^* ab)$ each evolving according to the unitary $U_k$ satisfying the left Hudson–Parthasarathy (H–P) QSDE (as opposed to the conventional right H–P QSDE in (2.2)):

$$\begin{aligned} dU_k(t)= & {} U_k(t)\left( \tilde{L}^{(k)*} (d\mathcal {A}_1(t),d\mathcal {A}_2(t))^{\top }- \tilde{L}^{(k)\top } (d\mathcal {A}_1(t),d\mathcal {A}_2(t))^{*} \right. \nonumber \\&\quad \left. +\left( \imath H_{ab}^{(k)}-(1/2)\tilde{L}^{(k){*}} \tilde{L}^{(k)}\right) dt\right) , \end{aligned}$$

Here, we are using the left QSDE following the convention used in [12] (see Remark 3 therein) so that the Heisenberg picture dynamics of an operator x is given by $x(t)=U_k(t)xU_k(t)^*$. We shall use the results of [12] to show, in a similar treatment to Sect. 3.2 therein, that in the limit as $k \rightarrow \infty$:

$$\begin{aligned} \lim _{k \rightarrow \infty }\sup _{0 \le t \le T}\Vert U_k(t)^*\phi -U(t)^* \phi \Vert =0\;\; \forall \phi \in \mathcal {H}_0 \otimes \mathcal {F}_2 \end{aligned}$$

(3.16)

for any fixed time $T>0$, where $\mathcal {H}_0$ is an appropriate Hilbert subspace of $\mathcal {H}$ (to be precisely specified in the next paragraph) for a limiting unitary U(t) (again as a left H–P QSDE) satisfying:

$$\begin{aligned} dU(t)= & {} U(t)\left( \left( \frac{\imath 2\Delta _2+\gamma _2}{\imath 2\Delta _2-\gamma _2}-1\right) d\Lambda _{22} +\sqrt{\gamma _1}a^*d\mathcal {A}_1(t)-\sqrt{\gamma _1}ad\mathcal {A}_1(t)^* \right. \nonumber \\&-\imath \sqrt{\gamma _2}(\imath \Delta _2 - \frac{\gamma _2}{2} )^{-1}(\alpha a^*+\beta ^* a)d\mathcal {A}_2(t)+\nonumber \\&\imath \frac{2\sqrt{\gamma _2}}{\imath 2\Delta _2-\gamma _2}(\alpha ^* a + \beta a^*)d\mathcal {A}_2(t)^*+ (\imath \Delta _1-\frac{\gamma _1}{2})a^*a dt+\nonumber \\&\left. (\imath \Delta _2-\frac{\gamma _2}{2})^{-1}(\alpha a^*+\beta ^* a)(\alpha ^*a+\beta a^*)dt \right) , \end{aligned}$$

(3.17)

on $\mathcal {H}_0 \otimes \mathcal {F}$. Note that (3.17) is a left H–P QSDE corresponding to the right form in Sect. 2.1.4 by noting that we may write:

$$\begin{aligned}&(\imath \Delta _1-\frac{\gamma _1}{2})a^*a+ (\imath \Delta _2-\frac{\gamma _2}{2})^{-1}(\alpha a^*+\beta ^* a)(\alpha ^*a+\beta a^*)\\&=\imath \left( \Delta _1 a^*a-\frac{\Delta _2}{\Delta _2^2+(\frac{\gamma _2}{2})^2}(\alpha a^*+\beta ^* a)(\alpha ^*a+\beta a^*)\right) -\frac{1}{2}(\tilde{L}_1^{*} \tilde{L}_1+\tilde{L}_2^{*} \tilde{L}_2), \end{aligned}$$

with $\tilde{L}_1=\sqrt{\gamma _1}a$ and $\tilde{L}_2=\imath \sqrt{\gamma _2}(-\imath \Delta _2 - \frac{\gamma _2}{2} )^{-1}(\alpha ^* a +\beta a^*)$. As such it satisfies the H–P Condition 1 of [12].

Let $\phi _0,\phi _1,\ldots$ be the standard orthogonal bases of $\ell ^2$, i.e., $\phi _l$ is an infinite sequence (indexed starting from 0) of complex numbers with all zeros except a 1 in the lth place. First, let us specify that $\mathcal {H}_0=\ell ^2 \otimes \mathbb {C}\phi _0$, and this is the subspace of $\mathcal {H}$ where the slow dynamics of the system will evolve. Next, we define a dense domain $\mathcal {D}=\mathrm{span}\{\phi _j \otimes \phi _l;\,j,l=0,1,2,\ldots \}$ of $\mathcal {H}$. The strategy is to show that [12, Assumptions 2–3] are satisfied from which the desired result will follow from [12, Theorem 3].

From the definition of $H_{ab}^{(k)}$, $\tilde{L}^{(k)}$ and $U_k$ given above, we can define the operators $Y,A,B,G_1,G_2$, and $W_{jl}\;(j,l=1,2)$ in [12, Assumption 1] as: $Y=(i\Delta _2-\frac{\gamma _2}{2})b^*b,A=i(\alpha a^*b+ \alpha ^* ab^* + \beta a ^*b^* + \beta ^* ab),B=(i\Delta _1-\frac{\gamma _1}{2}) a^*a, G_1=\sqrt{\gamma _1}a^*,G_2=0,F_1=0,F_2=\sqrt{\gamma _2}b^*, W_{jl}=\delta _{jl}$. Then, we can define the operators $K^{(k)}$, $L_j^{(k)}$ in this assumption as:

$$\begin{aligned} K^{(k)}= & {} k^2 Y+kA+B;\;L_j^{(k)}=kF_j+G_j\;(j=1,2). \end{aligned}$$

Let $P_0$ be the projection operator to $\mathcal {H}_{0}$. Let us now address Assumption 2. From our definition of $\mathcal {H}_0$, it is clear that we have that (a) $P_0\mathcal {D} \subset \mathcal {D}$. Any element of $P_0 \mathcal {D}$ is of the form $f \otimes \phi _{0}$ for some $f \in \mathrm{span}\{\phi _l;\,l=0,1,2,\ldots \}$; therefore, since $Y=(\imath \Delta _2-\frac{\gamma _2}{2})b^*b$ and $b \phi _{0}=0$, we find that (b) $YP_0 d=0$ for all $d \in \mathcal {D}$. Define the operator $\tilde{Y}$ on $\mathcal {D}$ by $\tilde{Y} f \otimes \phi _0=0$ and $\tilde{Y} f \otimes \phi _l= l^{-1}(\imath \Delta _2-\frac{\gamma _2}{2})^{-1} f \otimes \phi _l$ for $l=1,2,\ldots$ ($\tilde{Y}$ can then be defined to all of $\mathcal {D}$ by linear extension). From the definition of Y and $\tilde{Y}$, it is easily inspected that (c1) $Y\tilde{Y} f=\tilde{Y} Y f=P_1 f$ for all $f \in \mathcal {D}$, where $P_1=I-P_0$ (i.e., the projection onto the subspace of $\mathcal {H}$ complementary to $\mathcal {H}_0$). Moreover, because of the simple form of $\tilde{Y}$, it is also readily inspected that (c2) $\tilde{Y}$ has an adjoint $\tilde{Y}^*$ with a dense domain that contains $\mathcal {D}$. Since $F_1=0$, we have that (d1) $F_1^* P_0=0$ on $\mathcal {D}$, while since $F_2^* f \otimes \phi _0 = \sqrt{\gamma _2}b f \otimes \phi _0 = 0\;\forall f \in \ell ^2$, we also have (d2) $F_2^* P_0 =0$ on $\mathcal {D}$. Finally, from the expression for A and the orthogonality of the bases $\phi _0,\phi _1,\ldots$, a little algebra reveals that (e) $P_0 A P_0 d =0$ for all $d \in \mathcal {D}$. From (a), (b), (c1–c2), (d1–d2), and (e), we have now verified that Assumption 2 is satisfied.

Finally, we verify that the limiting operator coefficients $K,L_1,L_2,M_1,M_2$, $N_{jk}\;(i,j=1,2)$ (as operators on $\mathcal {H}_0$) of Assumption 3 coincide with the corresponding coefficients of (3.17). These operator coefficients are defined as $K=P_0(B-A \tilde{Y} A)P_0$, $L_j=P_0(G_j-A \tilde{Y} F_j)P_0$, $M_j=-\sum _{r=1}^{2}P_0 W_{jr}(G_r^*-F_r^*\tilde{Y} A)P_0$ and $N_{jl}=\sum _{r=1}^{2}P_0 W_{jr}(F_r^*\tilde{Y} F_l + \delta _{rl})P_0$. From these definitions and some straightforward algebra, we find that for all $f \in \mathrm{span}\{\phi _l;\,l=0,1,2,\ldots \}$:

$$\begin{aligned} K f \otimes \phi _0= & {} \left( (\imath \Delta _1-\frac{\gamma _1}{2})a^*a+(\imath \Delta _2-\frac{\gamma _2}{2})^{-1}(\alpha a^*+\beta ^* a)(\alpha ^*a+\beta a^*) \right) f \otimes \phi _0,\\ L_1 f \otimes \phi _0= & {} \sqrt{\gamma _1}a^* f \otimes \phi _0,\\ L_2 f \otimes \phi _0= & {} -\imath \sqrt{\gamma _2}(i\Delta _2 - \frac{\gamma _2}{2} )^{-1}(\alpha a^*+\beta ^* a) f \otimes \phi _0,\\ M_1 f \otimes \phi _0= & {} -\sqrt{\gamma _1}a f \otimes \phi _0,\\ M_2 f \otimes \phi _0= & {} \sqrt{\gamma _2}\left( \imath \Delta _2-\frac{\gamma _2}{2} \right) ^{-1}(\alpha ^* a + \beta a^*) f \otimes \phi _0, \end{aligned}$$

and

$$\begin{aligned}&N_{11}f \otimes \phi _0 = f \otimes \phi _0,\;N_{12}f \otimes \phi _0 =0,\;N_{21}f \otimes \phi _0=0,\\&N_{22}f \otimes \phi _0=\frac{\gamma _2+\imath 2\Delta _2}{-\gamma _2+\imath 2\Delta _2}f \otimes \phi _0. \end{aligned}$$

Therefore, we see that U(t) may be written as:

$$ dU(t)=U(t)\left( \sum _{j,l=1}^{2}(N_{jl}-\delta _{jl})d\Lambda _{jl} +\sum _{j=1}^{2} M_{j}d\mathcal {A}_j^*(t)+\sum _{j=1}^{2}L_j d\mathcal {A}_j(t) +K dt\right) . $$

Since we have already verified that (3.17) is bona fide left QSDE equation, it now follows that Assumption 3 of [12] is satisfied. Now (3.16) follows from [12, Theorem 3] and the proof is complete.

Moreover, we can observe from the derivation above that the coupling of a to $\mathcal {A}_2(t)$ after adiabatic elimination will not change if a is also coupled to other cavities modes $b_3,\ldots ,b_m$ via an interaction Hamiltonian of the form $\sum _{i=j}^{m}(\alpha _{j1} ab_j^*+ \alpha _{j1}^* a^*b_j + \alpha _{j2} a^*b_j^* + \alpha _{j2}^* a b_j)$, and each additional mode may also be linearly coupled to distinct bosonic fields $\mathcal {A}_3,\ldots ,\mathcal {A}_m$, respectively, as long as these other modes are not interacting with b and with one another (this amounts to just introducing additional operators $F_j,G_j$, $j \ge 3$, etc.). Moreover, under these conditions, one can also adiabatically eliminate any of the additional modes and the only effect will be the presence of additional sum terms in U(t) that do not involve b, $\mathcal {A}_1(t)$ and $\mathcal {A}_2(t)$. $\Box$

Appendix B: Proof of Theorem 3.4

For this proof, it will be convenient to interchange some rows and columns of the model matrix M to form another model matrix $\tilde{M}$ to avoid complicated bookkeeping and thus reduce unnecessary clutter. Hereby, “rows” and “columns” we mean, respectively, block rows and block columns of M formed with respect to its specified partitioning. This interchange is as follows.

First, we permute rows of M such that the first $2n-1$ rows from top to bottom are the rows labeled (while column labels are kept fixed as they are) $s_{00}$, $s_{12}$, $s_{21}$, $s_{13}$, $s_{31}$, $\ldots$, $s_{1n}$, $s_{n1}$, the next $2(n-2)$ rows, respectively, are the rows labeled $s_{23}$, $s_{32}$, $s_{24}$, $s_{42}$, $\ldots$, $s_{2n}$, $s_{n2}$, and so on in the same pattern until we get to the last n rows that are, respectively, those rows labeled $s_{11}$, $s_{22}$, $\ldots$, $s_{nn}$. Call the intermediate matrix resulting from this row permutation $\hat{M}$. Then fixing the row labels of $\hat{M}$, we permute its columns such that the first $2n-1$ columns from left to right are, respectively, the columns of $\hat{M}$ labeled $r_{00}$, $r_{12}$, $r_{21}$, $r_{13}$, $r_{31}$, $\ldots$, $r_{1n}$, $r_{n1}$, the next $2(n-2)$ columns are, respectively, the columns labeled $r_{23}$, $r_{32}$, $r_{24}$, $r_{42}$, $\ldots$, $r_{2n}$, $r_{n2}$, and so on in the same pattern until the final n columns that are, respectively, the columns labeled $r_{11}$, $r_{22}$, $\ldots$, $r_{nn}$. The resulting matrix after this permutation of columns is $\tilde{M}$.

It is important to note here that since the same permutation is applied to the rows and columns, M and $\tilde{M}$ are model matrix representations of the same physical system. That is to say that if M is the model matrix of $G=(S,L,H)$, then $\tilde{M}$ is the model matrix of $\tilde{G}=(PSP^{\top },PL,H)$ for some suitable constant real permutation matrix P, while it is clear that G and $\tilde{G}$ are representations of the same physical system. Thus with the same internal connections made, a reduced model matrix for $\tilde{M}$ is also a reduced model matrix for M, up to a possible relabeling of uneliminated ports.

Let $\tilde{L}=PL$ and $\tilde{S}=PSP^{\top }$. Then, $\tilde{L}$ can be partitioned as $\tilde{L}=(\tilde{L}_\mathrm{i}^{\top },\tilde{L}_\mathrm{e}^{\top })^{\top }$, where $\tilde{L}_\mathrm{i}$ is the first $n(n-1)+1$ rows of $\tilde{L}$, while $\tilde{L}_\mathrm{e}$ is the last n rows of $\tilde{L}$. They are of the form:

$$\begin{aligned} \tilde{L}_\mathrm{i}= & {} (L_{12}^{\top },L_{21}^{\top },L_{13}^{\top },L_{31}^{\top },\ldots ,L_{1n}^{\top },L_{n1}^{\top },L_{23}^{\top },L_{32}^{\top },L_{24}^{\top },L_{42}^{\top },\ldots ,L_{2n}^{\top },L_{n2}^{\top },\ldots ,\\&L_{(n-2)(n-1)}^{\top },L_{(n-1)(n-2)}^{\top },L_{(n-2)n}^{\top },L_{n(n-2)}^{\top },L_{(n-1)n}^{\top },L_{n(n-1)}^{\top })^{\top },\\ \tilde{L}_\mathrm{e}= & {} (L_{11}^{\top },L_{22}^{\top },\ldots ,L_{nn}^{\top })^{\top }. \end{aligned}$$

Similarly, $\tilde{S}$ can be partitioned as $\tilde{S}=\left[ \begin{array}{cc} \tilde{S}_\mathrm{ii} &{} \tilde{S}_\mathrm{ie} \\ \tilde{S}_\mathrm{ei}&{} \tilde{S}_\mathrm{ee} \end{array} \right]$, with $\tilde{S}_\mathrm{ii}$ and $\tilde{S}_\mathrm{ee}$ being block diagonal:

$$\begin{aligned} \tilde{S}_\mathrm{ii}= & {} \mathrm{diag}(S_{12},S_{21},S_{13},S_{31},\ldots ,S_{1n},S_{n1},S_{23},S_{32},S_{24},S_{42},\ldots ,S_{2n},S_{n2},\ldots ,\\&S_{(n-2)(n-1)},S_{(n-1)(n-2)}, S_{(n-2)n},S_{n(n-2)},S_{(n-1)n},S_{n(n-1)}),\\ \tilde{S}_\mathrm{ee}= & {} \mathrm{diag}(S_{11},S_{22},\ldots ,S_{nn}), \end{aligned}$$

and $\tilde{S}_\mathrm{ei}$ and $\tilde{S}_\mathrm{ie}$ both being zero matrices. Then, $\tilde{M}$ has a partitioning of the form (3.13) by identifying $S_\mathrm{ii}$, $S_\mathrm{ie}$, $S_\mathrm{ei}$, $S_\mathrm{ee}$, $L_\mathrm{i}$ and $L_\mathrm{e}$ with $\tilde{S}_\mathrm{ii}$, $\tilde{S}_\mathrm{ie}$, $\tilde{S}_\mathrm{ei}$, $\tilde{S}_\mathrm{ee}$ and $\tilde{L}_\mathrm{i}$, and $\tilde{L}_\mathrm{e}$, respectively. The reduced model matrix resulting from the subsequent simultaneous elimination of all internal edges $(s_{jk},r_{kj})$ $j,k=1,\ldots ,n,j\ne k$, can be conveniently determined by using the adjacency matrix $\eta$ defined by:

$$\begin{aligned} \eta= & {} \mathrm{diag}\left( \left[ \begin{array}{cc} 0 &{} I_{c_{12}} \\ I_{c_{21}} &{} 0\end{array}\right] , \left[ \begin{array}{cc} 0 &{} I_{c_{13}} \\ I_{c_{31}} &{} 0\end{array}\right] , \ldots , \right. \\&\quad \left. \left[ \begin{array}{cc} 0 &{} I_{c_{1n}} \\ I_{c_{n1}} &{} 0\end{array}\right] ,\left[ \begin{array}{cc} 0 &{} I_{c_{23}} \\ I_{c_{32}} &{} 0 \end{array}\right] , \left[ \begin{array}{cc} 0 &{} I_{c_{24}} \\ I_{c_{42}} &{} 0\end{array}\right] , \right. \\&\quad \left. \ldots , \left[ \begin{array}{cc} 0 &{} I_{c_{2n}} \\ I_{c_{n2}} &{} 0\end{array}\right] ,\ldots , \left[ \begin{array}{cc} 0 &{} I_{c_{(n-1)n}} \\ I_{c_{(n-1)n}} &{} 0\end{array}\right] \right) . \end{aligned}$$

(Recall that $c_{jk}=c_{kj}$). Hence, according to Theorem 3.3, the reduced model matrix $\tilde{M}_\mathrm{red}$ obtained after elimination of the internal edges $\{(s_{jk},r_{kj});\, j,k=1,\ldots ,n,j\ne k\}$ has parameters given by (recalling that $\tilde{S}_\mathrm{ei}$ and $\tilde{S}_\mathrm{ie}$ are zero matrices):

$$\begin{aligned} \tilde{S}_\mathrm{red}= & {} \tilde{S}_\mathrm{ee} + \tilde{S}_\mathrm{ei}(\eta -\tilde{S}_\mathrm{ii})^{-1}\tilde{S}_\mathrm{ie}= \tilde{S}_\mathrm{ee},\\ \tilde{L}_\mathrm{red}= & {} \tilde{L}_\mathrm{e}+\tilde{S}_\mathrm{ei}(\eta -\tilde{S}_\mathrm{ii})^{-1}\tilde{L}_\mathrm{i} = \tilde{L}_\mathrm{e},\\ \tilde{H}_\mathrm{red}= & {} \sum _{k=1}^n H_k + \sum _{j=\mathrm{i,e}} \mathfrak {I}\{\tilde{L}_j^{*}\tilde{S}_{j\mathrm{i}}(\eta -\tilde{S}_\mathrm{ii})^{-1}\tilde{L}_\mathrm{i}\} \\= & {} \sum _{k=1}^n H_k + \mathfrak {I}\{\tilde{L}_\mathrm{i}^{*}\tilde{S}_{\mathrm{ii}}(\eta -\tilde{S}_\mathrm{ii})^{-1}\tilde{L}_\mathrm{i}\} \\= & {} \sum _{k=1}^n H_k + \mathfrak {I}\{\tilde{L}_\mathrm{i}^{*}\eta (\eta -\tilde{S}_\mathrm{ii})^{-1}\tilde{L}_\mathrm{i}\} \\= & {} \sum _{k=1}^n H_k + \sum _{j=1}^{n-1} \sum _{k=j+1}^{n} \mathfrak {I}\biggl \{[\begin{array}{cc} L_{jk}^{*}&L_{kj}^{*}\end{array}] \left[ \begin{array}{cc} I &{} -S_{jk} \\ -S_{kj} &{} I \end{array} \right] ^{-1}\left[ \begin{array}{c} L_{jk} \\ L_{kj} \end{array} \right] \biggr \}. \end{aligned}$$

Since M and $\tilde{M}$ are model matrix representations of the same physical system and the external fields have the same ordering and labeling in both representations, the reduced model matrix of $M_\mathrm{red}$ and $\tilde{M}_\mathrm{red}$ of M and $\tilde{M}$, respectively, after elimination of internal edges $(s_{jk},r_{kj})$, coincide. Hence, also the linear quantum stochastic systems $G_\mathrm{red}$ and $\tilde{G}_\mathrm{red}$ associated with M and $\tilde{M}$, respectively, coincide. This completes the proof. $\Box$

Appendix C: Proof of Lemma 3.1

We begin by noting that

$$\begin{aligned} \mathfrak {I}\biggl \{\frac{S_{12}}{1-S_{12}S_{21}}K_1^{*}K_2+ \frac{S_{21}}{1-S_{12}S_{21}}K_1^{\top } K_2^{\#}\biggr \} =\frac{1}{2\imath }[\begin{array}{cc} -K_1^{*}\Delta ^*&K_1^{\top }\Delta \end{array}]\left[ \begin{array}{c} K_2 \\ K_2^{\#}\end{array} \right] , \end{aligned}$$

with $\Delta =\frac{S_{21}}{1-S_{21}S_{12}}-\frac{S_{12}^*}{1-S_{21}^*S_{12}^*}=2\frac{S_{21}-S_{12}^*}{|1-S_{21}S_{12}|^2}$ (exploiting the fact that $S_{12}S_{12}^*=1=S_{21}S_{21}^*$). Now, set $K_1=[\begin{array}{cc} \kappa&\imath \kappa \end{array}]$ for an arbitrary nonzero real constant $\kappa$, and note that $S_{21}S_{12} \ne 1$ implies that $\Delta \ne 0$ and:

$$\begin{aligned}{}[\begin{array}{cc} -K_1^{*}\Delta ^*&K_1^{\top }\Delta \end{array}]^{-1}= & {} \left[ \begin{array}{cc} -\kappa \Delta ^* &{} \kappa \Delta \\ \imath \kappa \Delta ^* &{} \imath \kappa \Delta \end{array}\right] ^{-1} \\= & {} -\frac{1}{2\imath \kappa ^2 |\Delta |^2}\left[ \begin{array}{cc} \imath \kappa \Delta &{} -\kappa \Delta \\ -\imath \kappa \Delta ^* &{} -\kappa \Delta ^* \end{array}\right] , \end{aligned}$$

and therefore for any real matrix V, $2\imath [\begin{array}{cc} -K_1^{*}\Delta ^*&K_1^{\top }\Delta \end{array}]^{-1}V=\left[ \begin{array}{c} Z \\ Z^{\#}\end{array}\right]$ for some complex row vector Z. Therefore, given any R, we see that we may solve the equation

$$ [\begin{array}{cc} -K_1^{*}\Delta ^*&K_1^{\top }\Delta \end{array}]\left[ \begin{array}{c} K_2 \\ K_2^{\#}\end{array} \right] =2\imath R, $$

for $K_2$ and this solution is as given in the statement of the corollary.

Alternatively, we could also have started by setting $K_2=[\begin{array}{cc} \kappa&\imath \kappa \end{array}]$ and analogously solving for $K_1$ for a given R. It is then an easy exercise that the solution for $K_1$ in this case is as stated in the corollary. $\Box$

Appendix D: Proof of Corollary 3.1

With $c_{jk}$, $S_{jk}$, $R_{jk}$, and $K_{jk}$, $j,k=1,\ldots ,n$, as defined in the statement of the corollary, from Theorem 3.4 and Lemma 3.1 we have that $S_\mathrm{red}=I_{nm}$, $L_\mathrm{red}=(L_{11}^{\top },L_{22}^{\top },\ldots ,L_{nn}^{\top })^{\top }$ with $L_{jj}=K_jx_j$, and

$$\begin{aligned} H_\mathrm{red}= & {} \sum _{j=1}^n H_j+ \sum _{j=1}^{n-1}\sum _{k=j+1}^{n} \mathfrak {I}\biggl \{ [\begin{array}{cc} L_{jk}^{*}&L_{kj}^{*} \end{array}] \left[ \begin{array}{cc} 1 &{} -S_{jk} \\ -S_{kj} &{} 1 \end{array} \right] ^{-1} \left[ \begin{array}{c} L_{jk} \\ L_{kj} \end{array}\right] \biggr \}. \end{aligned}$$

Expanding, we have:

$$\begin{aligned} H_\mathrm{red}= & {} (1/2)\sum _{j=1}^n x_j^{\top } R_{j} x_j + \sum _{j=1}^{n-1}\sum _{k=j+1}^{n} \mathfrak {I}\biggl \{ \frac{1}{1-S_{jk}S_{kj}} \\&\; \times (L_{jk}^{*}L_{jk}+S_{jk}L_{jk}^{*}L_{kj}+ S_{kj}L_{kj}^{*}L_{jk}+ L_{kj}^{*}L_{kj}) \biggr \}\\= & {} (1/2)\sum _{j=1}^{n} x_j^{\top } \biggl (R_{j}+ 2 \mathrm{sym}\biggl ( \sum _{k=1,k \ne j}^{n} \mathfrak {I}\{\frac{K_{jk}^{*}K_{jk}}{1-S_{jk}S_{kj}}\}\biggr )\biggr ) x_j\\&\; + \sum _{j=1}^{n-1}\sum _{k=j+1}^n x_j^{\top }\mathfrak {I}\{\frac{S_{jk}}{1-S_{jk}S_{kj}} K_{jk}^{*}K_{kj}+ \frac{S_{kj}}{1-S_{jk}S_{kj}} K_{jk}^{\top } K_{kj}^{\#} \}x_k\\= & {} (1/2)\sum _{j=1}^{n} x_j^{\top } R_{jj} x_j + \sum _{j=1}^{n-1}\sum _{k=j+1}^n x_j^{\top }\bigl (R_{jk}- \mathfrak {I}\{K_{j}^{\top } K_{k}^{\#}\}\bigr )x_k\\= & {} (1/2)x^{\top } R x - \sum _{j=1}^{n-1}\sum _{k=j+1}^{n}x_j^{\top } \mathfrak {I}\{K_{j}^{\top } K_{k}^{\#}\}x_k, \end{aligned}$$

where $\mathrm{sym}(A)=(1/2)(A+A^{\top })$, and $R=[R_{jk}]_{j,k=1,\ldots ,n}$ and $R_{kj}=R_{jk}^{\top }$. From this, it is clear that using the concatenation product we can decompose $G_\mathrm{red}$ as $G_\mathrm{red}=(0,0,H_\mathrm{red}) \boxplus \boxplus _{j=1}^{n} (I_m,L_{jj},0)$. Let $G_{\mathrm{red},0}=(0,0,H_\mathrm{red})$ and $G_{\mathrm{red},j}=(I_m,L_{jj},0)$, $j=1,\ldots ,n$. Now, using the series product rule, we easily compute that

$$\begin{aligned} G_\mathrm{net}= & {} G_{\mathrm{red},0} \boxplus (G_{\mathrm{red},n} \triangleleft \ldots \triangleleft G_{\mathrm{red},2} \triangleleft G_{\mathrm{red},1}) \\= & {} \biggl (0,0,(1/2)x^{\top } R x -\sum _{j=1}^{n-1}\sum _{k=j+1}^{n}x_j^{\top } \mathfrak {I}\{K_j^{\top } K_k^{\#}\} x_k\biggr ) \boxplus \\&\biggl (I_{m},[\begin{array}{cccc} K_1&K_2&\ldots&K_n \end{array}]x, \sum _{j=1}^{n-1}\sum _{k=j+1}^{n}x_j^{\top } \mathfrak {I}\{K_j^{\top } K_k^{\#}\} x_k\biggr ) \\= & {} \biggl ( I_{m},[\begin{array}{cccc} K_1&K_2&\ldots&K_n \end{array}]x, (1/2)x^{\top } R x\biggr ). \end{aligned}$$

Therefore, $G_\mathrm{net}$ realizes a linear quantum stochastic system with parameters $S_\mathrm{net}$, $L_\mathrm{net}$ and $H_\mathrm{net}$, as claimed.$\Box$