Minimal resources identifiability and estimation of quantum channels

Abstract

We characterize and discuss the identifiability condition for quantum process tomography, as well as the minimal experimental resources that ensure a unique solution in the estimation of quantum channels, with both direct and convex optimization methods. A convenient parametrization of the constrained set is used to develop a globally converging Newton-type algorithm that ensures a physically admissible solution to the problem. Numerical simulation is provided to support the results and indicate that the minimal experimental setting is sufficient to guarantee good estimates.

Appendix: global convergence of the Newton algorithm

Let us consider the identification problem in (20) with the ML functional (24). To prove the convergence of the corresponding Newton algorithm, we need of the following result.

Proposition 5

Consider a function $f: X \subset \mathbb {R}^n\rightarrow \mathbb {R}$ twice differentiable on $X$ with $H_x$ the Hessian of $f$ at $x$. Suppose moreover that $f$ is strongly convex on a set $D\subset X$, i.e., there exists a constant $m> 0$ such that $H_x\ge mI$ for $x\in D$, and $H_x$ is Lipschitz continuous on $D$. Let $\{x_i\}\in D$ be the sequence generated by the Newton algorithm. Under these assumptions, Newton’s algorithm with backtracking converges globally. More specifically, $\{x_i\}$ decreases in linear way for a finite number of steps and converges in a quadratic way to the minimum point after the linear stage.

Proof

See [12, 9.5.3, p. 488]. $\square$

We proceed in the following way: Identify a compact set $D$ such that $\underline{\theta }_l \in D$ and prove that the Hessian is coercive and Lipschitz continuous on $D$. We then apply Proposition A.1 in order to prove the convergence

Since $\underline{\theta }_0\in \hbox {int} \left( \mathcal {C} \right)$ we consider the set

$$\begin{aligned} D:=\{\underline{\theta }\in \mathbb {R}^{d^4-d^2} \;|\; G_q(\underline{\theta })\le G_q(\underline{\theta }_0)\}. \end{aligned}$$

(38)

The presence of the backtracking stage in the algorithm guarantees that the sequence $G_q(\underline{\theta }_0),G_q(\underline{\theta }_1), \ldots$ is decreasing. Thus, $\underline{\theta }_l\in D,\,\forall l\ge 0$.

Proposition 6

The following facts hold:

• 1. $D$ is a compact set.

• 2. $H_{\underline{\theta }}$ is coercive and bounded on $D$, namely there exist $s,S>0$ such that

  $$\begin{aligned} sI\le H_{\underline{\theta }} \le \hbox {SI} \quad \forall \; \underline{\theta }\in D. \end{aligned}$$

  (39)

• 3. $H_{\underline{\theta }}$ is Lipschitz continuous on $D$.

Proof

(1) $D$ is contained into the bounded set $\mathcal {C}$. Since $D$ is a finite-dimensional space, it is sufficient to show that

$$\begin{aligned} \lim _{\underline{\theta }\rightarrow \partial \mathcal {C}} G_q(\underline{\theta })=+\infty . \end{aligned}$$

(40)

Here, we have three kind of boundary: $\partial \mathcal {I}\cap \hbox {int} \left( A_+ \right) ,\; \hbox {int} \left( \mathcal {I} \right) \cap \partial A_+$ and $\partial \mathcal {I}\cap \partial A_+$. Notice that, $\log \det (\chi (\underline{\theta }))$ takes finite values on $\partial \mathcal {J}\cap \hbox {int} \left( A_+ \right)$. Accordingly, taking (21) into account,

$$\begin{aligned} \lim _{\underline{\theta }\rightarrow \partial \mathcal {I}\cap \hbox {int} \left( A_+ \right) }G_q(\underline{\theta })=q \lim _{\underline{\theta }\rightarrow \partial \mathcal {I}\cap \hbox {int} \left( A_+ \right) } J(\underline{\theta })=+\infty . \end{aligned}$$

(41)

Then, $\hbox {int} \left( \mathcal {I} \right) \cap \partial A_+$ is the set of $\underline{\theta }$ for which $J$ is bounded and there exists at least one eigenvalue of $\chi (\underline{\theta })$ equal to zero. Thus,

$$\begin{aligned} \lim _{\underline{\theta }\rightarrow \hbox {int} \left( \mathcal {I} \right) \cap \partial A_+}G_q(\underline{\theta })=-\lim _{\underline{\theta }\rightarrow \hbox {int} \left( \mathcal {I} \right) \cap \partial A_+}\log \det (\chi (\underline{\theta }))=+\infty . \end{aligned}$$

(42)

Finally, from (41) and (42) it follows that $G_q(\underline{\theta })$ diverges as $\underline{\theta }$ approach $\partial \mathcal {I}\cap \partial A_+$.

(2) First, observe that $D\subset \hbox {int} \left( \mathcal {C} \right)$. Since $D$ is a compact set, there exists $s>0$ such that $\chi (\underline{\theta })^{-1}\ge sI \;\; \forall \; \underline{\theta }\in D$. Define

$$\begin{aligned}&\delta _{jk}:=\frac{1-f_{jk}}{[1-\mathrm{tr}(\chi (\underline{\theta })B_{jk})]^2} + \frac{f_{jk}}{[\mathrm{tr}(\chi (\underline{\theta })B_{jk})]^2}>0\nonumber \\&[M_{jk}]_{r,s}:=\mathrm{tr}(Q_rB_{jk})\mathrm{tr}(Q_sB_{jk}) \end{aligned}$$

where $M_{jk}$ is a positive semidefinite matrix with rank equal to one. Accordingly, the Hessian of the functional in (24) is

$$\begin{aligned}{}[H_{\underline{\theta }}]_{r,s}&= q \sum _{j,k}\delta _{jk} [M_{jk}]_{r,s}+\nonumber \\&+\,\mathrm{tr}[\chi (\underline{\theta })^{-\frac{1}{2}}Q_r\chi (\underline{\theta })^{-1}Q_s\chi (\underline{\theta })^{-\frac{1}{2}}]\nonumber \\&\ge q\sum _{j,k} \delta _{jk}[M_{jk}]_{r,s} +s\mathrm{tr}[Q_r\chi (\underline{\theta })^{-1}Q_s]\\&\ge q\sum _{j,k}\delta _{jk} [M_{jk}]_{r,s}+s^2\mathrm{tr}[Q_rQ_s]\nonumber \\&\ge q\sum _{j,k} \delta _{jk}[M_{jk}]_{r,s}+s^2\langle Q_r,Q_s\rangle . \end{aligned}$$

Since $\{Q_l\}_{l=1}^{12}$ are orthonormal matrices and $\delta _{jk} M_{jk}\ge 0$, we have that

$$\begin{aligned} H_{\underline{\theta }}\ge q\sum _{j,k} \delta _{jk}M_{jk}+s^2I \ge s^2 I. \end{aligned}$$

(43)

Notice that, $H_{\underline{\theta }}$ is continuous on $\hbox {int} \left( \mathcal {C} \right)$. Since $D \subset \hbox {int} \left( \mathcal {C} \right)$, it follows that $H_{\underline{\theta }}$ is continuous on the compact $D$. Hence, there exists $S>0$ such that $H_{\underline{\theta }} \le SI\,\forall \; \underline{\theta }\in D$. We conclude that $H_{\underline{\theta }}$ is coercive and bounded on $D$.

(3) $H_{\underline{\theta }}$ is continuous on $D$ and $\Vert H_{\underline{\theta }} \Vert \le S\,\forall \; \underline{\theta }\in D$, therefore $H_{\underline{\theta }}$ is Lipschitz continuous on $D$. $\square$

Since all the hypothesis of the Proposition A.1 are satisfied, we have the following proposition.

Proposition 7

The sequence $\{\underline{\theta }_l\}_{l\ge 0}$ generated by the Newton algorithm of Section 4.3 converges to the unique minimum point $\hat{\underline{\theta }}^q\in \hbox {int} \left( \mathcal {C} \right)$ of $G_q$.