Optimization of Partially Isolated Quantum Harmonic Oscillator Memory Systems by Mean Square Decoherence Time Criteria^∗

The quantum communication and quantum information processing technologies, which undergo extensive theoretical and practical development, substantially rely on the possibility to manipulate and engineer quantum mechanical systems with novel properties exploiting nonclassical resources. The latter come from the noncommutative operator-valued structure of quantum variables and quantum states reflected in the Heisenberg and Schrödinger pictures of quantum dynamics and quantum probability, and the theory of quantum measurement [7]. One of the aspects of these approaches is that the fragile nature of quantum dynamics and quantum states, involving the atomic and subatomic scales [13], complicates the state initialization for such systems and their isolation from the environment in a controlled fashion. In particular, the controlled isolation is important in the context of quantum computation paradigms [16] where unitary transformations, performed with closed quantum systems [21] between the measurements, play a crucial role.

The absence of dissipation makes the perfectly reversible unitary dynamics of an isolated quantum system (including the preservation of the Hamiltonian and of the system variables or states in the zero-Hamiltonian case) particularly useful for storing quantum information. The storage phase is preceded by the “writing” stage and followed by the “reading” stage which can employ, for example, photonics-based platforms using light-matter interaction effects (see [4, 6, 28, 29, 30] and references therein). However, the ability to retain quantum states or dynamic variables over the course of time, which is required for the storage phase, is corrupted by the unavoidable coupling of the quantum system to its environment such as external fields.

The system-field interaction, even when it is weak, gives rise to quantum noise which makes the system drift away in a dissipative fashion from its initial condition as opposed to the ideal case of isolated zero-Hamiltonian system dynamics. The resulting open quantum systems are modelled by Hudson-Parthasarathy quantum stochastic differential equations (QSDEs) [10, 18] which are linear for open quantum harmonic oscillators (OQHOs) [11, 17, 19, 31] and quasi-linear for finite-level quantum systems [3, 24]. While the latter are particularly relevant for modelling qubit registers in the quantum computing context, continuous variables systems (such as the OQHOs with quantum positions and momenta) are also employed in quantum information theory including the Gaussian quantum channel models [8].

Using the quantum calculus framework, an approach has recently been proposed in [26, 27] for both classes of open quantum systems to their performance as a quantum memory in the storage phase. The system performance index is a decoherence time horizon at which a weighted mean-square deviation of the system variables from their initial values reaches a given fidelity threshold (note that different yet related decoherence measures are also discussed in [25] and references therein). The memory decoherence time (or its high-fidelity asymptotic approximation) can therefore be maximized over the energy and coupling parameters of a quantum network in order to improve its ability to approximately retain the initial conditions.

In the present paper, we follow the approach of [26] to OQHOs as Heisenberg picture quantum memory systems with position-momentum variables, whose internal dynamics and interaction with the environment are governed by linear QSDEs. Using the previously defined memory decoherence time, we apply this approach to a partially isolated subsystem of the oscillator, which is affected by the external fields only indirectly through another subsystem. The partial subsystem isolation from the external fields and the related system decomposition hold under a certain rank condition on the system-field coupling matrix and form an important special case which was not considered in [26]. This setting leads to a qualitatively different short-horizon asymptotic behaviour of the mean-square deviation of the subsystem variables from their initial values and yields a longer decoherence time in the high-fidelity limit. The maximization of the resulting approximate decoherence time over the energy parameters for improving the quantum memory performance is discussed for a coherent feedback interconnection of such systems involving the direct energy coupling and field-mediated coupling [32] between the constituent OQHOs.

The paper is organised as follows. Section II provides a minimum background material on QSDEs for open quantum stochastic systems. Section III specifies the class of OQHOs with position-momentum system variables and linear-quadratic energetics. Section IV considers weighted deviations of the system variables from their initial conditions and discusses a partially isolated subsystem along with the system decomposition. Section V reviews the mean-square quantification of such deviations and the memory decoherence time and studies the short-horizon and high-fidelity asymptotic behaviour of these functionals for the partially isolated subsystem. Section VI solves the approximate decoherence time maximization problem for a partially isolated subsystem of a coherent feedback interconnection of two OQHOs with direct energy and indirect field-mediated coupling.

As mentioned above, in comparison with the idealisation of completely isolated quantum dynamics, more realistic settings consider open quantum systems which interact with the environment, such as quantum fields or classical measuring devices. In the framework of the Hudson-Parthasarathy quantum stochastic calculus [10, 18], an open quantum system subject to external quantum fields, as shown in Fig. 1,

The CCRs (7) are typical for unbounded operators of position-momentum type [21] on a dense domain of an infinite-dimensional Hilbert space. Such quantum variables, when they are considered as the system variables $X_{1},\ldots,X_{n}$ with an even

$$n:=2\nu,$$

(8)

are provided by the quantum mechanical positions $q_{1},\ldots,q_{\nu}$ and momenta $p_{1},\ldots,p_{\nu}$ which can be implemented as the multiplication and differential operators

$$X_{2k-1}:=q_{k},\qquad X_{2k}:=p_{k}:=-i\partial_{q_{k}},\qquad k=1,\ldots,\nu,$$

(9)

on the Schwartz space of rapidly decreasing functions [20] on ${\mathbb{R}}^{\nu}$. At any time $t\geqslant 0$, these operators satisfy the one-point CCRs

$$[X(t),X(t)^{\mathrm{T}}]=2i\Theta,\qquad\Theta:=\frac{1}{2}I_{\nu}\otimes\mathbf{J}.$$

(10)

As continuous quantum variables, the conjugate position-momentum pairs (9) are qualitatively different from those in finite-level quantum systems [3, 24] and result from quantizing the classical positions and momenta of Hamiltonian mechanics [1]. They are employed as system variables in OQHOs [11, 17, 19, 31]. For an OQHO, the Hamiltonian $H$ and the system-field coupling operators $L_{1},\ldots,L_{m}$ are quadratic and linear functions of the system variables specified by an energy matrix $R=R^{\mathrm{T}}\in{\mathbb{R}}^{n\times n}$ and a coupling matrix $M\in{\mathbb{R}}^{m\times n}$ as

$$H:=\frac{1}{2}X^{\mathrm{T}}RX,\qquad L:=MX.$$

(11)

The general QSDEs (1) then acquire the form of linear QSDEs

$$\mathrm{d}X=AX\mathrm{d}t+B\mathrm{d}W,\qquad\mathrm{d}Y=CX\mathrm{d}t+D\mathrm{d}W,$$

(12)

where the matrices $A\in{\mathbb{R}}^{n\times n}$, $B\in{\mathbb{R}}^{n\times m}$, $C\in{\mathbb{R}}^{r\times n}$ are computed as

$$A:=A_{0}+\widetilde{A},\qquad B:=2\Theta M^{\mathrm{T}},\qquad C:=2DJM,$$

(13)

with

$$A_{0}:=2\Theta R,\qquad\widetilde{A}:=2\Theta M^{\mathrm{T}}JM,$$

(14)

in terms of the CCR matrices $J$, $\Theta$ from (6), (10) and the energy and coupling matrices $R$, $M$ from (11). In view of (14), the dynamics matrix $A$ in (13) depends linearly on $R$ and quadratically on $M$. Moreover, in addition to the noncommutative operator-valued nature of the quantum variables, the specific representation of the coefficients of the QSDEs (12) in terms of the commutation, energy and coupling parameters imposes quantum physical realizability constraints [11] in contrast to classical SDEs [12].

Despite the qualitative differences, OQHOs share some common features with classical linear stochastic systems due to the linearity of the QSDEs (12). In particular, the solution of the first QSDE is given by

$$X(t)=\mathrm{e}^{tA}X(0)+Z(t),\qquad Z(t):=\int_{0}^{t}\mathrm{e}^{(t-s)A}B\mathrm{d}W(s)$$

(15)

for any $t\geqslant 0$ and consists of the system responses to the initial condition $X(0)$ and the driving quantum Wiener process $W$ over the time interval $[0,t]$, with $Z(0)=0$. Here, the quantum process $Z$ of $n$ time-varying self-adjoint operators $Z_{1},\ldots,Z_{n}$ is adapted to the filtration of the Fock space $\mathfrak{F}$, and hence, $X(0)$ and $Z(t)$ commute with each other as operators acting on different spaces ($\mathfrak{H}_{0}$ and $\mathfrak{F}$):

$$[X(0),Z(t)^{\mathrm{T}}]=0,\qquad t\geqslant 0.$$

(16)

Furthermore, if the system-field quantum state on the space (2) has the form

$$\rho:=\rho_{0}\otimes\upsilon,$$

(17)

where $\rho_{0}$ is the initial system state on $\mathfrak{H}_{0}$, and $\upsilon$ is the vacuum field state [18] on $\mathfrak{F}$, then $Z$ is a Gaussian quantum process (see, for example, [23, Section 3]), which is statistically independent of $X(0)$ and has zero mean:

$$\mathbf{E}Z(t)=(\mathrm{Tr}(\upsilon Z_{k}(t)))_{1\leqslant k\leqslant n}=0.$$

(18)

Here, use is made of the quantum expectation $\mathbf{E}\zeta:=\mathrm{Tr}(\rho\zeta)$ of an operator $\zeta$ on the system-field space (2) over the quantum state (17), which reduces in (18) to the averaging over the vacuum state $\upsilon$ since the operators $Z_{k}$ act on the Fock space $\mathfrak{F}$. In combination with the independence between $X(0)$ and $Z(t)$, (18) implies that

$$\mathbf{E}(X(0)Z(t)^{\mathrm{T}})=\mathbf{E}X(0)\mathbf{E}Z(t)^{\mathrm{T}}=0,$$

(19)

where $\mathbf{E}X(0)=(\mathrm{Tr}(\rho_{0}X_{k}(0)))_{1\leqslant k\leqslant n}$. The above properties of $Z$ hold regardless of a particular structure of the initial system state $\rho_{0}$.

The process $Z$ in (15), caused by the interaction of the OQHO with the external fields, plays the role of a noise which corrupts the ability of the system as a quantum memory to retain the initial condition $X(0)$. Similarly to [26, 27], we describe the quantum memory performance in terms of a quadratic form

$$Q(t):=\xi(t)^{\mathrm{T}}\Sigma\xi(t)$$

(20)

of the deviation

$$\xi(t):=X(t)-X(0)=\alpha_{t}X(0)+Z(t)$$

(21)

of the system variables at time $t\geqslant 0$ from their initial values, where

$$\alpha_{t}:=\mathrm{e}^{tA}-I_{n},$$

(22)

with $\xi(0)=0$, $Q(0)=0$ and $\alpha_{0}=0$. Here, $\Sigma$ is a real positive semi-definite symmetric matrix of order $n$ factorised as

$$\Sigma:=F^{\mathrm{T}}F,\qquad F\in{\mathbb{R}}^{s\times n},\qquad s:=\mathrm{rank}\Sigma\leqslant n,$$

(23)

so that the rows of $F$ specify the coefficients of $s$ independent linear combinations of the deviations in (21) forming the vector

$$\eta(t):=F\xi(t)=FX(t)-FX(0)=F\alpha_{t}X(0)+FZ(t),$$

(24)

in terms of which the quantum process $Q$ in (20) is represented as

$$Q(t):=\eta(t)^{\mathrm{T}}\eta(t).$$

(25)

As discussed below, the matrix $F$ can be used for selecting those linear combinations (in particular, a subset) of the system variables, whose dynamics are affected by the external fields only indirectly and thus pertain to a partially isolated subsystem of the OQHO.

Since $\mathrm{rank}M\leqslant\min(m,n)\leqslant m$, then a sufficient condition for (26) is provided by the inequality $n>m$. Furthermore, if the null spaces of the matrices $F$, $G$ in (28) satisfy $\ker F\subset\ker G$ and hence, $G=NF$ for some matrix $N\in{\mathbb{R}}^{s\times s}$, then the ODE in (28) becomes autonomous: $\dot{\varphi}=N\varphi$, in which case, the dynamics of $\varphi$ are completely isolated from the external fields.

More generally, the matrix $F$ from Lemma 1 can be augmented by a matrix $T\in{\mathbb{R}}^{(n-s)\times n}$ to a nonsingular $(n\times n)$-matrix:

$$\det S\neq 0,\qquad S:={\begin{bmatrix}F\\ T\end{bmatrix}}\in{\mathbb{R}}^{n\times n}.$$

(32)

By partitioning the inverse matrix $S^{-1}$ into blocks $S_{1}\in{\mathbb{R}}^{n\times s}$ and $S_{2}\in{\mathbb{R}}^{n\times(n-s)}$ as $S^{-1}:=\begin{bmatrix}S_{1}&S_{2}\end{bmatrix}$ and introducing a quantum process $\psi$ of $n-s$ time-varying self-adjoint operators on (2), so that

$$X=S^{-1}\zeta=S_{1}\varphi+S_{2}\psi,\qquad\zeta:={\begin{bmatrix}\varphi\\ \psi\end{bmatrix}},\qquad\psi:=TX,$$

(33)

it follows that the first QSDE in (12) can be decomposed into an ODE and a QSDE with respect to the transformed system variables as

$$\mathop{\varphi}^{\centerdot}=a_{11}\varphi+a_{12}\psi,\qquad\mathrm{d}\psi=(a_{21}\varphi+a_{22}\psi)\mathrm{d}t+b\mathrm{d}W,$$

(34)

where

$$a:=SAS^{-1}:={\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}}={\begin{bmatrix}GS_{1}&GS_{2}\\ TAS_{1}&TAS_{2}\end{bmatrix}},\qquad b:=TB.$$

(35)

Therefore, the internal dynamics of the OQHO can be represented as a feedback interconnection of linear systems $\Phi$ and $\Psi$, with the corresponding vectors $\varphi$, $\psi$ of system variables, where only $\Psi$ interacts directly with the quantum Wiener process $W$, as shown in Fig. 2.

With a slight abuse of notation, the transfer functions of these systems are computed in terms of the matrices (35) as

	$\displaystyle\Phi(u)$	$\displaystyle=(uI_{s}-a_{11})^{-1}a_{12},$		(36)
	$\displaystyle\Psi(u)$	$\displaystyle=\begin{bmatrix}\Psi_{1}(u)&\Psi_{2}(u)\end{bmatrix}=(uI_{n-s}-a_{22})^{-1}\begin{bmatrix}a_{21}&b\end{bmatrix}$		(37)

and relate the Laplace transforms

	$\displaystyle\widehat{\varphi}(u)$	$\displaystyle:=\int_{0}^{+\infty}\mathrm{e}^{-ut}\varphi(t)\mathrm{d}t,\qquad\widehat{\psi}(u):=\int_{0}^{+\infty}\mathrm{e}^{-ut}\psi(t)\mathrm{d}t,$		(38)
	$\displaystyle\widehat{W}(u)$	$\displaystyle:=\int_{0}^{+\infty}\mathrm{e}^{-ut}\mathrm{d}W(t)$		(39)

(with the integral in (39) being of the Ito type in contrast to those in (38)) of the quantum processes $\varphi$, $\psi$, $W$ for all $u\in\mathbb{C}$ with sufficiently large $\mathrm{Re}u>0$ for convergence of the integrals as

	$\displaystyle\widehat{\varphi}(u)$	$\displaystyle=\chi_{1}(u)\zeta(0)+\Phi(u)\widehat{\psi}(u),$		(40)
	$\displaystyle\widehat{\psi}(u)$	$\displaystyle=\chi_{2}(u)\zeta(0)+\Psi(u){\begin{bmatrix}\widehat{\varphi}(u)\\ \widehat{W}(u)\end{bmatrix}}$
		$\displaystyle=\chi_{2}(u)\zeta(0)+\Psi_{1}(u)\widehat{\varphi}(u)+\Psi_{2}(u)\widehat{W}(u).$		(41)

Here, $\chi_{1}$, $\chi_{2}$ are $(s\times n)$ and $(n-s)\times n$-blocks of the transfer function $\chi(u):={\small\begin{bmatrix}\chi_{1}(u)\\ \chi_{2}(u)\end{bmatrix}}=(uI_{n}-a)^{-1}$ associated with the response of the quantum process $\zeta$ in (33) to its initial condition $\zeta(0)$. The equations (40), (41) can be solved for $\widehat{\varphi}$ as

	$\displaystyle\widehat{\varphi}(u)$	$\displaystyle=(\chi_{1}(u)+\Gamma(u)\chi_{2}(u))\zeta(0)+\Gamma(u)\Psi_{2}(u)\widehat{W}(u)$
		$\displaystyle=\begin{bmatrix}\chi_{1}(u)+\Gamma(u)\chi_{2}(u)&\Gamma(u)\Psi_{2}(u)\end{bmatrix}{\begin{bmatrix}\zeta(0)\\ \widehat{W}(u)\end{bmatrix}},$		(42)

where $\Gamma$ is an auxiliary transfer function associated with (36), (37) by

$$\Gamma(u):=\Phi(u)(I_{n-s}-\Psi_{1}(u)\Phi(u))^{-1}.$$

(43)

The Laplace transform of the quantum process $\eta$ in (24) is then found by substituting (42) into

$$\widehat{\eta}(u):=\int_{0}^{+\infty}\mathrm{e}^{-ut}\eta(t)\mathrm{d}t=\widehat{\varphi}(u)-\frac{1}{u}\varphi(0),$$

(44)

where $\varphi(0)=FX(0)$ from (27) is the subvector of $\zeta(0)$ in (33). Note that the resulting frequency-domain representation (44) of $\eta$ does not depend on a particular choice of the matrix $T$ in (32). In view of (42), the subsystem $\Phi$ with the vector $\varphi$ of transformed system variables can be approximately isolated from the external fields $W$ by making the transfer function $\Gamma\Psi_{2}$ “small” in a suitable sense. By (43), this requires “smallness” not only of $\Phi$ and $\Psi_{2}$, but also of $\Psi_{1}\Phi$ due to the presence of the feedback loop in Fig. 2, which is typical for small-gain theorem arguments (see, for example, [2] and references therein).

Following [26, 27], for any given time $t\geqslant 0$, we will quantify the “size” of the deviation $\eta(t)$ in (24) by a mean-square functional

$$\Delta(t):=\mathbf{E}Q(t)=\langle\Sigma,\mathrm{Re}\Upsilon(t)\rangle=\|F\alpha_{t}\sqrt{P}\|^{2}+\langle\Sigma,\mathrm{Re}V(t)\rangle.$$

(45)

This quantity provides an upper bound on the tail probabilities for the positive semi-definite self-adjoint quantum variable $Q(t)$ in (20), (25):

$$\mathbf{P}_{t}([z,+\infty))\leqslant\frac{\Delta(t)}{z},\qquad z>\Delta(t),$$

(46)

where Markov’s inequality [22] is applied to the probability distribution $\mathbf{P}_{t}(\cdot)$ of $Q(t)$ on $[0,+\infty)$ obtained by averaging its spectral measure [7] over the quantum state (17). In (45), use is made of the Frobenius norm $\|\cdot\|$ and inner product $\langle\cdot,\cdot\rangle$ of matrices [9] along with the one-point second-moment matrix of the process $\xi$: $\Upsilon(t):=\mathbf{E}(\xi(t)\xi(t)^{\mathrm{T}})=\alpha_{t}\Pi\alpha_{t}^{\mathrm{T}}+V(t)$. The latter is computed in [26] by using (16), (19), (21), (22) and the second-moment matrix

$$\Pi:=\mathbf{E}(X(0)X(0)^{\mathrm{T}})=P+i\Theta,\qquad P:=\mathrm{Re}\Pi$$

(47)

of the initial system variables $X_{1}(0),\ldots,X_{n}(0)$, including their CCR matrix $\Theta$ from (10), and the quantum covariance matrix

$$V(t):=\mathbf{E}(Z(t)Z(t)^{\mathrm{T}})=\int_{0}^{t}\mathrm{e}^{sA}B\Omega B^{\mathrm{T}}\mathrm{e}^{sA^{\mathrm{T}}}\mathrm{d}s,\qquad t\geqslant 0$$

(48)

of the process $Z$ in (15), with $\Omega$ the quantum Ito matrix from (5). The matrix $V(t)$ in (48), which coincides with the controllability Gramian of the pair $(A,B\sqrt{\Omega})$ over the time interval $[0,t]$, satisfies the initial value problem for the Lyapunov ODE

$$\mathop{V}^{\centerdot}(t)=AV(t)+V(t)A^{\mathrm{T}}+\mho,\qquad t\geqslant 0,\quad V(0)=0,$$

(49)

where

$$\mho:=B\Omega B^{\mathrm{T}}.$$

(50)

From (49), the time derivatives $V^{(k)}:=\mathrm{d}^{k}V/\mathrm{d}t^{k}$ can be computed recursively as $V^{(k+1)}=AV^{(k)}+V^{(k)}A^{\mathrm{T}}+\delta_{k0}\mho$ for any $k\geqslant 0$, where $\delta_{jk}$ is the Kronecker delta. In particular, the first three derivatives at $t=0$ are

	$\displaystyle\dot{V}(0)$	$\displaystyle=\mho,$		(51)
	$\displaystyle\ddot{V}(0)$	$\displaystyle=A\mho+\mho A^{\mathrm{T}},$		(52)
	$\displaystyle\dddot{V}(0)$	$\displaystyle=A^{2}\mho+\mho(A^{\mathrm{T}})^{2}+2A\mho A^{\mathrm{T}}.$		(53)

In what follows, we will use the short-horizon asymptotic behaviour of (45) described below.

Due to the special choice of the matrix $F$ in Lemmas 1 and 2, the leading term in (54) is quadratic (rather than linear) in time. This is a consequence of the partial isolation of the subsystem $\Phi$ from the external fields (see (58) and Fig. 2) and makes the mean-square deviation functional $\Delta$ grow slower (at least at short time horizons) compared to the case of arbitrary matrices $F$ considered in [26].

From the viewpoint of optimizing a network of OQHOs as a quantum memory system, the minimization (at a suitably chosen time $t>0$) of the mean-square deviation functional $\Delta(t)$ in (45) over admissible energy and coupling parameters of the network also minimizes the tail probability bounds (46) and provides a relevant performance criterion for such applications. This minimization improves the ability of the OQHO as a quantum memory to retain its initial system variables and is closely related (by duality) to maximizing the decoherence time

$$\tau(\epsilon):=\inf\{t\geqslant 0:\ \Delta(t)>\epsilon\|F\sqrt{P}\|^{2}\}$$

(60)

proposed in [26, 27] as a horizon by which the selected system variables do not deviate too far from their initial values. In accordance with (23), (47), the quantity $\mathbf{E}((FX(0))^{\mathrm{T}}FX(0))=\langle\Sigma,\Pi\rangle=\|F\sqrt{P}\|^{2}$ in (60) provides a reference scale, with respect to which the dimensionless parameter $\epsilon>0$ specifies a relative error threshold for $FX(t)$ to approximately reproduce $FX(0)$. Since the columns of the matrix $F$ in (23) are linearly dependent if $s<n$, we assume that

$$F\sqrt{P}\neq 0$$

(61)

in order to avoid the trivial case $\tau(\epsilon)=0$. In the partial isolation setting of Lemmas 1 and 2, where $F$ satisfies (29) (as opposed to the case of $FB\neq 0$ studied in [26, 27]), the asymptotic behaviour of (60) in the high-fidelity limit (of small values of $\epsilon$) is as follows.

As a consequence of the partial subsystem isolation, the asymptotic behaviour (63) is qualitatively different and yields a longer decoherence time compared to the case of $FB\neq 0$ in [26], where $\tau(\epsilon)$ is asymptotically linear with respect to $\epsilon$. In the framework of (63), the maximization of $\tau(\epsilon)$ at a given small value of $\epsilon$ can be replaced with its “approximate” version

$$\widehat{\tau}(\epsilon)\longrightarrow\sup.$$

(66)

With $F$ and $P$ being fixed, (66) is equivalent to the minimization of the denominator

$$\|G\sqrt{P}\|=2\|F\Theta R\sqrt{P}\|$$

(67)

and is organised as a convex quadratic optimization problem over the energy matrix $R$.

Consider the approximate memory decoherence time maximization (66) for a coherent feedback interconnection of two OQHOs from [25, Section 8] and [26], which interact with external input bosonic fields and are coupled to each other through a direct energy coupling and an indirect field-mediated coupling [32]; see Fig. 3.

In this setting, the quantum Wiener processes $W^{(1)}$, $W^{(2)}$ of the external fields of even dimensions $m_{1}$, $m_{2}$ are in the vacuum states $\upsilon_{1}$, $\upsilon_{2}$ on symmetric Fock spaces $\mathfrak{F}_{1}$, $\mathfrak{F}_{2}$, respectively, so that $W^{(1)}$, $W^{(2)}$ commute with and are statistically independent of each other. The augmented quantum Wiener process $W:={\small\begin{bmatrix}W^{(1)}\\ W^{(2)}\end{bmatrix}}$ of dimension $m:=m_{1}+m_{2}$ on the composite Fock space $\mathfrak{F}:=\mathfrak{F}_{1}\otimes\mathfrak{F}_{2}$ has the quantum Ito matrix $\Omega={\small\begin{bmatrix}\Omega_{1}&0\\ 0&\Omega_{2}\end{bmatrix}}$ in (5) coming from the individual Ito tables $\mathrm{d}W^{(k)}\mathrm{d}W^{(k)}{}^{\mathrm{T}}=\Omega_{k}\mathrm{d}t$, where $J={\small\begin{bmatrix}J_{1}&0\\ 0&J_{2}\end{bmatrix}}$, $\Omega_{k}:=I_{m_{k}}+iJ_{k}$ and $J_{k}:=I_{m_{k}/2}\otimes\mathbf{J}$, with the matrix $\mathbf{J}$ from (6). The component OQHOs are endowed with initial spaces $\mathfrak{H}_{k}$ and vectors $X^{(k)}$ of even numbers $n_{k}:=2\nu_{k}$ of dynamic variables on the composite system-field space (2), where $\mathfrak{H}_{0}:=\mathfrak{H}_{1}\otimes\mathfrak{H}_{2}$. Accordingly, the vector $X:={\small\begin{bmatrix}X^{(1)}\\ X^{(2)}\end{bmatrix}}$ of $n:=n_{1}+n_{2}$ system variables with $\nu=\nu_{1}+\nu_{2}$ in (8) for the augmented OQHO satisfies (10) with the CCR matrix

$$\Theta:={\begin{bmatrix}\Theta_{1}&0\\ 0&\Theta_{2}\end{bmatrix}}$$

(68)

which consists of the individual CCR matrices $\Theta_{k}=\frac{1}{2}I_{\nu_{k}}\otimes\mathbf{J}$:

$$[X^{(1)},X^{(2)}{}^{\mathrm{T}}]=0,\qquad[X^{(k)},X^{(k)}{}^{\mathrm{T}}]=2i\Theta_{k},\qquad k=1,2.$$

(69)

In addition to the individual Hamiltonians $H_{k}:=\frac{1}{2}X^{(k)}{}^{\mathrm{T}}R_{k}X^{(k)}$ of the OQHOs, parameterized by their energy matrices $R_{k}=R_{k}^{\mathrm{T}}\in{\mathbb{R}}^{n_{k}\times n_{k}}$, the direct energy coupling (see Fig. 3) contributes the term

$$H_{12}:=X^{(1)}{}^{\mathrm{T}}R_{12}X^{(2)}=X^{(2)}{}^{\mathrm{T}}R_{21}X^{(1)}$$

(70)

specified by

$$R_{12}=R_{21}^{\mathrm{T}}\in{\mathbb{R}}^{n_{1}\times n_{2}}$$

(71)

in view of the commutativity in (69). Since there is also an additional indirect coupling of the OQHOs, which is mediated by their output fields $Y^{(1)}$, $Y^{(2)}$ of even dimensions $r_{1}$, $r_{2}$, the resulting interconnection is governed by the QSDEs [26]

	$\displaystyle\mathrm{d}X^{(k)}$	$\displaystyle=(A_{k}X^{(k)}+F_{k}X^{(3-k)})\mathrm{d}t+B_{k}\mathrm{d}W^{(k)}+E_{k}\mathrm{d}Y^{(3-k)},$		(72)
	$\displaystyle\mathrm{d}Y^{(k)}$	$\displaystyle=C_{k}X^{(k)}\mathrm{d}t+D_{k}\mathrm{d}W^{(k)},\qquad k=1,2.$		(73)

The matrices $A_{k}\in{\mathbb{R}}^{n_{k}\times n_{k}}$, $B_{k}\in{\mathbb{R}}^{n_{k}\times m_{k}}$, $C_{k}\in{\mathbb{R}}^{r_{k}\times n_{k}}$, $E_{k}\in{\mathbb{R}}^{n_{k}\times r_{3-k}}$, $F_{k}\in{\mathbb{R}}^{n_{k}\times n_{3-k}}$ are parameterized as

	$\displaystyle A_{k}$	$\displaystyle=2\Theta_{k}(R_{k}+M_{k}^{\mathrm{T}}J_{k}M_{k}+N_{k}^{\mathrm{T}}\widetilde{J}_{3-k}N_{k}),\quad B_{k}=2\Theta_{k}M_{k}^{\mathrm{T}},$		(74)
	$\displaystyle C_{k}$	$\displaystyle=2D_{k}J_{k}M_{k},\qquad E_{k}=2\Theta_{k}N_{k}^{\mathrm{T}},\qquad F_{k}=2\Theta_{k}R_{k,3-k},$		(75)

and the matrices $D_{k}\in{\mathbb{R}}^{r_{k}\times m_{k}}$ consist of $r_{k}\leqslant m_{k}$ conjugate rows of permutation matrices of orders $m_{k}$ with $D_{k}\Omega_{k}D_{k}^{\mathrm{T}}=I_{r_{k}}+i\widetilde{J}_{k}$, where $\widetilde{J}_{k}:=D_{k}J_{k}D_{k}^{\mathrm{T}}=I_{r_{k}/2}\otimes\mathbf{J}$. Here, $M_{k}\in{\mathbb{R}}^{m_{k}\times n_{k}}$, $N_{k}\in{\mathbb{R}}^{r_{3-k}\times n_{k}}$ are the matrices of coupling of the $k$th OQHO to its external input field $W^{(k)}$ and the output $Y^{(3-k)}$ of the other OQHO, respectively. By (72), (73), the composite OQHO in Fig. 3 is governed by the first QSDE in (12) with the matrices

$$A={\begin{bmatrix}A_{1}&F_{1}+E_{1}C_{2}\\ F_{2}+E_{2}C_{1}&A_{2}\end{bmatrix}},\qquad B={\begin{bmatrix}B_{1}&E_{1}D_{2}\\ E_{2}D_{1}&B_{2}\end{bmatrix}}$$

(76)

and, by (74), (75), has the following energy and coupling matrices:

$$R=R_{*}+{\begin{bmatrix}0&R_{12}\\ R_{12}^{\mathrm{T}}&0\end{bmatrix}},\qquad M={\begin{bmatrix}M_{1}&D_{1}^{\mathrm{T}}N_{2}\\ D_{2}^{\mathrm{T}}N_{1}&M_{2}\end{bmatrix}}.$$

(77)

Here, the matrix

$$R_{*}:={\begin{bmatrix}R_{1}&N_{1}^{\mathrm{T}}D_{2}J_{2}M_{2}-M_{1}^{\mathrm{T}}J_{1}D_{1}^{\mathrm{T}}N_{2}\\ N_{2}^{\mathrm{T}}D_{1}J_{1}M_{1}-M_{2}^{\mathrm{T}}J_{2}D_{2}^{\mathrm{T}}N_{1}&R_{2}\end{bmatrix}}$$

(78)

is associated with the field-mediated coupling between the constituent OQHOs, their coupling to the external fields, and the individual Hamiltonians. Therefore, if the matrices $M_{k}$, $N_{k}$, $R_{k}$ are fixed for all $k=1,2$, and hence, so also is $R_{*}$ in (78), then the energy matrix $R$ in (77) can only be varied over a proper affine subspace in the subspace of real symmetric matrices of order $n$ by varying the direct energy coupling matrix $R_{12}$. This leads to the following formulation of the approximate memory decoherence time maximization problem (66):

$$\widehat{\tau}(\epsilon)\longrightarrow\sup\qquad{\rm over}\ R_{12}\in{\mathbb{R}}^{n_{1}\times n_{2}}.$$

(79)

This setting is similar to that in [26], except that we are concerned here with a subsystem of the closed-loop OQHO which is partially isolated from the external fields $W^{(1)}$, $W^{(2)}$.