Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

Keywords

Quantum non-demolition measurements; Measurement-based feedback; Photon-number states (Fock states); Quantum filter; Strict control Lyapunov function; Markov chain; Feedback stabilization.

Introduction

Manipulating quantum systems allows one to accomplish tasks far beyond the reach of classical devices. Quantum information is paradigmatic in this sense: quantum computers will substantially outperform classical machines for several problems . Though significant progress has been made recently, severe difficulties still remain, amongst which decoherence is certainly the most important. Large systems consisting of many qubits must be prepared in fragile quantum states, which are rapidly destroyed by their unavoidable coupling to the environment. Measurement-based feedback and coherent feedback are possible routes towards the preparation, protection and stabilization of such states. For coherent feedback strategy, the controller is also a quantum system coupled to the original one (see and the references therein). This paper is devoted to measurement-based feedback where the controller and the control input are classical objects . The results presented here are directly inspired by a recent experiment demonstrating that such a quantum feedback scheme achieves the on-demand preparation and stabilization of non-classical states of a microwave field.

Following and relying on continuous-time Lyapunov techniques exploited in , an initial measurement-based feedback was proposed in . This feedback scheme stabilizes photon-number states (Fock states) of a microwave field (see e.g. for a physical description of such cavity quantum electrodynamics (CQED) systems). The controller consists of a quantum filter that estimates the state of the field from discrete-time measurements performed by probe atoms, and secondly a stabilizing state-feedback that relies on Lyapunov techniques. The discrete-time behavior is crucial for a possible real-time implementation of such controllers. Closed-loop simulations reported in have been confirmed by the stability analysis performed in . In the experimental implementation , the state-feedback has been improved by considering a strict Lyapunov function: this ensures better convergence properties by avoiding the passage by high photon numbers during the transient regime. The goal of this paper is to present, for a class of discrete-time quantum systems, the mathematical methods underlying such improved Lyapunov design. Our main result is given in Theorem 4.2 where closed-loop convergence is proved in presence of delays and measurement imperfections.

The state-feedback scheme may be applied to generic discrete-time finite-dimensional quantum systems using controlled measurements in order to deterministically prepare and stabilize the system at some pre-specified target state. The dynamics of these systems may be expressed in terms of a (classical) nonlinear controlled Markov chain, whose state space consists of the set of density matrices on some Hilbert space. The jumps in this Markov chain are induced by quantum measurements and the associated jump probabilities are state-dependent. These systems are subject to a discrete-time sequence of positive operator valued measurements (POVMs ) and we use these POVMs to stabilize the system at the target state. By controlled measurements, we mean that at each time-step the chosen POVM is not fixed but is a function of some classical control signal $u$, similar to . However, we assume that when the control $u$ is zero, the chosen POVM performs a quantum non-demolition (QND) measurement for some orthonormal basis that includes the target state. The feedback-law is based on a Lyapunov function that is a linear combination of a set of martingales corresponding to the open-loop QND measurements. This Lyapunov function determines a “distance” between the target state and the current state. The parameters of this Lyapunov function are given by inverting Metzler matrices characterizing the impact of the control input on the Kraus operators defining the Markov processes and POVMs. The (graph theoretic) properties of the Metzler matrices are used to construct families of open-loop supermartingales that become strict supermartingales in closed-loop. This fact provides directly the convergence to the target state without using the invariance principle.

A common problem that occurs in quantum feedback control is that of delays between the measurement process and the control process . In this paper we demonstrate, using a predictive quantum filter, that the proposed scheme works even in the presence of delays. Convergence analysis is done for perfect and imperfect measurements. For imperfect measurements, the dynamics of the system are governed by a nonlinear Markov chain given in .

In both the perfect and imperfect measurement situations we prove a robustness property of the feedback algorithm: the convergence of the closed-loop system is ensured even when the feedback law is based on the state of a quantum filter that is not initialized correctly. This robustness property, is similar in spirit to the separation principle proven in . We use the fact that the state space is a convex set and the target state, being a pure state, is an extreme point of this convex state space and therefore cannot be expressed as a convex combination of any other states. One then uses the linearity of the conditional expectation to prove the robustness property. Our result is only valid for target quantum states that are pure states.

The paper is organized as follows. In Section 2, we describe the finite dimensional Markov model together with the main modeling assumptions in the case of perfect measurements and study the open-loop behavior (Theorem 2.1) which can be seen as a non-deterministic protocol for preparing a finite number of isolated and orthogonal quantum states. In Section 3, we present the main ideas underlying the construction of these control-Lyapunov functions $W_{\epsilon}$: a Metzler matrix attached to the second derivative of the measurement operators and a technical lemma assuming this Metzler matrix is irreducible. Finally, Theorem 3.1 describes the stabilizing state feedback derived from $W_{\epsilon}$. The same analysis is done for the case of imperfect measurements in Section 4. For the state estimations used in the feedback scheme we propose a brief discussion on the quantum filters and prove a rather general robustness property for perfect measurements in Section 3 with Theorem 3.2 and for imperfect ones in Section 4 with Theorem 4.2. Section 5 is devoted to the experimental implementation that has been done at Laboratoire Kastler-Brossel of Ecole Normale Supérieure de Paris. Closed-loop simulations and experimental data complementary to those reported in are presented.

System model and open-loop dynamics

The random evolution of the state $\rho_{k}\in{\mathcal{D}}$ at time-step $k$ is modeled through the following dynamics:

We now state some assumptions that we will be using in the remainder of this paper.

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

All $M_{\mu}^{u}$ are $C^{2}$ functions of $u$.

2 Convergence of the open-loop dynamics

When the control input vanishes ($u\equiv 0$), the dynamics are simply given by

Consider a Markov process $\rho_{k}$ obeying the dynamics of (3) with an initial condition $\rho_{0}$ in ${\mathcal{D}}$. Then with probability one, $\rho_{k}$ converges to one of the $d$ states $\left|n\right>\left<n\right|$ with $n\in\{1,\ldots,d\}$ and the probability of convergence towards the state $\left|n\right>\left<n\right|$ is given by $\left<n\right|\rho_{0}\left|n\right>.$

This Theorem is already proved in and also in in a slightly different formulation. A direct proof is based on the following Lyapunov function:

Since $\left<n\right|\rho\left|n\right>$ is a martingale for any $n$ and since

Feedback stabilization with perfect measurements

In Subsection 3.1, we give an overview of the control method and then in Subsections 3.2 and 3.3, we prove the main results. Finally in Subsection 3.4, we prove a robustness principle that explains how we can ensure convergence even if the initial state $\rho_{0}$ is unknown.

Theorem 2.1 shows that the open-loop dynamics are stable in the sense that in each realization $\rho_{k}$ converges (non-deterministically) to one of the pure states $\left|n\right>\left<n\right|$ with probability $\left<n\right|\rho_{0}\left|n\right>$. The control goal is to make this convergence deterministic toward a chosen $\bar{n}\in\{1,\ldots,d\}$ playing the role of controller set point. We build on the ideas in to design a controller that is based on a strict Lyapunov function for the target state. In this paper we assume arbitrary controlled Kraus operators $M^{u}_{\mu}$ that cannot be decomposed into QND measurement operators $M^{0}_{\mu}$ followed by a unique controlled unitary operator $D_{u}$ with $D_{0}=\text{\bf{I}}$ as assumed in where $M^{u}_{\mu}\equiv D_{u}M_{\mu}^{0}$. It can be argued that the control we are proposing is non-Hamiltonian control , as the control parameter $u$ is not necessarily a parameter in the interaction Hamiltonian and could indeed be any parameter of an auxiliary system such as the measurement device.

Here $\bar{u}$ is some small positive number that needs to be determined. Because $0\in[-\bar{u},\bar{u}]$ and the control $u_{k}$ is chosen to minimize $V$ at each step, we directly have that $V$ is a closed-loop supermartingale, i.e.,

What remains to be done is to choose an appropriate open-loop supermartingale $V$ so that the set $I_{\infty}$ is restricted to the target state $\{\left|\bar{n}\right>\left<\bar{n}\right|\}$. The Lyapunov function $\Gamma$ defined in (4) does not discriminate between the different basis vectors. We therefore use the Lyapunov function $V_{\epsilon}(\rho)=V_{0}(\rho)-\frac{\epsilon}{2}\sum_{n=1}^{d}\left<n\right|\rho\left|n\right>^{2}$ where $V_{0}(\rho)=\sum_{n=1}^{d}\sigma_{n}\left<n\right|\rho\left|n\right>$ and $\epsilon$ is a small positive constant. The weights $\sigma_{n}$ are strictly positive numbers except for $\sigma_{\bar{n}}=0$. This function $V_{\epsilon}$ is clearly a concave function of the open-loop martingales $\left<n\right|\rho\left|n\right>$ and therefore is an open-loop supermartingale. Moreover, the weights $\sigma_{n}$ can be used to quantify the distance of the state $\rho$ from the target state $\left|\bar{n}\right>\left<\bar{n}\right|$ (c.f. Figure 2 below which shows how $\sigma_{n}$ are chosen for the experimental setting).

A state $\rho$ is in the set $I_{\infty}$ if and only if for all $u\in[-\bar{u},\bar{u}]$, we have

Also from the fact that $V_{\epsilon}$ is an open-loop supermartingale, we have for all $\rho\in{\mathcal{D}}$

This idea can easily be extended to the situations where the delay $\tau$ is non zero. Take $(\rho_{k},u_{k-1},\ldots,u_{k-\tau})$ as state at step $k$: denote by $\chi=(\rho,\beta_{1},\ldots,\beta_{\tau})$ this state where $\beta_{r}$ stands for the control input $u$ delayed $r$ steps. Then the state form of the delayed dynamics (2) is governed by the following Markov chain

Finally, we address the situation where the initial state of the system is not fully known but only estimated by $\rho^{\text{\tiny est}}_{0}$. We show under some assumptions on the initial condition that, the feedback law based on the state $\rho^{\text{\tiny est}}_{k}$ of the miss-initialized filter still ensures the convergence of $\rho^{\text{\tiny est}}_{k}$ as well as the well-initialized conditional state $\rho_{k}$ towards $\left|\bar{n}\right>\left<\bar{n}\right|$. This demonstrates how the control algorithm is robust to uncertainties in the initialization of the estimated state of the quantum system.

The construction of the control Lyapunov function relies on two lemmas.

Consider the $d\times d$ matrix $R$ defined by

For $n_{1}\neq n_{2}$, $R_{n_{1},n_{2}}\geq 0$. Thus $R$ is a Metzler matrix A Metzler matrix is a matrix such that all the off-diagonal components are non-negative.. Let us prove that the sum of each row vanishes. This results from identity $\sum_{\mu}{M_{\mu}^{u}}^{\dagger}M_{\mu}^{u}={\text{\bf{I}}}$. Deriving twice versus $u$ the relation

To the Metzler matrix $R$ defined in Lemma 3.1, we associate its directed graph denoted by $G.$ This graph admits $d$ vertices labeled by $n\in\{1,\ldots,d\}$. To each strictly positive off-diagonal element of the matrix $R$, say, on the $n_{1}$’th row and the $n_{2}$’th column we associate an edge from vertex $n_{1}$ towards vertex $n_{2}.$

Assume the directed graph $G$ of the matrix $R$ defined in Lemma 3.1 is strongly connected, i.e., for any $n,n^{\prime}\in\{1,\ldots,d\}$, $n\neq n^{\prime}$, there exists a chain of $r$ distinct elements $(n_{j})_{j=1,\ldots,r}$ of $\{1,\ldots,d\}$ such that $n_{1}=n$, $n_{r}=n^{\prime}$ and for any $j=1,\ldots r-1$, $R_{n_{j},n_{j+1}}\neq 0$. Take $\bar{n}\in\{1,\ldots,d\}$. Then, there exist $d-1$ strictly positive real numbers $e_{n}$, $n\in\{1,\ldots,d\}\setminus\{\bar{n}\}$, such that

3 The global stabilizing feedback

The main result of this section is expressed through the following theorem.

where $Q_{1}$ is defined in (5). Thus, $W_{\epsilon}(\chi_{k})$ is a supermartingale. According to Theorem A.1, the $\omega$-limit set $I_{\infty}$ of $\chi_{k}$ is included in

4 Quantum filter and robustness property of filter

When the measurement process is fully efficient (i.e., the detectors are ideal with detection efficiency equal to one and they observe all the quantum jumps) and the jump model (2) admits no error, the Markov process (8) represents a natural choice for estimating the hidden state $\rho.$ Indeed, the estimate $\rho^{\text{\tiny est}}$ of $\rho$ satisfies the following recursive dynamics

where the measurement outcome $\mu_{k}$ is driven by (2). In practice, the control $u_{k}$ defined in Theorem 3.1 could only depend on this estimation $\rho^{\text{\tiny est}}_{k}$ replacing $\rho_{k}$ in $\chi_{k}$. If $\rho^{\text{\tiny est}}_{0}=\rho_{0}$, then $\rho^{\text{\tiny est}}_{k}\equiv\rho_{k}$ and Theorem 3.1 ensures convergence towards the target state. Otherwise, the following result guaranties the convergence of such observer/controller scheme when $\ker(\rho^{\text{\tiny est}}_{0})\subset\ker(\rho_{0})$.

Consider the recursive Equation (2) and assume that the assumptions of Theorem 3.1 are satisfied. For each measurement outcome $\mu_{k}$ given by (2), consider the estimation $\rho^{\text{\tiny est}}_{k}$ given by (11) with an initial condition $\rho^{\text{\tiny est}}_{0}$. Set $u_{k}=f(\rho^{\text{\tiny est}}_{k},u_{k-1},\ldots,u_{k-\tau})$ where $f$ is given in Theorem 3.1. Then there exists $u^{*}>0$ such that, for all $\bar{u}\in]0,u^{*}]$ and $\epsilon\in\left]0,\min_{n\neq\bar{n}}\left(\frac{\lambda_{n}}{R_{n,n}}\right)\right]$, $\rho_{k}$ and $\rho^{\text{\tiny est}}_{k}$ converge almost surely towards the target state $\left|\bar{n}\right>\left<\bar{n}\right|$ as soon as $\ker(\rho^{\text{\tiny est}}_{0})\subset\ker(\rho_{0})$.

Imperfect Measurements

We now consider the feedback control problem in the presence of classical measurement imperfections with the possibility of detection errors. This model is a direct generalization of the ones used in (see also e.g., for an introduction to quantum filtering). The imperfections in the measurement process are described by a classical probabilistic model relying on a left stochastic matrix $(\eta_{\mu^{\prime},\mu})$, $\mu^{\prime}\in\{1,\ldots,m^{\prime}\}$ and $\mu\in\{1,\ldots,m\}$: $\eta_{\mu^{\prime},\mu}\geq 0$ and for any $\mu$, $\sum_{\mu^{\prime}=1}^{m^{\prime}}\eta_{\mu^{\prime},\mu}=1$. The integer $m^{\prime}$ corresponds to the number of imperfect outcomes and $\eta_{\mu^{\prime},\mu}$ is the probability of having the imperfect outcome $\mu^{\prime}$ knowing the perfect one $\mu$. Set

Since $\rho_{k}$ follows (2), $\widehat{\rho}_{k}$ is also governed by a recursive equation :

We now consider the dynamics of the filter state $\widehat{\rho}$ in the presence of delays in the feedback control. Similar to the case with perfect measurements, let $\widehat{\chi}_{k}=(\widehat{\rho}_{k},\beta_{1,k},\ldots,\beta_{\tau,k})$ be the filter state at step $k$, where $\beta_{l,k}=u_{k-l}$ is the feedback control at time-step $k$ delayed $l$ steps. Then the delay dynamics are determined by the following Markov chain

For all $n_{1}\neq n_{2}$ in $\{1,\ldots,d\},$ there exists $\mu^{\prime}\in\{1,\ldots,m^{\prime}\}$ such that $\sum_{\mu=1}^{m}\eta_{\mu^{\prime},\mu}|c_{\mu,n_{1}}|^{2}\neq\sum_{\mu=1}^{m}\eta_{\mu^{\prime},\mu}|c_{\mu,n_{2}}|^{2}.$

We now state the analogue of Theorem 3.1 in the case of imperfect measurements.

where $\mu^{\prime}_{k}$ corresponds to the imperfect outcome detected at step $k$. Such $\mu^{\prime}_{k}$ is correlated to the perfect and hidden outcome $\mu_{k}$ of (2) through the classical stochastic process attached to $(\eta_{\mu^{\prime},\mu})$: for each $\mu_{k}$, $\mu^{\prime}_{k}$ is a random variable to be equal to $\mu^{\prime}\in\{1,\ldots,m^{\prime}\}$ with probability $\eta_{\mu^{\prime},\mu_{k}}$. In practice, the control $u_{k}$ defined in Theorem 4.1 could only depend on this estimation $\widehat{\rho}^{\text{\tiny est}}_{k}$ replacing $\widehat{\rho}_{k}$ in $\widehat{\chi}_{k}=(\widehat{\rho}_{k},u_{k-1},\ldots,u_{k-\tau})$. The following result guaranties the convergence of the feedback scheme when $\ker(\widehat{\rho}^{\text{\tiny est}}_{0})\subset\ker(\rho_{0})$.

Consider the recursive Equation (2) and take assumptions of Theorem 4.1. Consider the estimation $\widehat{\rho}^{\text{\tiny est}}_{k}$ given by (15) with an initial condition $\widehat{\rho}^{\text{\tiny est}}_{0}$. Set $u_{k}=f(\widehat{\rho}^{\text{\tiny est}}_{k},u_{k-1},\ldots,u_{k-\tau})$ where $f$ is given by Theorem 4.1. Then there exists $u^{*}>0$ such that, for all $\bar{u}\in]0,u^{*}]$ and $\epsilon\in\left]0,\min_{n\neq\bar{n}}\left(\frac{\lambda_{n}}{R_{n,n}}\right)\right]$, $\rho_{k}$ and $\widehat{\rho}^{\text{\tiny est}}_{k}$ converge almost surely towards the target state $\left|\bar{n}\right>\left<\bar{n}\right|$ as soon as $\ker(\widehat{\rho}^{\text{\tiny est}}_{0})\subset\ker(\rho_{0})$.

The photon box

In this section, we give the explicit expression of the feedback controller which has been experimentally tested in Laboratoire Kastler-Brossel (LKB) at Ecole Normal Supérieure (ENS) de Paris. We briefly summarize how the control design elucidated in this paper is applied to the LKB experiment. We refer the interested reader to for more details. This feedback controller has been obtained by the Lyapunov design discussed in previous sections.

2 The controlled Markov process and quantum filter

After taking into account our full knowledge about the experiment, we finally get the following state estimate at step $k+1$:

The decoherence manifests itself through spontaneous loss or capture of a photon to or from the environment which is described as follows:

where $L_{0}=\text{\bf{I}}-\theta(1/2+n_{\text{th}})\,\text{\bf{N}}-(\theta n_{\text{th}}/2)\,\text{\bf{I}}$, $L_{-}=\sqrt{\theta(1+n_{\text{th}})}\,\text{\bf{a}},$ and $L_{+}=\sqrt{\theta n_{\text{th}}}\,\text{\bf{a}}^{\dagger},$ with $\theta\ll 1,$ a and $\text{\bf{a}}^{\dagger}$ photon annihilation and creation operators ($\text{\bf{a}}\left|n_{\text{ph}}\right>=\sqrt{n_{\text{ph}}}\left|n_{\text{ph}}\!-\!1\right>$ and $\text{\bf{a}}^{\dagger}\left|n_{\text{ph}}\right>=\sqrt{n_{\text{ph}}\!+\!1}\left|n_{\text{ph}}\!+\!1\right>$) and with $\text{\bf{N}}=\text{\bf{a}}^{\dagger}\text{\bf{a}}$ the photon number operator ($\text{\bf{N}}\left|n_{\text{ph}}\right>=n_{\text{ph}}\left|n_{\text{ph}}\right>$). Besides, $n_{\text{th}}$ is the mean number of photons in the cavity mode at thermal equilibrium with its environment.

The evolution of the state $\rho$ after the control injection is modeled through

with $D_{u}=\exp(u\text{\bf{a}}^{\dagger}-u\text{\bf{a}}).$ In reality, the control at step $k$, $u_{k},$ is subject to a delay of $\tau>0$ steps which corresponds to the number of flying atoms between the cavity $C$ and the detector $D$.

In the real experiment, the atom source is probabilistic and is characterized by a truncated Poisson probability distribution $P_{a}(n_{a})\geq 0$ to have $n_{a}\in\{0,1,2\}$ atom(s) in a sample (we neglect events with more than 2 atoms). This expands the set of the possible detection outcomes to $m=7$ values $\mu\in\{\o,g,e,gg,eg,ge,ee\}$, related to the following measurement operators, $L_{\o}=\sqrt{P_{a}(0)}\,\text{\bf{I}},$ $L_{g}=\sqrt{P_{a}(1)}\,\cos(\phi_{\text{\bf{N}}})$, $L_{e}=\sqrt{P_{a}(1)}\,\sin(\phi_{\text{\bf{N}}})$, $L_{gg}=\sqrt{P_{a}(2)}\,\cos^{2}(\phi_{\text{\bf{N}}})$, $L_{ee}=\sqrt{P_{a}(2)}\,\sin^{2}(\phi_{\text{\bf{N}}})$ and $L_{ge}=L_{eg}=\sqrt{P_{a}(2)}\,\cos(\phi_{\text{\bf{N}}})\sin(\phi_{\text{\bf{N}}})$ where $\phi_{\text{\bf{N}}}=\frac{\phi_{r}+\phi_{0}({\text{\bf{N}}}+1/2)}{2},$ $\phi_{r}$ and $\phi_{0}$ are physical parameters.

The real measurement process is not perfect: the detection efficiency is limited to $\varepsilon<1$ and the state detection errors are non-zero ($0<\eta_{e/g}<1)$. These imperfections are taken into account by considering the left stochastic matrix $\eta_{\mu^{\prime},\mu}$ which is given in . Consequently, the optimal state estimate after measurement outcome $\mu^{\prime}$ gets the following form:

The measurement operators $(L_{\mu})_{1\leq\mu\leq m}$ are diagonal in the Fock basis $\{\left|n_{\text{ph}}\right>\}$, illustrating their quantum non-demolition nature with respect to the photon number operator and thus fulfilling Assumption 1. Besides, Assumption 4 can also be fulfilled by a proper choice of the experimental parameters $\phi_{{r}}$ and $\phi_{0}$.

3 Feedback controller

4 Simulations and experimental results

Conclusion

We have proposed a Lyapunov design for state-feedback stabilization of a discrete-time finite-dimensional quantum system with QND measurements. Extensions of this design are possible in different directions such as

replacing the continuous and one-dimensional input $u$ by a multi-dimensional one $(u_{1},\ldots,u_{p})$;

assuming that $u$ belongs to a finite set of discrete values;

taking an infinite dimensional state space as in where the truncation to finite photon numbers is removed;

considering continuous-time systems similar to the ones investigated in ;

ensuring convergence towards a sub-space instead of a pure-state and thus achieving a goal similar to error correction code as already proposed in .

Acknowledgements: the authors thank Michel Brune, Serge Haroche and Jean-Michel Raimond for enlightening discussions and advices.

References

Appendix A Appendix

The following theorem is just an application of Theorem $1$ in [18, Ch. 8].

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

where $Q(X)$ is a non-negative continuous function of $X,$ then the $\omega$-limit set $\Omega$ (in the sense of almost sure convergence) of $X_{k}$ is contained by the following set $I_{\infty}=\{X|\quad Q(X)=0\}.$

Consider the function $Q_{1}$ defined by (5) and $\widetilde{u}>0$. Then there exists $C>0$ such that for all $(\rho,\beta_{1})\in{\mathcal{D}}\times[-\widetilde{u},\widetilde{u}]$ satisfying $Q_{1}(\rho,\beta_{1})=0$, there exists $n\in\{1,\ldots,d\}$ such that $\rho_{n,n}=\left<n\right|\rho\left|n\right>\geq 1-C|\beta_{1}|$.

we have by continuity for $C$ tending to $+\infty$ and for all $n$ and $\mu$: $|c_{\mu,n}|^{2}\rho^{*}_{n,n}=\left(\sum_{n^{\prime}}|c_{\mu,n^{\prime}}|^{2}\rho^{*}_{n^{\prime},n^{\prime}}\right)\rho^{*}_{n,n}$. Thus there exists $n^{*}\in\{1,\ldots,d\}$ such that $\rho^{*}=\left|n^{*}\right>\left<n^{*}\right|$ (see the proof of Theorem 2.1). Since $\rho^{*}_{n^{*},n^{*}}=1$, for $C$ large enough, $\rho^{C}_{n^{*},n^{*}}>1/2$ and thus

Taking $n\neq n^{*}$, by Assumption 2, there exists $\mu$ such that $|c_{\mu,n}|^{2}\neq|c_{\mu,n^{*}}|^{2}$. Replacing (22) in (21) yields: $\left(|c_{\mu,n}|^{2}-|c_{\mu,n^{*}}|^{2}+\beta_{1}^{C}~{}\tfrac{b_{\mu,n^{*}}(\rho^{C},\beta_{1}^{C})}{\rho^{C}_{n^{*},n^{*}}}\right)\rho^{C}_{n,n}=\beta_{1}^{C}~{}b_{\mu,n}(\rho^{C},\beta_{1}^{C}).$ Thus, there exists $C_{0}>0$, such that for $n\neq n^{*}$ and $C$ large enough $\rho^{C}_{n,n}\leq C_{0}|\beta_{1}^{C}|.$ But $\rho^{C}_{n^{*},n^{*}}=1-\sum_{n\neq n^{*}}\rho^{C}_{n,n}\geq 1-C_{0}(d-1)|\beta_{1}^{C}|$. This is in contradiction with $\rho^{C}_{n^{*},n^{*}}\leq 1-C|\beta_{1}^{C}|$ as soon as $C>C_{0}(d-1)$.