Design of Strict Control-Lyapunov Functions for Quantum Systems with QND Measurements

Introduction

Feedback stabilization of quantum states is closely related to the concept of Quantum Non-Demolition (QND) measurement . Indeed, as soon as we are interested in applying a measurement-based feedback to stabilize a quantum state, we need to make sure that the measurement itself is not changing the desired target state. This means that the measurement procedure is QND with respect to the projection over the target state. In fact, very often a well-chosen QND measurement protocol can itself be considered as a preparation tool for various quantum states. However, this preparation is generally non-deterministic and one can not make sure to converge towards the desired state except by repeating the experiment many times. The feedback can be applied here to make this process deterministic .

This paper is a generalization of the feedback law, proposed in and experimentally tested in , to generic discrete-time quantum systems where, between two successive QND measurements, a controlled unitary evolution can be applied. The dynamics of such discrete-time quantum systems are governed by non linear Markov chains. In the feedback laws were obtained by maximizing the fidelity with respect to the target state at each time-step: this means that the feedback strategy was based on the same Lyapunov function given by the fidelity between the current and the target state. We propose here a systematic and explicit method to design a new family of control-Lyapunov functions. The main interest of these new Lyapunov functions relies on the crucial fact that the increases of their expectation values at step $k+1$ knowing the state $\rho_{k}$ at step $k$ remain strictly positive when $\rho_{k}$ does not coincide with the target state. In closed-loop, these Lyapunov functions become strict and the convergence analysis is notably simplified since invariance principles are not necessary. The construction of these strict Lyapunov functions is based on the Hamiltonian $H$ underlying the controlled unitary evolution and relies on the connectivity of the graph attached to $H$. They are obtained by inverting a Laplacian matrix derived from $H$ and the quantum states that are untouched by the QND measurements.

In section 2, we describe the finite dimensional Markovian model together with the main modeling assumptions. Section 3 is devoted to the open-loop behavior (Theorem 3.1) that can be seen as a non-deterministic protocol for preparing a finite number of isolated and orthogonal quantum states. In Section 4, we present the main ideas underlying the construction of these strict control-Lyapunov functions $W_{\epsilon}$. Then we define the connectivity graph, the Laplacian matrix attached to $H$ and two technical lemmas used during this construction. Finally Theorem 4.1 describes the stabilizing feedback derived from $W_{\epsilon}$. Closed-loop simulations corresponding to an experimental setup at Ecole Normale Supérieure are sketched in section 5.

The authors thank M. Brune, I. Dotsenko, S. Gleyzes, S. Haroche and J.M. Raimond for enlightening discussions and references. Advices of L. Praly concerning control-Lyapunov functions are also acknowledged.

The non-linear Markov model

The system state is described by the density operator $\rho$ belonging to ${\mathcal{D}}({\mathcal{H}})$ the set of positive, Hermitian matrices of trace one:

The random evolution of the state $\rho_{k}\in{\mathcal{D}}({\mathcal{H}})$ at time step $k$ is modeled through the following Markov process:

$\mu_{k}$ is a random variable taking values $\mu$ in $\{1,\cdots,m\}$ with probability $p_{\mu,\rho_{k}}=\text{Tr}\left(M_{\mu}\rho_{k}M_{\mu}^{\dagger}\right)$,

defined for $\rho\in{\mathcal{D}}({\mathcal{H}})$ such that $\text{Tr}\left(M_{\mu}\rho M_{\mu}^{\dagger}\right)\neq 0$.

We suppose throughout this paper that the two following assumptions are verified by the system under consideration.

For all $n_{1}\neq n_{2}$ in $\{1,\cdots,d\},$ there exists a $\mu\in\{1,\cdots,m\}$ such that $|c_{\mu,n_{1}}|^{2}\neq|c_{\mu,n_{2}}|^{2}.$

Assumption 1 means that the considered measurement process achieves a Quantum Non Demolition (QND) measurement for the physical observables given by orthogonal projections over the states \bigl{\{}\left|n\right>\,|\quad n\in\{1,\cdots,d\}\bigr{\}}. This implies that for $u_{k}\equiv 0$, any $\rho=\left|n\right>\left<n\right|$ corresponding to the orthogonal projector on the basis vector $\left|n\right>$, $n\in\{1,\ldots,d\}$, is a fixed point of the Markov process (1). Since the operators $M_{\mu}$ must satisfy $\sum_{\mu=1}^{m}M_{\mu}^{{\dagger}}M_{\mu}={{1\!\!1}}$, we have, according to assumption 1, $\sum_{\mu=1}^{m}|c_{\mu,n}|^{2}=1$ for all $n\in\{1,\cdots,d\}.$

Assumption 2 means that there exists a $\mu$ such that the statistics when $u_{k}\equiv 0$ for obtaining the measurement result $\mu$ are different for the fixed points $\left|n_{1}\right>\left<n_{1}\right|$ and $\left|n_{2}\right>\left<n_{2}\right|$. This follows by noting that $\text{Tr}\left(M_{\mu}\left|n\right>\left<n\right|M_{\mu}^{\dagger}\right)=|c_{\mu,n}|^{2}$ for $n\in\{1,\cdots,d\}$.

Convergence of the open-loop dynamics

where $\mu_{k}$ is a random variable with discrete values in $\{1,\ldots,m\}$. The probability $p_{\mu,\rho_{k}}$ to have $\mu_{k}=\mu$ depends on $\rho_{k}$: $p_{\mu,\rho_{k}}=\text{Tr}\left(M_{\mu}\rho_{k}M_{\mu}^{\dagger}\right)$. We have then the following theorem characterizing the open-loop asymptotic behavior:

Consider a Markov process $\rho_{k}$ obeying the dynamics of (2) with an initial condition $\rho_{0}$ in ${\mathcal{D}}({\mathcal{H}})$. Then

with probability one, $\rho_{k}$ converges to one of the $d$ states $\left|n\right>\left<n\right|$ with $n\in\{1,\cdots,d\}.$

the probability of convergence towards the state $\left|n\right>\left<n\right|$ depends only on the initial condition $\rho_{0}$ and is given by

The proof is a generalization of the one given in .

For any $n\in\{1,\ldots,d\}$, $\text{Tr}\left(\left|n\right>\left<n\right|\rho\right)$ is a martingale. This results from

where we have used the facts that $M_{\mu}$ and $\left|n\right>\left<n\right|$ commute and that $\sum_{\mu=1}^{m}M_{\mu}^{\dagger}M_{\mu}={1\!\!1}$. Take the function

where $f(x)=\tfrac{x^{2}}{2}$. The function $f$ being convex and each $\text{Tr}\left(\left|n\right>\left<n\right|\rho\right)$ being a martingale, we infer that $V(\rho)$ is a sub-martingale

For each $\mu$, let $\theta_{\mu}=\text{Tr}\left(M_{\mu}\rho_{k}M_{\mu}^{\dagger}\right)$, $x_{\mu}=\tfrac{\text{Tr}\left(\left|n\right>\left<n\right|M_{\mu}\rho_{k}M_{\mu}^{\dagger}\right)}{\text{Tr}\left(M_{\mu}\rho_{k}M_{\mu}^{\dagger}\right)}$ if $\theta_{\mu}>0$ and $x_{\mu}=0$ otherwise. Identity (4) yields

defined only when $\text{Tr}\left(M_{\mu}\rho M_{\mu}^{\dagger}\right)\text{Tr}\left(M_{\nu}\rho M_{\nu}^{\dagger}\right)>0$. Since this mapping is positive and bounded by $\rho\mapsto\text{Tr}\left(M_{\mu}\rho M_{\mu}^{\dagger}\right)\text{Tr}\left(M_{\nu}\rho M_{\nu}^{\dagger}\right)$, it can be extended by continuity to any $\rho\in{\mathcal{D}}({\mathcal{H}})$ by taking a null value when $\text{Tr}\left(M_{\mu}\rho M_{\mu}^{\dagger}\right)\text{Tr}\left(M_{\nu}\rho M_{\nu}^{\dagger}\right)=0$. Thus

is continuously defined for any $\rho\in{\mathcal{D}}({\mathcal{H}})$ and still satisfies

By Theorem A.1 of the Appendix, the $\omega$-limit set $\Omega$ (in the sense of almost sure convergence), for the trajectories $\rho_{k}$, is a subset of the set $\{\rho\in{\mathcal{D}}({\mathcal{H}})|\quad Q(\rho)=0\}.$

Let us consider a density matrix $\rho_{\infty}$ in the $\omega$-limit set. Therefore $Q(\rho_{\infty})=0$ implies

for all $\mu$ and $\nu$ in $I_{\rho_{\infty}}$ and for each basis element $\left|n\right>$. Since $\text{Tr}\left(\rho_{\infty}\right)=1,$ there is at least one $\bar{n}$ such that $\left<\bar{n}\right|\rho_{\infty}\left|\bar{n}\right>>0$. Then Equation (7) simplifies to

for all $\mu,\nu\in I_{\rho_{\infty}}$. Summing over $\nu\in I_{\rho_{\infty}}$, we find

For $\nu\notin I_{\rho_{\infty}}$, $c_{\nu,\bar{n}}=0$ since $\sum_{n=1}^{d}|c_{\nu,n}|^{2}\left<n\right|\rho_{\infty}\left|n\right>=0$, each $\left<n\right|\rho_{\infty}\left|n\right>\geq 0$ and $\left<\bar{n}\right|\rho_{\infty}\left|\bar{n}\right>>0$. Thus $\sum_{\nu\in I_{\rho_{\infty}}}|c_{\nu,\bar{n}}|^{2}=\sum_{\nu=1}^{m}|c_{\nu,\bar{n}}|^{2}=1$. Finally, we have

as soon as $\left<\bar{n}\right|\rho_{\infty}\left|\bar{n}\right>>0$.

Assume now that exist $\bar{n}_{1}\neq\bar{n}_{2}$ in $\{1,\ldots,d\}$ such that $\left<\bar{n}_{1}\right|\rho_{\infty}\left|\bar{n}_{1}\right>>0$ and $\left<\bar{n}_{2}\right|\rho_{\infty}\left|\bar{n}_{2}\right>>0$. Then (8) implies that

By Assumption 2, there exists a $\bar{\mu}\in\{1,\cdots,m\}$ such that $|c_{\bar{\mu},\bar{n}_{1}}|^{2}\neq|c_{\bar{\mu},\bar{n}_{2}}|^{2}$. These terms cannot be simultaneously zero, thus $\text{Tr}\left(M_{\bar{\mu}}\rho_{\infty}M_{\bar{\mu}}^{\dagger}\right)>0$, $\bar{\mu}\in I_{\rho_{\infty}}$. This is in contradiction with (9). This closes the proof of the assertion: the $\omega$-limit set is reduced to the set fixed point $\left|n\right>\left<n\right|,$ with $n\in\{1,\cdots,d\}$.

Feedback stabilization

To improve convergence and avoid such constant feedback zone, we propose to modify $V_{\bar{n}}$ using the other open-loop martingales $V_{n}(\rho)=\left<n\right|\rho\left|n\right>$ and the sub-martingale $V(\rho)=\sum_{n=1}^{d}(\left<n\right|\rho\left|n\right>)^{2}$ used during the proof of theorem 3.1. The goal of such modification is to get control Lyapunov functions still admitting a unique global maximum at $\left|\bar{n}\right>\left<\bar{n}\right|$ but being strongly convex versus $u$ around when $\rho$ is close to any $\left|n\right>\left<n\right|$, $n\neq\bar{n}$.

Let us continue by some definitions and lemmas that underlay the construction of these strict control-Lyapunov functions $W_{\epsilon}$.

2 Connectivity graph and Laplacian matrix

To the Hamiltonian operator $H$ defining the controlled unitary evolution $U_{u}=e^{-\imath uH}$, we associate its undirected connectivity graph denote by $G^{H}$. This graph admits $d$ vertices labeled by $n\in\{1,\ldots,d\}$. Two different vertices $n_{1}\neq n_{2}$ ($n_{1},n_{2}\in\{1,\ldots,d\}$) are linked by an edge, if and only if, $\left<n_{1}\right|H\left|n_{2}\right>\neq 0$. Attached to $H$, we also associate $R^{H}$, the real symmetric matrix $d\times d$ (Laplacian matrix) with entries

Note that $R^{H}$ is symmetric and the sum of the entries for any column and any row of $R^{H}$ is equal to zero. Therefore, the vector $(1\cdots 1)^{T}$ is in the kernel of $R^{H}$. The diagonal (resp.non-diagonal) components of $R^{H}$ are positive (resp. negative). Therefore $R^{H}$ is a Laplacian matrix (see [2, Ch. 4]). The connectivity graph associated to $R^{H}$ coincides with $G^{H}$. Since this graph is supposed connected, classical results of graph theory (see, e.g., [2, Theorem $3.1$]) imply that the ”constant” vector $(1,\cdots,1)^{T}$ spans the kernel of $R^{H}$. Therefore, the dimension of the image of $R^{H}$ is equal to $d-1$. Since $R^{H}$ is symmetric, its image coincides with the orthogonal to its kernel. For the sake of completeness, here we give a simple proof of this statement. Indeed, for any vector $X$ in the kernel of $R^{H}$, we have

This implies $R^{H}_{n_{1},n_{2}}(X_{n_{1}}-X_{n_{2}})^{2}=0$, $\forall n_{1},n_{2}\in\{1,\cdots,d\}$. As the graph of $R^{H}$ is connected, we necessarily have $X_{n_{1}}=X_{n_{2}}$ for all $n_{1},n_{2}\in\{1,\cdots,d\}.$ Thus any vector orthogonal to $(1\cdots 1)^{T}$ is in the image of $R^{H}$. The vector $\lambda$ is orthogonal to $(1,\cdots,1)^{T}$. ∎

Then for any $n\in\{1,\ldots,d\}/\{\bar{n}\}$ we have

Consequently for any $l\in\{1,\ldots,d\}$, we have

since $\text{Tr}\left(\left|l\right>\left<l\right|[H,\left|n\right>\left<n\right|]\right)=-\text{Tr}\left([\left|l\right>\left<l\right|,\left|n\right>\left<n\right|]H\right)=0$ because $\left|l\right>\left<l\right|$ commutes with $\left|n\right>\left<n\right|$), and since

Thus up to third order terms in $u$, we have

3 The global stabilizing feedback

The main result is expressed through the following theorem.

where the control-Lyapunov function $W_{\epsilon}(\rho)$ is defined by

with the parameter $\epsilon>0$ not too large to ensure that

Then, for any $\rho_{0}\in{\mathcal{D}}({\mathcal{H}})$, the closed-loop trajectory $\rho_{k}$ converges almost surely to the pure state $\left|\bar{n}\right>\left<\bar{n}\right|$.

The proof relies on the fact that $W_{\epsilon}(\rho)$ is a strict Lyapunov function for the closed-loop system.

We define the following functions of $\rho_{k}$,

These functions are both positive continuous functions of $\rho_{k}$ (the continuity of these functions can be proved in the same way as the proof of the continuity of $Q(\rho_{k})$ in Theorem 3.1). By Theorem A.1 of the appendix, the $\omega$-limit set $\Omega$ is included in the following set

Closed-loop simulations for the Photon-Box

The Hamiltonian $H=\imath(\bf a^{\dagger}-a)$ yields the unitary operator $U_{u}=e^{u(\bf a^{\dagger}-a)}$ (also known as displacement operator). Its graph $G^{H}$ is connected. The associated Laplacian matrix $R^{H}$ admits a simple tri-diagonal structure with diagonal elements $R^{H}_{n,n}=4n+2$, upper diagonal elements $R^{H}_{n-1,n}=-2n$ and under diagonal elements $R^{H}_{n+1,n}=-2n-2$ (up to some truncation distortion for $n=n^{\text{\tiny max}}-1,n^{\text{\tiny max}}$).

For a goal photon number $\bar{n}\in\{0,\ldots,n^{\text{\tiny max}}-1\}$, we propose the following setting for the $\lambda_{n}$ and $\epsilon$ defining $W_{\epsilon}$ of Theorem 4.1:

Figure 1 corresponds to the values of $\sigma_{n}$ found with $\lambda_{n}$ given in above with $\bar{n}=3$ and $n^{\text{\tiny max}}=10$. We remark that $\sigma_{\bar{n}}$ is maximal.

Since $(\left<n\right|H\left|n\right>)^{2}-\left<n\right|H^{2}\left|n\right>=-2n-1$, the constraint on $\epsilon>0$ imposed by Theorem 4.1 reads $\forall n\neq\bar{n}$, $\epsilon<\frac{1}{2n+1}$, i.e., $\epsilon<\frac{1}{2n^{\text{\tiny max}}+1}$.

The maximization defining the feedback in Theorem 4.1 could be problematic in practice. The unitary propagator $U_{u}$ does not admit in general a simple analytic form, as it is the case here. Simulations below show that we can replace $U_{u}$ by ist quadratic approximation valid for $u$ small and derived from Baker-Campbell-Hausdorff:

Concluding remarks

The method proposed here to derive strict control-Lyapunov could certainly be extended to

prove exponential closed-loop convergence for the feedback law given by theorem 4.1.

the general situation where the control $u$ appears directly in the Kraus operators $M_{\mu}(u)$ instead of being separated from the QND measures and attached to a unitary evolution applied after each measurement.

continuous-time quantum systems subject to QND measurements such as those considered in and described in detail in .

infinite-dimensional quantum systems as in that consider the photon-box system without truncation to a finite number of photons.

References

Appendix A Appendix

Let $X_{k}$ be a Markov chain on the compact state space $S.$ Suppose, there exists a non-negative function $V(X)$ satisfying

where $Q(X)$ is a positif continuous function of $X,$ then the $\omega$-limit set $\Omega$ (in the sense of almost sure convergence) of $X_{k}$ is included in the following set

The proof is just an application of the Theorem $1$ in [6, Ch. 8], which shows that $Q(X_{k})$ converges to zero for almost all paths. It is clear that the continuity of $Q(X)$ with respect to $X$ and the compactness of $S$ implies that the $\omega$-limit set of $X_{k}$ is necessarily included into the set I. ∎