On exponential stabilization of two-qubit systems

TL;DR

This paper develops feedback control laws for two-qubit systems under continuous measurements, ensuring exponential convergence to Bell states, with explicit formulas and numerical validation.

Contribution

It introduces explicit feedback control laws for stabilizing two-qubit systems to Bell states under continuous measurement, with conditions for exponential convergence.

Findings

Explicit feedback laws ensure almost sure exponential stabilization.

Single-channel case achieves asymptotic convergence.

Numerical simulations validate the control strategies.

Abstract

In this paper, we consider a two-qubit system undergoing continuous-time measurements. In presence of multiple channels, we provide sufficient conditions on the continuous feedback control law ensuring almost sure exponential convergence to a predetermined Bell state. This is obtained by applying stochastic tools, Lyapunov methods and geometric control tools. With one channel, we establish asymptotic convergence towards a predetermined Bell state. In both cases, we provide explicit expressions of feedback control laws satisfying the above-mentioned conditions. Finally, we demonstrate the effectiveness of our methodology through numerical simulations.

Equations

$$d{\rho }_t=F_0({\rho }_t)dt+\sum _{k=1}^n{F}_k({\rho }_t)dt+\sum _{k=1}^n\sqrt{{\eta }_k}{G}_k({\rho }_t)d{W}_k(t),$$

$$\begin{array}{rl}{F}_0(\rho )& :=-i[H_0,\rho ]-i{\sum }_{j=1}^m{u}_j(\rho )[H_j,\rho ],\\ F_k(\rho )& :=L_k\rho L_k-L_k^2\rho /2-\rho L_k^2/2,\\ G_k(\rho )& :=L_k\rho +\rho L_k-2\mathrm{Tr}(L_k\rho )\rho .\end{array}$$

$${\Psi }_{\pm }=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\otimes \left[\begin{array}{c}1\\ 0\end{array}\right]\pm \left[\begin{array}{c}0\\ 1\end{array}\right]\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\right)=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 0\\ 0\\ \pm 1\end{array}\right],$$

$${\Phi }_{\pm }=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ 0\end{array}\right]\otimes \left[\begin{array}{c}0\\ 1\end{array}\right]\pm \left[\begin{array}{c}0\\ 1\end{array}\right]\otimes \left[\begin{array}{c}1\\ 0\end{array}\right]\right)=\frac{1}{\sqrt{2}}\left[\begin{array}{c}0\\ 1\\ \pm 1\\ 0\end{array}\right].$$

$$d_B({\rho }_a,{\rho }_b):=\sqrt{2-2\mathrm{Tr}\left(\sqrt{\sqrt{{\rho }_b}{\rho }_a\sqrt{{\rho }_b}}\right)}.$$

$$d_B({\rho }_a,E):=\underset{\rho \in E}{\min }{d}_B({\rho }_a,\rho ).$$

$$B_r(E):=\{\rho \in \mathcal{S}|d_B(\rho ,E)<r\}.$$

$$\mathbb{P}\left({\rho }_t\in B_r(\bar{E})\text{ for }t\ge 0\right)\ge 1-\epsilon ,$$

$$\mathbb{P}\left(\underset{t\to \infty }{\lim }{d}_B({\rho }_t,\bar{E})=0\right)=1,$$

$$\mathbb{E}(d_B({\rho }_t,\bar{E}))\le \alpha d_B({\rho }_0,\bar{E})e^{-\beta t},$$

$$\underset{t\to \infty }{\mathrm{limsup}}\frac{1}{t}\log d_B({\rho }_t,\bar{E})<0,a.s.$$

$$\begin{array}{rl}\mathcal{L}V(q,t):=\frac{\partial V(q,t)}{\partial t}& +\sum _{i=1}^p\frac{\partial V(q,t)}{\partial q_i}{f}_i(q)+\frac{1}{2}\mathrm{Tr}\left(g^{\ast }(q)\frac{{\partial }^{2}V(q,t)}{\partial q_i\partial q_j}g(q)\right).\end{array}$$

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

$$\begin{array}{rl}d{\Lambda }_2({\rho }_t)& =-4\sqrt{{\eta }_1{M}_1}{\Lambda }_1({\rho }_t){\Lambda }_2({\rho }_t)d{W}_1(t)+4\sqrt{{\eta }_2{M}_2}\Delta ({\rho }_t)d{W}_2(t).\end{array}$$

$$|\mathbf{Re}\{{\rho }_{2,3}\}|\le {\Lambda }_2(\rho )/2,|\mathbf{Re}\{{\rho }_{1,4}\}|\le {\Lambda }_1(\rho )/2,$$

$$\begin{array}{rl}{V}_{\Psi }(\rho )& =1/{\Lambda }_2(\rho ),\text{mbox}if{\Lambda }_2(\rho )>\epsilon .\end{array}$$

$${\mathcal{S}}_{\Psi }\cap {\mathcal{R}}_{\pm }={\mathbf{\Psi }}_{\pm },{\mathcal{S}}_{\Phi }\cap {\mathcal{R}}_{\pm }={\mathbf{\Phi }}_{\pm }.$$

$$\begin{array}{rl}d\Gamma ({\rho }_t)& =2\sqrt{{\eta }_2{M}_2}(1-{\Gamma }^2({\rho }_t))d{W}_2(t)-8\sqrt{{\eta }_1{M}_1}\Delta ({\rho }_t)d{W}_1(t).\end{array}$$

$$d\Gamma ({\rho }_t)=2\sqrt{{\eta }_2{M}_2}(1-{\Gamma }^2({\rho }_t))d{W}_2(t).$$

$$\begin{array}{rl}{V}_{+}(\rho )& =1/(1+\Gamma (\rho )),\text{mbox}if1+\Gamma (\rho )>\epsilon ;\\ V_{-}(\rho )& =1/(1-\Gamma (\rho )),\text{mbox}if1-\Gamma (\rho )>\epsilon .\end{array}$$

$$V(\rho )=\sqrt{V_z(\rho )+V_x(\rho )}$$

$$\mathbb{E}(V({\rho }_t))=V({\rho }_0)-\bar{C}{\int }_0^t\mathbb{E}(V({\rho }_s))ds.$$

$$C_1{d}_B(\rho ,\bar{E})\le V(\rho )\le C_2{d}_B(\rho ,\bar{E}),$$

$$\mathbb{E}(d_B({\rho }_t,\bar{E}))\le \frac{C_2}{C_1}{d}_B({\rho }_0,\bar{E})e^{-\bar{C}t},\forall {\rho }_0\in \mathcal{S},$$

$$\underset{t\to \infty }{\mathrm{limsup}}\frac{1}{t}\log d_B({\rho }_t,\bar{E})\le -\bar{C},a.s.$$

$$V(\rho )=1/(1-X_{\bar{\mathbf{\rho }}}(\rho )),\text{mbox}if{X}_{\bar{\mathbf{\rho }}}(\rho )<1-\epsilon .$$

$$\begin{array}{rl}{\dot{\rho }}_v(t)=F_0({\rho }_v& (t))+\sum _{j=1}^2{\hat{F}}_j({\rho }_v(t))\sum _{j=1}^2\sqrt{{\eta }_j}{G}_j({\rho }_v(t))v_j(t),\end{array}$$

$$\begin{array}{rl}{\hat{F}}_1(\rho ):=& (1-{\eta }_1)(L_1\rho L_1-\rho )+2{\eta }_1\mathrm{Tr}(L_1\rho )G_1(\rho ),\\ {\hat{F}}_2(\rho ):=& (1-{\eta }_2)(L_2\rho L_2-\rho )+2{\eta }_2\mathrm{Tr}(L_2\rho )G_2(\rho ),\end{array}$$

$$\begin{array}{rl}{\dot{X}}_{\bar{\mathbf{\rho }}}({\rho }_v(t))=& {\Theta }_u({\rho }_v(t))+\sum _{j=1}^2{\Theta }_j({\rho }_v(t))+2X_{\bar{\mathbf{\rho }}}({\rho }_v(t))\sum _{j=1}^2\sqrt{{\eta }_j{M}_j}{P}_j({\rho }_v(t))v_j(t),\end{array}$$

$${\Theta }_1(\rho )$$

$${\Theta }_2(\rho )$$

Full text

1 Introduction

In view of the rapid development of quantum information science [1], the generation of quantum entangled states [2] has become essential in a variety of applications such as quantum teleportation, quantum cryptography and quantum computation. The simplest entangled states are the Bell states, which are pure states corresponding to maximal quantum entanglement of two spin-$\frac{1}{2}$ systems (i.e. a two-qubit system).

Two-qubit systems undergoing continuous-time measurements represent a particular example of open quantum systems whose evolution is described by stochastic master equations. The problem of controlling open quantum systems by feedback has received attention in the control community mainly starting from the years 2000s [3, 4, 5, 6, 7, 8, 9]. This field is a branch of the stochastic control theory whose first main ideas have been developed by Belavkin in [10].

Concerning stabilization of two-qubit systems, some interesting results have been derived in [5] and [8]. In [5], the authors design a switching quantum feedback controller that asymptotically stabilizes the system towards two specific Bell states. Then, in [8], the methods in [3] are adopted in order to construct a continuous feedback controller stabilizing the target Bell state starting from almost any initial pure state when the measurement is perfect.

In this paper we generalize the methods in [11, 12] in order to apply them to the feedback stabilization problem for two-qubit systems with multiple quantum channels. In this case, unlike the above-mentioned papers, the associated measurement operators may be degenerate rendering the stabilization problem more complicated. Here, we derive some general conditions on the feedback laws and the control Hamiltonians enforcing the exponential convergence towards the target Bell state. In addition, when only one quantum channel is available, we design a continuous feedback law which asymptotically stabilizes the system to an arbitrary Bell state. This control law is adapted from the switching feedback law proposed in [5], although our stability analysis is radically different.

This paper is organized as follows. In Section 2, we introduce the stochastic model describing two-qubit systems with multiple quantum channels in presence of imperfect measurements. In Section 3, we introduce the notions of stochastic stability needed throughout the paper. In Section 4, we show the exponential convergence of the system with two quantum channels and with zero control input towards the set of Bell states. We consider a class of appropriate control Hamiltonians and further propose necessary conditions on the continuous feedback law ensuring the exponential stabilization. In Section 5, explicit feedback laws are exhibited which asymptotically stabilize the system with only one quantum channel to the target state. Simulation results are provided in Section 6.

Notations: The imaginary unit is denoted by $i$. We take $\mathds{1}$ as the identity matrix. We denote the conjugate transpose of a matrix $A$ by $A^{*}.$ $A_{i,j}$ represents the element of the matrix $A$ at $i$-th row and $j$-th column. The function $\mathrm{Tr}(A)$ corresponds to the trace of a square matrix $A.$ The commutator of two square matrices $A$ and $B$ is denoted by $[A,B]:=AB-BA.$ $A\otimes B$ represents the Kronecker product of $A$ and $B$. For $x\in\mathbb{C}$, $\mathbf{Re}\{x\}$ is the real part of $x$ and $\mathbf{Im}\{x\}$ is the imaginary part of $x$.

2 System description

The dynamics of a two-qubit system undergoing continuous-time measurements with $n$ channels can be described by a matrix-valued stochastic differential equation of the form [3, 13, 14, 15]:

[TABLE]

where the quantum state is described by the density operator $\rho_{t}$, which belongs to the compact space $\mathcal{S}:=\{\rho\in\mathbb{C}^{4\times 4}|\,\rho=\rho^{*},\mathrm{Tr}(\rho)=1,\rho\geq 0\}$. Here $W_{t}=(W_{k}(t))_{1\leq k\leq n}$ is a $n$-dimensional standard Wiener process with the natural filtration $\mathcal{F}_{t},$ and $W_{k}$ are independent, i.e., for $i,j=1,\dots,n,$ one has $\left\langle W_{i}(t),W_{j}(t)\right\rangle=\delta_{i,j}t.$ The filtered probability space associated with the above evolution is $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P}).$ The measurement efficiency for the $k$-th channel is given by $\eta_{k}\in(0,1]$. The functions $F_{0}$, $F_{k}$ and $G_{k}$ are given by the following expressions

[TABLE]

The function $u$ appearing in $F_{0}$ denotes the feedback law taking values in $\mathbb{R}^{m},$ while $H_{0}=H^{*}_{0}\in\mathbb{C}^{4\times 4}$ is the free Hamiltonian, $H_{j}=H^{*}_{j}\in\mathbb{C}^{4\times 4}$ are the control Hamiltonians and $L_{k}=L^{*}_{k}\in\mathbb{C}^{4\times 4}$ are the measurement operators. If the feedback $u$ is in $\mathcal{C}^{1}(\mathcal{S},\mathbb{R}^{m})$, the existence and uniqueness of the solution of (1) as well as the strong Markov property of the solution are ensured by the results established in [5].

In the following sections, we consider the feedback stabilization of the above system towards one of the four Bell states $\boldsymbol{\Psi}_{\pm}=\Psi_{\pm}\Psi^{*}_{\pm}$ and $\boldsymbol{\Phi}_{\pm}=\Phi_{\pm}\Phi^{*}_{\pm}$ where

[TABLE]

These states are assumed to be common eigenstates of the measurement operators $L_{k}$.

3 Preliminary stochastic tools

In this section, we introduce some basic definitions which are fundamental for the rest of the paper.

Stochastic stability

We introduce some notions of stochastic stability needed throughout the paper by adapting classical notions (see e.g. [16, 17]) to our setting. In order to provide them, we first present the definition of Bures distance [2].

Definition 3.1**.**

The Bures distance between two quantum states $\rho_{a}$ and $\rho_{b}$ in $\mathcal{S}$ is defined as

[TABLE]

Also, the Bures distance between a quantum state $\rho_{a}$ and a set $E\subseteq\mathcal{S}$ is defined as

[TABLE]

Given $E\subseteq\mathcal{S}$ and $r>0$, we define the neighborhood $B_{r}(E)$ of $E$ as

[TABLE]

Definition 3.2**.**

Let $\bar{E}$ be an a.s. invariant set of system (1), then $\bar{E}$ is said to be

locally stable in probability*, if for every $\varepsilon\in(0,1)$ and for every $r>0$, there exists $\delta=\delta(\varepsilon,r)$ such that,*

[TABLE]

whenever $\rho_{0}\in B_{\delta}(\bar{E})$. 2. 2.

almost surely asymptotically stable*, if it is locally stable in probability and,*

[TABLE]

whenever $\rho_{0}\in\mathcal{S}$. 3. 3.

exponentially stable in mean*, if for some positive constants $\alpha$ and $\beta$,*

[TABLE]

whenever $\rho_{0}\in\mathcal{S}$. The smallest value $-\beta$ for which the above inequality is satisfied is called the average Lyapunov exponent. 4. 4.

almost surely exponentially stable*, if*

[TABLE]

whenever $\rho_{0}\in\mathcal{S}$. The left-hand side of the above inequality is called the sample Lyapunov exponent of the solution.

Note that any equilibrium $\bar{\rho}$ of (1), that is any quantum state satisfying $\sum^{n}_{k=0}F_{k}(\bar{\rho})=0$ and $G_{k}(\bar{\rho})=0$ for $k\in\{1,\dots,n\}$, is a special case of invariant set.

Infinitesimal generator and Itô’s formula

Given a stochastic differential equation $dq_{t}=f(q_{t})dt+g(q_{t})dW_{t}$, where $q_{t}$ takes values in $Q\subset\mathbb{R}^{p},$ the infinitesimal generator is the operator $\mathscr{L}$ acting on twice continuously differentiable functions $V:Q\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ in the following way

[TABLE]

Itô’s formula describes the variation of the function $V$ along solutions of the stochastic differential equation and is given as follows $dV(q,t)=\mathscr{L}V(q,t)dt+\frac{\partial V(q,t)}{\partial q_{i}}g(q)dW_{t}.$

With these basic tools, we are ready to analyse the stabilization problem of Equation (1). From now on, the operator $\mathscr{L}$ is associated with the equation (1) and we choose appropriate measurement operators so that Bell states are equilibria of (1). In particular, we set $\bar{E}:=\{\boldsymbol{\Psi}_{\pm},\boldsymbol{\Phi}_{\pm}\}$.

4 System with two quantum channels

In this section, we study the dynamics of the system (1) with two quantum channels, i.e., with two measurement operators $L_{1}$ and $L_{2}$ and, consequently, two diffusion terms. Consider the Pauli matrices

[TABLE]

We set $L_{1}=\sqrt{M_{1}}L_{z}$ with $M_{1}>0$ and $L_{2}=\sqrt{M_{2}}L_{x}$ with $M_{2}>0$, where $L_{z}:=\sigma_{z}\otimes\sigma_{z}$ and $L_{x}:=\sigma_{x}\otimes\sigma_{x}$. Here $M_{1},M_{2}>0$ are the strengths of the interaction between the light and the atoms. We also take $H_{0}=\omega L_{z}$ with $\omega\geq 0$ and use only one control Hamiltonian $H_{1}$. Note that the four Bell states coincide with the common eigenstates of the chosen operators $L_{1}$ and $L_{2}$.

4.1 Quantum State Reduction

Before attacking the exponential stabilization problem of the system (1), we analyze the asymptotic behavior of the system (1) with $u(\rho)\equiv 0$. This highlights the importance of the diffusion terms to design the feedback laws as already observed in [11, 12, 18].

We define $\Lambda_{1}(\rho):=\rho_{1,1}+\rho_{4,4}$, $\Lambda_{2}(\rho):=\rho_{2,2}+\rho_{3,3}$, $\mathcal{S}_{\Psi}:=\{\rho\in\mathcal{S}|\,\Lambda_{2}(\rho)=0\},$ $\mathcal{S}_{\Phi}:=\{\rho\in\mathcal{S}|\,\Lambda_{1}(\rho)=0\}$ and $\Delta(\rho):=\Lambda_{1}(\rho)\mathbf{Re}\{\rho_{2,3}\}-\Lambda_{2}(\rho)\mathbf{Re}\{\rho_{1,4}\}$. Then we state the following two lemmas, inspired by analogous result in [16, 17], identifying invariant subsets of the system.

Lemma 4.1**.**

Assume $u\equiv 0$. Then $\mathcal{S}_{\Psi}$ and $\mathcal{S}_{\Phi}$ are a.s. invariant for Equation (1). If the initial state satisfies $\rho_{0}\notin\mathcal{S}_{\Psi}$ or $\rho_{0}\notin\mathcal{S}_{\Phi}$, then $\mathbb{P}(\rho_{t}\notin\mathcal{S}_{\Psi},\forall\,t\geq 0)=1$ or $\mathbb{P}(\rho_{t}\notin\mathcal{S}_{\Phi},\forall\,t\geq 0)=1$ respectively.

Proof.

For $u\equiv 0,$ the dynamics of $\Lambda_{2}(\rho)$ is given by

[TABLE]

Since $\rho\geq 0$ one has

[TABLE]

so that $\Delta(\rho)=0$ for all $\rho\in\mathcal{S}_{\Psi}\cup\mathcal{S}_{\Phi}$, which yields the first part of the lemma for $\mathcal{S}_{\Psi}$.

Let us now prove the second part of the lemma. Given $\varepsilon>0$, consider any $\mathcal{C}^{2}$ function on $\mathcal{S}$ such that

[TABLE]

A simple computation shows that $\mathscr{L}V_{\Psi}(\rho)\leq RV_{\Psi}(\rho)$ if $\Lambda_{2}(\rho)>\varepsilon$ for some positive constant $R$. To conclude the proof in the case $\rho_{0}\notin\mathcal{S}_{\Psi}$, one just applies the same arguments as in [12, Lemma 4.1]. Roughly speaking, setting $f(\rho,t)=e^{-Rt}V_{\Psi}(\rho)$ one has $\mathscr{L}f\leq 0$ whenever $\Lambda_{2}>\varepsilon$. From this fact one proves that the probability of reaching $\mathcal{S}_{\Psi}$ in a finite fixed time $T$ is proportional to $\varepsilon$ and, being the latter arbitrary, it must be [math]. This conclude the proof in the case $\rho_{0}\notin\mathcal{S}_{\Psi}$. The same arguments lead to the result in the case $\rho_{0}\notin\mathcal{S}_{\Phi}$. $\square$

We denote $\Gamma(\rho):=2\mathbf{Re}\{\rho_{2,3}\}+2\mathbf{Re}\{\rho_{1,4}\}$, $\mathcal{R}_{+}:=\{\rho\in\mathcal{S}|\,\Gamma(\rho)=1\}$ and $\mathcal{R}_{-}:=\{\rho\in\mathcal{S}|\,\Gamma(\rho)=-1\}.$

Lemma 4.2**.**

Assume $u\equiv 0$. If $\rho_{0}\in\mathcal{S}_{\Psi}\setminus\{\boldsymbol{\Psi}_{\pm}\}$ or $\rho_{0}\in\mathcal{S}_{\Phi}\setminus\{\boldsymbol{\Phi}_{\pm}\}$, then $\mathbb{P}(\rho_{t}\neq\boldsymbol{\Psi}_{\pm},\forall\,t\geq 0)=1$ or $\mathbb{P}(\rho_{t}\neq\boldsymbol{\Phi}_{\pm},\forall\,t\geq 0)=1$ respectively.

Proof.

As $u\equiv 0,$ then $\boldsymbol{\Psi}_{\pm}$ and $\boldsymbol{\Phi}_{\pm}$ are four equilibria of Equation (1). Moreover, we have the following relations

[TABLE]

The dynamic of $\Gamma(\rho)$ is given by

[TABLE]

Due to Lemma 4.1, for all $\rho\in\mathcal{S}_{\Psi}\cup\mathcal{S}_{\Phi}$, we have

[TABLE]

Given $\varepsilon>0$, consider any $\mathcal{C}^{2}$ functions on $\mathcal{S}$ such that

[TABLE]

Then we have $\mathscr{L}V_{+}(\rho)\leq R_{+}V_{+}(\rho)$ if $1+\Gamma(\rho)>\varepsilon$ and $\mathscr{L}V_{-}(\rho)\leq R_{-}V_{-}(\rho)$ if $1-\Gamma(\rho)>\varepsilon$ for some positive constants $R_{+}$ and $R_{-}.$ To conclude the proof, one just applies the same arguments as in the previous lemma and [12, Lemma 4.1]. $\square$

For the system (1), the asymptotic convergence towards $\bar{E}=\{\boldsymbol{\Psi}_{\pm},\boldsymbol{\Phi}_{\pm}\}$ has been proved in [19]. We now show the exponential convergence towards $\bar{E}$ in mean and almost surely.

Theorem 4.3** (Quantum state reduction).**

For system (1), with $u\equiv 0$ and $\rho_{0}\in\mathcal{S},$ the set $\bar{E}$ is exponentially stable in mean and a.s. with average and sample Lyapunov exponent less or equal than $-\min\{\eta_{1}M_{1},\eta_{2}M_{2}\}$. Moreover, the probability of convergence to $\bar{\boldsymbol{\rho}}\in\bar{E}$ is $\mathrm{Tr}(\rho_{0}\bar{\boldsymbol{\rho}})$.

Proof.

Consider the function

[TABLE]

as a candidate Lyapunov function, where $V_{z}(\rho):=\Lambda_{1}(\rho)\Lambda_{2}(\rho)$ and $V_{x}(\rho):=1-\Gamma^{2}(\rho)$. Note that $4V_{z}(\rho)=\mathrm{Tr}(L^{2}_{z}\rho)-\mathrm{Tr}^{2}(L_{z}\rho)$ and $V_{x}(\rho)=\mathrm{Tr}(L^{2}_{x}\rho)-\mathrm{Tr}^{2}(L_{x}\rho)$ can be considered as the “variance” functions of $L_{z}$ and $L_{x}$ respectively. Moreover, $V(\rho)=0$ if and only if $\rho\in\bar{E}$. Due to Lemmas 4.1 and 4.2, the complementary set of $\bar{E}$ is invariant. Since $V$ is twice continuously differentiable in this set, we can apply Itô’s formula. We have $\mathscr{L}V(\rho)\leq-\bar{C}V(\rho)$ with $\bar{C}:=\min\{\eta_{1}M_{1},\eta_{2}M_{2}\}$. For all $\rho_{0}\in\mathcal{S}$, we have

[TABLE]

In virtue of Grönwall inequality, we have $\mathbb{E}(V(\rho_{t}))\leq V(\rho_{0})e^{-\bar{C}t}.$ By a straightforward calculation, we can show that the candidate Lyapunov function is bounded from below and above by the Bures distance from $\bar{E}$,

[TABLE]

where $C_{1}=1/\sqrt{6}$ and $C_{2}=2\sqrt{2}$. It implies

[TABLE]

which means that the set $\bar{E}$ is exponentially stable in mean with average Lyapunov exponent less or equal than $-\bar{C}$.

Now we consider the stochastic process $Q(\rho_{t},t)=e^{\bar{C}t}V(\rho_{t})\geq 0$ whose infinitesimal generator is given by $\mathscr{L}Q(\rho,t)=e^{\bar{C}t}(\bar{C}\,V(\rho)+\mathscr{L}V(\rho))\leq 0.$ Hence, the process $Q(\rho_{t},t)$ is a positive supermartingale. Due to Doob’s martingale convergence theorem [20], the process $Q(\rho_{t},t)$ converges almost surely to a finite limit as $t$ tends to infinity. Consequently, $Q(\rho_{t},t)$ is almost surely bounded, that is $\sup_{t\geq 0}Q(\rho_{t},t)=A$, for some a.s. finite random variable $A$. This implies $\sup_{t\geq 0}V(\rho_{t})=Ae^{-\bar{C}t}$ a.s. Letting $t$ goes to infinity, we obtain $\limsup_{t\rightarrow\infty}\frac{1}{t}\log V(\rho_{t})\leq-\bar{C}$ a.s. By the inequality (3),

[TABLE]

which means that the set $\bar{E}$ is a.s. exponentially stable with sample Lyapunov exponent less or equal than $-\bar{C}$.

Finally, the fact that the probability of convergence to $\bar{\boldsymbol{\rho}}\in\bar{E}$ is $\mathrm{Tr}(\rho_{0}\bar{\boldsymbol{\rho}})$ may be proved by standard arguments (see e.g [12, Theorem 5.1]). The proof is complete. $\square$

4.2 Exponential stabilization by continuous feedback

In this section, we study the exponential stabilization of system (1) towards a target state $\bar{\boldsymbol{\rho}}\in\bar{E}$. We first establish a general result ensuring the exponential convergence towards $\bar{\boldsymbol{\rho}}$ under some assumptions on the feedback law and an additional local Lyapunov type condition. Next, we design a parametrized family of feedback control laws satisfying such conditions for some choice of the control Hamiltonian. Denote $X_{\bar{\boldsymbol{\rho}}}(\rho):=\mathrm{Tr}(\rho\bar{\boldsymbol{\rho}})$ and $\Theta_{u}(\rho):=u(\rho)\mathrm{Tr}(i[H_{1},\rho]\bar{\boldsymbol{\rho}})$.

Lemma 4.4**.**

Assume that the initial state satisfies $\rho_{0}\neq\bar{\boldsymbol{\rho}},$ $u\in\mathcal{C}^{1}(\mathcal{S}\setminus\bar{\boldsymbol{\rho}},\mathbb{R})$ and $|\Theta_{u}(\rho)|\leq C(1-X_{\bar{\boldsymbol{\rho}}}(\rho))$ for some $C>0$ . Then $\mathbb{P}(\rho_{t}\neq\bar{\boldsymbol{\rho}},\forall\,t\geq 0)=1.$

Proof.

Given $\varepsilon>0,$ we consider any $\mathcal{C}^{2}$ function on $\mathcal{S}$ such that

[TABLE]

Under the assumptions of the lemma, it is easy to check that $\mathscr{L}V(\rho)\leq KV(\rho)$ for some $K>0,$ whenever $X_{\bar{\boldsymbol{\rho}}}(\rho)<1-\varepsilon$. To conclude the proof, one just applies the same arguments as in Lemma 4.1 and [12, Lemma 4.1]. $\square$

Generally speaking, based on the support theorem [21], trajectories of Equation (1) may be interpreted as limits of solutions of the following deterministic equation

[TABLE]

with $v_{1}(t),v_{2}(t)\in\mathcal{V}$, where $\mathcal{V}$ is the set of all piecewise constant functions from $\mathbb{R}_{+}$ to $\mathbb{R}$, and

[TABLE]

with $F_{0}$, $G_{1}$ and $G_{2}$ defined as in (1). In particular, the set $\mathcal{S}$ is positively invariant for Equation (5).

Lemma 4.5**.**

Let $\bar{\boldsymbol{\rho}}=\xi\xi^{*}$ with $\xi\in\{\Psi_{\pm},\Phi_{\pm}\}.$ Assume that $u\in\mathcal{C}^{1}(\mathcal{S}\setminus\bar{\boldsymbol{\rho}},\mathbb{R})$, $u\neq 0$ on the set $\{\rho|\,X_{\bar{\boldsymbol{\rho}}}(\rho)=0\}$ and $\xi$ is not an eigenvector of $H_{1}^{2}$. Then for all $r>0$ and any given initial state $\rho_{0}\in\mathcal{S},$ $\mathbb{P}(\tau_{r}<\infty)=1,$ where $\tau_{r}:=\inf\{t\geq 0|\,\rho_{t}\in B_{r}(\bar{\boldsymbol{\rho}})\}$ and $\rho_{t}$ corresponds to the solution of system (1).

Proof.

The lemma holds trivially for $\rho_{0}\in B_{r}(\bar{\boldsymbol{\rho}})$, as in this case $\tau_{r}=0$. Let us thus suppose that $\rho_{0}\in\mathcal{S}\setminus B_{r}(\bar{\boldsymbol{\rho}})$. We show that there exists $T\in(0,\infty)$ and $\zeta\in(0,1)$ such that $\mathbb{P}_{\rho_{0}}(\tau_{r}<T)>\zeta$. For this purpose, we make use of the support theorem. Consider the following differential equation derived from (5),

[TABLE]

where $v_{1}(t),v_{2}(t)\in\mathcal{V}$ is the control input, and

[TABLE]

Denote that $\mathbf{P}_{1}:=\{\rho\in\mathcal{S}|\,P_{1}(\rho)=0\}$ and $\mathbf{P}_{2}:=\{\rho\in\mathcal{S}|\,P_{2}(\rho)=0\}$, we now analyze the following four different cases ,

If $\bar{\boldsymbol{\rho}}=\boldsymbol{\Psi}_{\pm}$, then $\mathbf{P}_{1}\cap\mathbf{P}_{2}=\mathcal{S}_{\Psi}\cap\mathcal{R}_{\pm}=\boldsymbol{\Psi}_{\pm}$; 2. 2.

If $\bar{\boldsymbol{\rho}}=\boldsymbol{\Phi}_{\pm}$, then $\mathbf{P}_{1}\cap\mathbf{P}_{2}=\mathcal{S}_{\Phi}\cap\mathcal{R}_{+}=\boldsymbol{\Phi}_{\pm}$.

Suppose $\bar{\boldsymbol{\rho}}=\boldsymbol{\Psi}_{+}$. Due to the assumption of the lemma on the feedback law and $H_{1}$, one can easily show that $X_{\bar{\boldsymbol{\rho}}}(\rho_{v}(t))>0$ for $t>0.$ For $t>0,$ we can thus take the feedback $v_{1}=KP_{1}(\rho)/X_{\bar{\boldsymbol{\rho}}}(\rho)$ and $v_{2}=KP_{2}(\rho)/X_{\bar{\boldsymbol{\rho}}}(\rho)$ with $K>0$ sufficiently large. The proposed control input $v$ guarantees that $\rho_{v}(t)\in B_{r}(\bar{\boldsymbol{\rho}})$ for $t\leq T$ with $T<\infty$. The other three Bell states can be treated in the same way. Now, considering the stochastic solution of (1), we deduce that $\mathbb{P}(\rho_{t}\in B_{r}(\bar{\boldsymbol{\rho}}))>0$ for $t\leq T$ from the support theorem [21].

By compactness of $\mathcal{S}\setminus B_{r}(\bar{\boldsymbol{\rho}})$ and the Feller continuity of $\rho_{t},$ we have $\sup_{\rho_{0}\in\mathcal{S}\setminus B_{r}(\bar{\boldsymbol{\rho}})}\mathbb{P}_{\rho_{0}}(\tau_{r}\geq T)\leq 1-\zeta<1,$ 111Note that $\mathbb{P}_{\rho_{0}}$ corresponds to the probability law of $\rho_{t}$ starting at $\rho_{0}$ and the associated expectation is denoted by $\mathbb{E}_{\rho_{0}}$. for some $\zeta>0.$ By Dynkin inequality [22],

[TABLE]

Then by Markov inequality, for all $\rho_{0}\in\mathcal{S}\setminus B_{r}(\bar{\boldsymbol{\rho}})$, we have

[TABLE]

which implies $\mathbb{P}_{\rho_{0}}(\tau_{r}<\infty)=1.$ The proof is complete. $\square$

By combining the previous lemmas and following arguments similar to [12, Theorem 6.2], we get the following general result concerning the exponential stabilization towards Bell states.

Theorem 4.6**.**

Assume that $\rho_{0}\in\mathcal{S}$ and the feedback control law satisfies the assumptions of Lemma 4.4 and Lemma 4.5. Additionally, suppose that there exists a positive-definite function $V(\rho)$ such that $V(\rho)=0$ if and only if $\rho=\bar{\boldsymbol{\rho}}$, and $V$ is continuous on $\mathcal{S}$ and twice continuously differentiable on the set $\mathcal{S}\setminus\bar{\boldsymbol{\rho}}$. Moreover, suppose that there exist positive constants $C$, $C_{1}$ and $C_{2}$ such that

1. (i)

$C_{1}\,d_{B}(\rho,\bar{\boldsymbol{\rho}})\leq V(\rho)\leq C_{2}\,d_{B}(\rho,\bar{\boldsymbol{\rho}})$, $\forall\,\rho\in\mathcal{S}$, and 2. (ii)

$\limsup_{\rho\rightarrow\bar{\boldsymbol{\rho}}}\frac{\mathscr{L}V(\rho)}{V(\rho)}=-C$.

Then, $\bar{\boldsymbol{\rho}}$ is a.s. exponentially stable for the system (1) with sample Lyapunov exponent less or equal than $-C-\frac{K}{2}$, where K:=\liminf_{\rho\rightarrow\bar{\boldsymbol{\rho}}}\big{(}g_{1}^{2}(\rho)+g_{2}^{2}(\rho)\big{)} with $g_{j}(\rho):=\sqrt{\eta_{j}}\frac{\partial V(\rho)}{\partial\rho}\frac{G_{j}(\rho)}{V(\rho)}$ for $j=1,2$.

Next, we derive a general condition on the feedback law and the control Hamiltonian which allows us to apply the previous theorem.

Theorem 4.7**.**

Let $\rho_{0}\in\mathcal{S}$ and $\bar{\boldsymbol{\rho}}\in\bar{E}$ be the target state. Suppose that the feedback law and control Hamiltonian satisfy Lemma 4.4, Lemma 4.5 and the following relation

[TABLE]

Then $\bar{\boldsymbol{\rho}}$ is almost surely exponentially stable with sample Lyapunov exponent less or equal than $-\bar{C}$ where $\bar{C}=\min\{\eta_{1}M_{1},\eta_{2}M_{2}\}.$

Proof.

To prove the theorem, we show that we can apply Theorem 4.6 with the Lyapunov function $V(\rho)=\sqrt{1-X_{\bar{\boldsymbol{\rho}}}(\rho)}$ with $\bar{\boldsymbol{\rho}}\in\bar{E}$. Note that $d_{B}(\rho,\bar{\boldsymbol{\rho}})\leq V(\rho)\leq\sqrt{2}d_{B}(\rho,\bar{\boldsymbol{\rho}}),$ we are then left to show the condition (ii). The infinitesimal generator of the Lyapunov function satisfies,

[TABLE]

Since $\rho\geq 0$, by estimating the right hand side of the above inequality, we obtain the following for all $\rho\in\mathcal{S}\setminus\bar{\boldsymbol{\rho}}$,

[TABLE]

Since $g_{1}^{2}(\rho)+g_{2}^{2}(\rho)\geq\bar{C}X^{2}_{\bar{\boldsymbol{\rho}}}(\rho)$ and by using the relation (6), we can apply Theorem 4.6 with $C=\bar{C}/2$ and $K=\bar{C}.$ The proof is hence complete. $\square$

The application of the previous results is given below.

Proposition 4.8**.**

Consider system (1) with $\rho_{0}\in\mathcal{S}$. Let $\bar{\boldsymbol{\rho}}\in\bar{E}$ be the target state. Define the control Hamiltonian as $H_{1}=\sigma_{z}\otimes\sigma_{y}-3(\mathds{1}\otimes\sigma_{y})$ and the feedback law as

[TABLE]

where $\gamma>0,$ $\beta>1$ and $\alpha>0$ sufficiently large. Then $\bar{\boldsymbol{\rho}}$ is almost surely exponentially stable with sample Lyapunov exponent less or equal than $-\bar{C}$ where $\bar{C}=\min\{\eta_{1}M_{1},\eta_{2}M_{2}\}.$

Proof.

Since $\rho\geq 0$, we can show that the feedback law and the control Hamiltonian satisfy the relation (6) and the assumptions of Lemma 4.4 and Lemma 4.5. The proof is complete by applying Theorem 4.7. $\square$

5 System with one quantum channel

In this section, our purpose is to stabilize the system (1) towards a target Bell state $\bar{\boldsymbol{\rho}}\in\bar{E}$ with only one measurement operator $L_{1}=\sqrt{M_{1}}L_{z}$. Unlike the previous case, the diffusion terms strengthen the exponential convergence towards $\mathcal{S}_{\Psi}\cup\mathcal{S}_{\Phi}$ instead of $\bar{E}$. Hence, we do not expect to obtain exponential stabilization towards $\bar{\boldsymbol{\rho}}$ by using the methods developed before. Hence, we focus on asymptotic stabilization of the system (1). Here, we take $H_{0}=\omega L_{z}$ with $\omega>0$.

The following result is analogous to Lemma 4.5.

Lemma 5.1**.**

Let $\bar{\boldsymbol{\rho}}=\xi\xi^{*}$ with $\xi\in\{\Psi_{\pm},\Phi_{\pm}\}.$ Assume that $u\in\mathcal{C}^{1}(\mathcal{S}\setminus\bar{\boldsymbol{\rho}},\mathbb{R}^{m})$, $u_{1}\neq 0$ on the set $\{\rho|\,X_{\bar{\boldsymbol{\rho}}}(\rho)=0\}$ and $\xi$ is not an eigenvector of $H_{1}^{2}$. Suppose moreover $u_{k}\equiv 0$ for $k>1$ on the above set. Then for all $r>0$ and any given initial state $\rho_{0}\in\mathcal{S},$ $\mathbb{P}(\tau_{r}<\infty)=1,$ where $\tau_{r}:=\inf\{t\geq 0|\,\rho_{t}\in B_{r}(\bar{\boldsymbol{\rho}})\}$ and $\rho_{t}$ corresponds to the solution of System (1).

By employing the first two steps of the proof of [12, Theorem 6.2], we can obtain the general result concerning the asymptotic stabilization of System (1) with only one quantum channel towards the target Bell state.

Theorem 5.2**.**

Assume that the feedback law $u$ satisfies the assumptions of Lemma 5.1. Additionally, suppose that there exists a positive function $V(\rho)$ such that $V(\rho)=0$ if and only if $\rho=\bar{\boldsymbol{\rho}}$, and $V$ is continuous on $\mathcal{S}$ and twice continuously differentiable on the set $\mathcal{S}\setminus\bar{\boldsymbol{\rho}}$. Moreover, suppose that there exist positive constants $C$, $C_{1}$ and $C_{2}$ such that

1. (i)

$C_{1}\,d^{p}_{B}(\rho,\bar{\boldsymbol{\rho}})\leq V(\rho)\leq C_{2}\,d^{p}_{B}(\rho,\bar{\boldsymbol{\rho}})$* with $p>0$, for all $\rho\in\mathcal{S}$, and* 2. (ii)

$\mathscr{L}V(\rho)\leq 0,$* for all $\rho\in B_{r}(\bar{\boldsymbol{\rho}})$ with $r>0$.*

Then, $\bar{\boldsymbol{\rho}}$ is a.s. asymptotically stable for the system (1).

Remark 5.3**.**

Theorem 5.2 ensures the global asymptotic stabilization of the system only providing local Lyapunov type condition. The further assumptions on $u_{k}$ and $H_{k}$ are used to avoid the presence of invariant subsets of $\mathcal{S}$. These conditions are not optimal and may be easily weakened. We believe that by applying [12, Proposition 4.5], we can relax these conditions for the case $\eta<1$. We note that we do not need to find a global Lyapunov condition or apply the LaSalle theorem as in [5, 8].

Next, we define the following continuously differentiable function on $[0,1]$,

[TABLE]

where $\epsilon\in(0,1/2)$. Denote $\Pi_{1}:=-\sigma_{y}\otimes\sigma_{z}-3(\sigma_{y}\otimes\mathds{1})$ and $\Pi_{2}:=\sigma_{y}\otimes\sigma_{z}-3(\sigma_{y}\otimes\mathds{1})$. Then we propose the following continuous feedback law and control Hamiltonians inspired by [5, Theorem 5.1].

Proposition 5.4**.**

Consider the system (1) with $\rho_{0}\in\mathcal{S}$. Let $\bar{\boldsymbol{\rho}}\in\bar{E}$ be the target state. Define $H_{1}=\sigma_{z}\otimes\sigma_{y}-3(\mathds{1}\otimes\sigma_{y})$ and the feedback laws in the following form

[TABLE]

where $|\gamma_{1}|=|\gamma_{2}|$ sufficient large. If

• •

$\bar{\boldsymbol{\rho}}=\boldsymbol{\Psi}_{\pm}$, take $\gamma_{1}=\pm\gamma_{2}$ and $H_{2}=\Pi_{1}$;

• •

$\bar{\boldsymbol{\rho}}=\boldsymbol{\Phi}_{\pm}$, take $\gamma_{1}=\pm\gamma_{2}$ and $H_{2}=\Pi_{2}$.

Then $\bar{\boldsymbol{\rho}}$ is a.s. asymptotically stable.

Proof.

We apply Theorem 5.2 with the Lyapunov function $V(\rho)=1-X_{\bar{\boldsymbol{\rho}}}(\rho)$. We can easily verify that the feedback law and control Hamiltonians satisfy the assumptions of Lemma 5.1, $d^{2}_{B}(\rho,\bar{\boldsymbol{\rho}})\leq V(\rho)\leq 2d^{2}_{B}(\rho,\bar{\boldsymbol{\rho}})$ in $\mathcal{S}$ and $\mathscr{L}V(\rho)\leq 0$ in a neighborhood of $\bar{\boldsymbol{\rho}}$. Hence, the proof is complete. $\square$

6 Simulations

In this section, we simulate the dynamics of two-qubit systems in order to illustrate our results.

The simulations in the case with two quantum channels are shown in Fig. 1, Fig. 2 and Fig. 3. Fig. 1 shows the case $u\equiv 0$; we observe that the expectation of the Lyapunov function $\mathbb{E}(V(\rho_{t}))$ is bounded by the exponential function $V(\rho_{0})e^{-\bar{C}t}$, and the expectation of the Bures distance $\mathbb{E}(d_{B}(\rho_{t},\bar{E}))$ is always bounded by $4\sqrt{3}d_{B}(\rho_{0},\bar{E})e^{-\bar{C}t}$ which confirms the results of Theorem 4.3. Then we set $\boldsymbol{\Psi}_{+}$ as the target state and $\boldsymbol{\Phi}_{-}$ as the initial state; the behavior of the system with the continuous feedback (7) is shown in Fig. 2. Similar simulations for the case with $\boldsymbol{\Phi}_{-}$ as the target state and $\boldsymbol{\Psi}_{+}$ as initial state are shown in Fig. 3.

The simulations in the case with only one quantum channel are shown in Fig. 4 and Fig. 5 for $\boldsymbol{\Psi}_{+}$ as the target state and $\boldsymbol{\Phi}_{-}$ as the target state respectively. Such simulations clearly confirm the validity of Proposition 5.4.

7 Conclusion and future works

In this paper, we have studied the asymptotic behavior of trajectories of two-qubit systems. In particular, for the case of two quantum channels, we have provided a general result concerning the feedback exponential stabilization towards the Bell states by applying local stochastic Lyapunov techniques and analyzing the asymptotic behavior of quantum trajectories. Furthermore, we constructed a parameterized continuous feedback law satisfying the conditions of our general results. Next, for the system with one quantum channel, by a similar analysis, we proposed a continuous feedback law stabilizing the system asymptotically.

Further research lines will address the possibility of extending our results to multi-qubit entanglement generation, or in presence of delays.