# Minimal resources identifiability and estimation of quantum channels

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## 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.

### Keywords

Quantum process tomography Maximum likelihood estimation Quantum information## 1 Introduction

Recent advances and miniaturization in laser technology and electronic devices, together with some profound results in quantum physics and quantum information theory, have generated in the last two decades, increasing interest in the promising field of *quantum information engineering*. The potential of these new technologies have been demonstrated by a number of theoretical and experimental results, including intrinsically secure quantum cryptography protocols, proof-of-principle implementation of quantum computing, as well as dramatic advances in controlled engineering of molecular dynamics, opto-mechanical devices, and a variety of other experimentally available systems. In this area, a key role is played by *control, estimation* and *identification* problems for quantum-mechanical systems [18, 24, 34, 35, 43].

In the spirit that has driven the contributions of the control-theoretical community so far [3, 4, 5, 17, 19, 22, 27, 28, 33, 39, 40, 41], namely of developing research which is both strongly motivated by emerging applications and mathematically rigorous, we consider an *identification problem arising in the reconstruction of quantum dynamical models from experimental data*. This is a key issue in many quantum information processing tasks [10, 11, 31, 32, 34]. For example, a precise knowledge of the behavior of a channel to be used for quantum computation or communications is needed in order to ensure that optimal encoding/decoding strategies are employed, and to verify that the noise thresholds for hierarchical error-correction protocols, or for effectiveness of quantum key distribution protocols, are met [11, 32]. In many cases of interest, for example, in free-space communication [42], channels are not stationary, and to ensure good performances, repeated and fast estimation steps would be needed as a prerequisite for adaptive encodings. In addition to this, when the goal is to embed, the system used for probing the channel in a moving vehicle or a satellite, one seeks the simplest implementation, or at least a compromise between estimation accuracy and the number of experimental resources needed. This work has been motivated by the interaction of the authors with experimental groups, and the issues we here address are directly relevant to the quantum communication applications described above.

In fact, all these applications call for an answer to the following *identifiability* problem: are the given experimental resources sufficient to uniquely and correctly reconstruct the unknown channel from the available data? Strictly related issues include determining the *minimal* resources needed for correct identification and the efficiency of the minimal resources with respect to “richer” experimental settings.

In this paper, we are concerned with *standard quantum process tomography* (see e.g., [31, 34] and references therein; *standard* is used to distinguish the methods we discuss here from those that use auxiliary quantum systems as resources, see discussion below) for estimating quantum channels. More precisely, we focus on the minimal experimental resources (or *quorum*, in the language of [21]) needed for the *identifiability* of a quantum channel. Remarkably, the same results applies to all the standard quantum process tomography methods, including “inversion” and convex optimization methods. Among these, we revisit as notable examples some typical applications of the commonly used maximum likelihood (ML) method. In doing so, we pursue a rigorous presentation of the results and we try, whenever possible, to make contact with ideas and methods of (classical) system identification.

As noticed before, it is also possible to estimate a quantum channel by using an ancillary system [8, 20, 38]. Such a method is named *ancilla-assisted process tomography*. In the large body of literature regarding both standard quantum process tomography and ancilla-assisted process tomography, the experimental resources are usually assumed to be given. Mohseni et al. [31] compare different strategies, but focus on the role of having entangled states as an additional resource. However, it can be argued (see e.g., [44]) that the information acquired through an ancilla-assisted method by measuring a certain number \(K\) of independent observables can be equivalently gathered by the method we describe, by properly choosing \(L\) probe states and \(M\) observables such that \(L\cdot M=K\). Accordingly, our result on the minimal resources for TP channel estimation can be easily adapted to this approach. The problem we study is close in spirit to that taken in [36] while studying minimal *state* tomography.

In the derivation of the main result, we employ a natural parametrization for the quantum channels that directly account for the *trace-preservation* constraint. The same parametrization can be used as a starting point for convex optimization methods by resorting to a *general* Newton-type algorithm with barriers that guarantees convergence to a physical (nonnegative) solution.

With respect to the existing (ML) approaches, we do not introduce the TP constraint through a Lagrange multiplier [31, 34], as we constrain the set of channels of the optimization problem to TP maps from the beginning. In a \(d\)-level quantum system, this allow for an immediate reduction from \(d^4\) to \(d^4-d^2\) free parameters in the estimation problem. Our analysis can be considered as complementary to the one presented in [9], where the TP assumption is relaxed to include losses. The Newton-type algorithm is then used for our numerical simulations, showing that using the *minimal setting* does not deteriorate the performance: the estimation accuracy is mainly dependent just on the total number of trials. As expected, simulations also confirm that the convex optimization methods allow for more robust solutions with respect to the positivity constraints and confirm the consistency property.

The paper is structured as follows: Sect. 2 serves as an introduction to quantum channels and their linear-algebraic representation. Section 3 employs these linear-algebraic tools in deriving the core *identifiability* result and in proving existence and uniqueness of the solution, as well as consistency, for *general* convex optimization methods. Section 4 is devoted to applications of these theoretical contributions. First, we show how our results can be used to prove existence, uniqueness and consistency of the solution for two well-known ML estimation problems. Next, we show that a Newton-type algorithm can be designed to compute the estimate for this class of identification methods. Finally, numerical simulations are presented.

## 2 A review of quantum channels and \(\chi \)-representation

*Choi representation*and its properties [15]. Consider a \(d\)-level quantum system with associated Hilbert space \(\mathcal {H}\) isomorphic to \({\mathbb C}^d\). If \(X\) is the matrix representation of a linear operator on \(\mathcal {H},\) we shall denote by \(X^\mathrm{T}\) its transpose, while \(X^\dag \) will denote its transpose-conjugate (i.e., the matrix representation of the adjoint operator). The

*state*of the system is described by a density matrix, namely by a positive semidefinite, unit-trace matrix

*pure*if \(\rho \) is an orthogonal projection matrix on a one-dimensional subspace. Measurable quantities or

*observables*are associated with Hermitian matrices \(X=\sum _kx_k\varPi _k,\) with \(\{\varPi _k\}\) the associated spectral family of orthogonal projections. Their spectrum \(\{x_k\}\) represents the possible outcomes, and the probability of observing the \(k\)th outcome can be computed as \(p_\rho (\varPi _k)=\mathrm{tr}(\varPi _k\rho )\). More generally, indirect or generalized measurements are associated with families of nonnegative operators \(\{M_k\}\) such that \(\sum _k M_k=I\) and \(p_\rho (M_k)=\mathrm{tr}(M_k\rho ),\) also known as

*Positive-Operator Valued Measurements (POVM).*In the rest of the paper, we consider projective measurements, but the same reasoning applies with no further complications to POVMs.

*Completely Positive*(CP), namely it must admit an Operator-Sum Representation (OSR)

*Kraus operators*. Moreover, such a map must be

*Trace Preserving*(TP), a necessary condition to map states to states. This is the case if

*representation*. Each Kraus operator \(K_j\in \mathbb {C}^{d\times d}\) can be expressed as a linear combination (with complex coefficients) of \(\{F_m\}_{m=1}^{d^2},\,F_m\) being the elementary matrix \(E_{jk},\) (whose entries are all zero except the one in position \(jk\) which is \(1\)) with \(m=(j-1)d+k\). Accordingly, the OSR (1) can be rewritten as

*partial trace*. Consider two finite-dimensional vector spaces \(\mathcal{V}_1\,\mathcal{V}_2,\) with \(\dim \mathcal{V}_1=n_1,\,\dim \mathcal{V}_2=n_2\). Let us denote by \(\mathcal{M}_{n}\) the set of complex matrices of dimension \(n\times n\). Let \(\{M_j\}\) be a basis for \(\mathcal{M}_{n_1},\) and \(\{N_j\}\) be a basis for \(\mathcal{M}_{n_2},\) representing linear maps on \(\mathcal{V}_1\) and \(\mathcal{V}_2,\) respectively. Consider \(\mathcal{M}_{n_1\cdot n_2}=\mathcal{M}_{n_1}\otimes \mathcal{M}_{n_2}\): it is easy to show that the \(n_1^2\times n_2^2\) linearly independent matrices \(\{M_j\otimes N_k\}\) form a basis for \(\mathcal{M}_{n_1\cdot n_2},\) where \(\otimes \) denotes the Kronecker product. Thus, one can express any \(X\in \mathcal{M}_{n_1\cdot n_2}\) as

*partial trace*over \(\mathcal{V}_2\) is the linear map

**Lemma 1**

*Proof*

*Remark*

The \(\chi \) representation is equivalent to the Choi representation for quantum channels, i.e., the one associated with the Choi matrix \(C_\mathcal{E}=\sum _{mn}E_{mn}\otimes \mathcal {E}({E_{mn})}\) [35]. In fact, either by direct computation or by confronting formula (6) with its equivalent for the Choi matrix \(C_\mathcal {E}\) (see e.g., [34], chapter 2), it is easy to see that \(C_\mathcal {E}=O\chi O^\dag ,\) where \(O\) is the unique unitary such that \(O( X\otimes Y) O^\dag =Y\otimes X\) [7].

Lemma 1 leads to a useful expression for the computation of the expectations.

**Corollary 1**

*Proof*

It suffices to substitute (6) in \(p_{\chi ,\rho }(\varPi )=\mathrm{tr}(\mathcal {E}(\rho )\varPi ),\) and use the identity \(\mathrm{tr}((X\otimes I) Y)=\mathrm{tr}(X\mathrm{tr}_2(Y))\). \(\square \)

The TP condition (5) can also be re-expressed directly in terms of the \(\chi \) matrix.

**Corollary 2**

*Proof*

## 3 Identifiability condition and minimal setting

### 3.1 The channel identification problem

*known pure state*\(\rho \) is fed to an unknown quantum channel \(\mathcal{E}\). The system in the

*output state*\(\mathcal {E}(\rho )\) is then subjected to a projective measurement of an

*observable*. By noting that an observable (being represented by an Hermitian matrix in our setting) admits a decomposition in orthogonal projections representing mutually incompatible quantum events, we can without loss of generality restrict ourselves to consider measurements associated with orthogonal projections \(\varPi =\varPi ^\dag =\varPi ^2\) (or, more generally, positive semidefinite \(M_k\le I\) for generalized measurements.) For each one of these, the outcome \(x\) is in the set \(\{0,1\},\) and can be interpreted as a sample of the (classical) random variable \(X\) which has distribution

*data*, namely

*standard quantum process tomography*problem [31, 32, 34], consists in estimating a matrix \(\hat{\chi }\) satisfying constraints (3), (4) such that the corresponding

*Kraus map*\({\mathcal {E}}_{\hat{\chi }}\) fits the experimental data in some optimal way.

### 3.2 Necessary and sufficient conditions for identifiability

The minimal experimental setting needed to uniquely identify an unknown quantum state has been object of study in [13], in the framework of informationally complete measurements, and it has been also studied in detail in [36]. Here, by exploiting the similarity of the \(\chi \) matrix with a density operator, we follow a similar path and characterize the minimal set of probe states/measurements for which a quantum channel is identifiable.^{1}

It is well known [34, 37] that by imposing linear constraints associated with the TP condition (5), or equivalently (8), one reduces the \(d^4\) real degrees of freedom of \(\chi \) to \(d^4-d^2\). We shall exploit this fact by directly parameterizing \(\chi \) in a “generalized” Pauli basis (also known as Gell-Mann matrices, Fano basis or coherence vector representation in the case of states [2, 6, 34]). Usually the TP constraint is not directly included in the standard tomography method [31], since in principle it should emerge from the physical properties of the channel, or it is imposed through a (nonlinear) Lagrange multiplier in the ML approach [34]. Here, in order to investigate the minimum number of probe (input) states and measured projectors needed to uniquely determine \(\chi \), it is convenient to include this constraint from the very beginning. Doing so, we lose the possibility of exploiting a Cholesky factorization in order to impose positive semidefiniteness of \(\chi \): nonetheless, we show in Sect. 4.3 that semidefiniteness of the solution can be imposed algorithmically by using a barrier method [12].

**Proposition 1**

\(g\) is injective if and only if \(\mathcal {S}_{TP}= \mathcal {B}\).

*Proof*

We anticipate here that \(g\) being injective is a necessary and sufficient condition for a priori identifiability of \(\chi ,\) and thus for having a unique solution of the problem for both inversion and convex optimization-based (e.g., ML) methods, up to some basic assumptions on the cost functional. The proof of these facts is given in full detail in Sects. 3.3 and 3.4.

As a consequence of these facts, we can determine the *minimal experimental resources*, in terms of input states and measured projectors, needed for faithfully reconstructing \(\chi \) from noiseless data \(\{f^\circ _{jk}\}\), where \(f^\circ _{jk}=p_{\chi ,\rho _k}(\varPi _j)\). In the light of Proposition 1, the minimal experimental setting is characterized by a choice of \(\{\varPi _j,\rho _k\}\) such that \(\mathcal{S}_{TP}=\mathcal{B}\). Recalling the definition of \(\mathcal{B},\) through (12), it is immediate to see that \(\mathcal{S}_{TP}=\mathcal{B}\) if and only if \(\mathrm{span}\{\varPi _j- \frac{r_j}{d}I_d\}=\mathrm{span}\{\sigma _j,j=1,\ldots ,d^2-1\}\) and \(\mathrm{span}\{\rho _k\}=\mathcal {H}_d\). Here, \(\mathcal {H}_d\) denotes the vector space of Hermitian matrices of dimension \(d\). We can summarize this fact as a corollary of Proposition 1.

**Corollary 3**

*at least*\(d^2\) linearly independent input states \(\{\rho _k\},\) and \(d^2-1\) measured \(\{\varPi _j\}\) such that

It is worth observing that the reduction in the number of the observables is a direct consequence of the imposed trace-preserving constraint. We call such a set a *minimal experimental setting*. Notice that, using the terminology of [21, 34], the minimal *quorum* of observables consists of \(d^2-1\) properly chosen elements. While in most of the literature at least \(d^2\) observables are considered [23, 31], we showed it is in principle possible to spare a measurement channel at the output. A physically inspired interpretation for this fact is that, since we a priori know, or assume, that the map is TP, measuring the component of the observables along the identity does not provide useful information. This is clearly not true if one relaxes the TP condition, as it has been done in [9]: in that case, by the same line of reasoning, \(d^2\) linearly independent observables are the necessary and sufficient for \(g\) to be injective.

### 3.3 Standard process tomography by inversion

*full column rank*, namely has rank \(d^4-d^2\). Hence, in principle, one can reconstruct \(\hat{\theta }\) as

### 3.4 Convex methods: general framework

**Proposition 2**

\(\mathcal{C}\) is a bounded set.

*Proof*

First, we remark that \(\mathcal{C}\) is neither closed nor open in general. Since \(\mathcal {C}\subset \mathcal{A}_+\), it is sufficient to show that \(\mathcal{A_+}\) is bounded or, equivalently, that a sequence \(\{\underline{\theta }_j\}_{j\ge 0}\), with \(\underline{\theta }_j\in \mathbb {R}^{d^4-d^2}\), and \(\Vert \underline{\theta }_j\Vert \rightarrow +\infty \), cannot belong to \(\mathcal{A_+}\). To this end, it is sufficient to show that, as \(\Vert \underline{\theta }_j\Vert \rightarrow +\infty \), the minimum eigenvalue of \(\chi (\underline{\theta }_j)\) tends to \(-\infty \) so that, for \(j\) large enough, \(\underline{\theta }_j\) does not satisfy condition \(\chi (\underline{\theta }_j)\ge 0\). Notice that the map \(\underline{\theta }\mapsto \chi (\underline{\theta })\) is affine. Moreover, since the \(Q_\ell \) are linearly independent, this map is injective. Accordingly, \(\Vert \chi (\underline{\theta }_j)\Vert \) approach infinity as \(\Vert \underline{\theta }_j\Vert \rightarrow +\infty \). Since \(\chi (\underline{\theta }_j)\) is a Hermitian matrix, \(\chi (\underline{\theta }_j)\) has an eigenvalue \(\lambda _j\) such that \(|\lambda _j|\rightarrow +\infty \) as \(\Vert \chi (\underline{\theta }_j)\Vert \rightarrow +\infty \). Recall that \(\chi (\underline{\theta }_j)\) satisfies (8) by construction which implies that \(\mathrm{tr}(\chi (\underline{\theta }_j))=d\) namely the sum of its eigenvalues is always equal to \(d\). Thus, there exists an eigenvalue of \(\chi (\underline{\theta }_j)\) which approaches \(-\infty \) as \(j\rightarrow +\infty \), which is in contrast with its positivity. So, \(\mathcal {C}\) is bounded. \(\square \)

Here we focus on the following issue: under which conditions on the experimental setting (or, mathematically, on the set \(\mathcal{B}\) defined above) do the optimization approach have a unique solution? In either of the cases above, \(\mathcal{C}\) is the intersection of convex nonempty sets: in fact, \(\mathcal{S}_{TP}\) and \(\chi \ge 0\) are convex and so must be the corresponding sets of \(\underline{\theta }\), and it is immediate to verify that \(\mathcal{I}\) is convex as well; all of these contain \(\underline{\theta }=0,\) corresponding to \(\frac{1}{d}I_{d^2},\) and hence, they are nonempty. In the light of this, it is possible to derive sufficient conditions on \(J\) for existence and uniqueness of the minimum in the presence of arbitrary constraint set \(\mathcal{C}\). Define \(\partial \mathcal{C}_0:=\partial \mathcal{C}\setminus (\partial \mathcal{C}\cap \mathcal{C})\).

**Proposition 3**

*Proof*

*consistency*of this method. First, notice that \(h\) depends on \(\{\varPi _j,\rho _k,f_{jk}\}\) where \(f_{jk}\) are the sample frequencies. We write \(h_{\underline{f}}\), where \([\underline{f} ]_{jk}=f_{jk}\), to make explicit the dependence of \(h\) on the matrix of \(f_{jk}\). Let \(N\) be the number of measures performed for each \(\rho _k\) and \(\varPi _j\) and denote the estimate (20) by \(\hat{\theta }_N\) to highlight its dependance on \(N\). Let \(\underline{\theta }^\circ \) be the “true” value of the parameter. We recall that the estimate is consistent if \(\forall \theta \in {\mathcal C}\),

**Proposition 4**

*Proof*

Notice that any \(h_{\underline{f}}(x)\) of the form \(D(\underline{f},x)\) where \(D\) is any (pseudo-)distance satisfies the requirements of this proposition.

## 4 Applications and numerical methods

In the previous section, we considered an identification paradigm for estimating quantum channels, deriving simple sufficient conditions on \(J\) which guarantee identifiability, as well as the uniqueness and the consistency of the corresponding estimate. In Sects. 4.1 and 4.2, we consider two ML problems widely considered in the literature, see for instance [23] and [26]. We directly show that these fit our framework and satisfy the sufficient conditions. In Sect. 4.3, we show that it is possible to design a globally convergent Newton algorithm for numerically computing the solution to problem (20) when the cost functional \(J\) satisfies the assumption in Proposition 3.3. In Sect. 4.4, we analyze the estimation performances of the *minimal setting* in the particular case of the ML method of Sect. 4.1.

### 4.1 ML binomial functional

^{2}the function

*Kullback–Leibler*(pseudo-)distance, [16], and \(y_f(j)\) denotes the \(j\)-th entry of \(y_f\). The second term on the r.h.s in (25) plays no role in the optimization problem (20). Accordingly, we can equivalently consider the functional \(J(\underline{\theta })=D(y_f,y_{g(\underline{\theta })})\) and, in view of Proposition 4, we conclude that the method is consistent.

Finally, notice that one can minimize with respect to the other argument of the Kullback–Leibler distance, i.e., consider the functional \(J(\underline{\theta })=D(y_{g(\underline{\theta })},y_f)\). Clearly, Propositions 3 and 4 are still valid. This method can be understood as a generalization of the *maximum entropy criterium* in the presence of a prior. In fact, when \(y_f(j)=\frac{1}{2}\) the minimization of \(J\) is equivalent to the maximization of the entropy functional \(\mathbb {H}(y_x)=\sum _{j=1}^{2ML} y_x(j)\log (y_x(j))\).

### 4.2 ML Gaussian functional

### 4.3 A convergent Newton-type algorithm

In Proposition 3, we characterized a general identification paradigm for estimating quantum channels and showed the existence and uniqueness of its solution. Now, we face the problem of (numerically) finding the solution \(\hat{\underline{\theta }}\) minimizing \(J\) over the prescribed set.

- 1.
Set the initial condition \(\underline{\theta }_0\in \hbox {int} \left( \mathcal {C} \right) \).

- 2.At each iteration, compute the Newton stepwhere$$\begin{aligned} \Delta \underline{\theta }_l =-H_{\underline{\theta }_l}^{-1} \nabla G_{\underline{\theta }_l}\in \mathbb {R}^{d^4-d^2} \end{aligned}$$(34)are the element in position \(s\) of the gradient (understood as column vector) and the element in position \((r,s)\) of the Hessian of \(G_q\) both computed at \(\underline{\theta }\).$$\begin{aligned} \left[ \,\nabla G_{\underline{\theta }}\,\right] _s&:= \frac{\partial {G_q(\underline{\theta })}}{\partial {\theta _{s}}}=q\frac{\partial {J(\underline{\theta })}}{\partial {\theta _{s}}}-\mathrm{tr}[\chi (\underline{\theta })^{-1}Q_s]\\ \left[ \,H_{\underline{\theta }}\,\right] _{r,s}&:= \frac{\partial {G_q(\underline{\theta })}}{\partial {\theta _{r}\theta _{s}}}=q \frac{\partial {J(\underline{\theta })}}{\partial {\theta _{r}\theta _{s}}} +\mathrm{tr}[\chi (\underline{\theta })^{-1}Q_r\chi (\underline{\theta })^{-1}Q_s] \end{aligned}$$
- 3.Set \(t^0_l = 1\), and let \(t^{p+1}_l=t^p_l/2\) until all the following conditions hold:where \(\gamma \) is a real constant, \(0<\gamma <\frac{1}{2}\).$$\begin{aligned}&\underline{\theta }_l + t^p_l \Delta \underline{\theta }_l\in \hbox {int} (\mathcal {C})\\&G_q( \underline{\theta }_l + t^p_l \Delta \underline{\theta }_l )< G_q(\underline{\theta }_l)+\gamma t^p_l \nabla G_{\underline{\theta }_l}^{\mathrm{T}} \Delta \underline{\theta }_l \end{aligned}$$
- 4.
Set \(\underline{\theta }_{l+1} = \underline{\theta }_l + t^p_l \Delta \underline{\theta }_l\in \hbox {int} \left( \mathcal {C} \right) \).

- 5.
Repeat steps 2, 3 and 4 until the condition \(\Vert \nabla G_{\underline{\theta }_l}\Vert < \epsilon \) is satisfied, where \(\epsilon \) is a (small) tolerance threshold, then set \(\hat{\underline{\theta }}^q= \underline{\theta }_l\).

- 1.
Set the initial conditions \(q_0>0\) and \(\underline{\theta }^{q_0}=\left[ \begin{array}{lll} 0 &{}\ldots &{} 0 \\ \end{array} \right] ^{\mathrm{T}}\in \hbox {int} \left( \mathcal {C} \right) \).

- 2.
Centering step: At the \(k\)-th iteration compute \(\hat{\underline{\theta }}^{q_k}\in \hbox {int} \left( \mathcal {C} \right) \) by minimizing \(G_{q_k}\) with starting point \(\hat{\underline{\theta }}^{q_{k-1}}\) using the Newton method previously presented.

- 3.
Set \(q_{k+1}=\mu q_{k}\).

- 4.
Repeat steps 2 and 3 until the condition \( \frac{d^2}{q_k}< \xi \) is satisfied, then set \(\hat{\underline{\theta }}= \hat{\underline{\theta }}^{q_k}\).

### 4.4 Numerical simulations

*IN method*to denote the standard quantum process tomography by inversion of Sect. 3.3.*ML method*to denote the ML method presented in Sect. 4.1. Here, we want to compare the performance of IN and ML method for the qubit case \(d=2\). Consider a set of CPTP map \(\{\chi _l\}_{l=1}^{100}\) randomly generated and the minimal setting (16). Once the number of measurements \(N\) for each couple \((\rho _k,\varPi _j)\) is fixed, we consider the following comparison procedure:At the \(l\)-th experiment, let \(\{c_{jk}^l\}\) be the data corresponding to the map \(\chi _l\). Then, compute the corresponding frequencies \(f_{jk}^l=c^l_{jk}/N\).

From \(\{f_{jk}^l\}\) compute the estimates \(\hat{\chi }^{IN}_l\) and \(\hat{\chi }^{ML}_l\) using IN and ML method, respectively.

- Compute the relative errors$$\begin{aligned} e_{IN}(l)=\frac{\Vert \hat{\chi }^{IN}_l-\chi _l\Vert }{\Vert \chi _l\Vert },\;e_{ML}(l)=\frac{\Vert \hat{\chi }^{ML}_l-\chi _l\Vert }{\Vert \chi _l\Vert }. \end{aligned}$$(36)
- When the experiments are completed, compute the mean of the relative error$$\begin{aligned} \mu _{IN}=\frac{1}{100}\sum _{l=1}^{100} e_{IN}(l),\;\mu _{ML}=\frac{1}{100}\sum _{l=1}^{100} e_{ML}(l). \end{aligned}$$(37)

Let \(\mathcal {T}_{M,L}\) denote the set of the experimental settings with \(L\) input states and \(M\) observables satisfying Proposition 1. Accordingly, the set of the minimal experimental settings is \(\mathcal {T}_{d^2-1,d^2}\). Here, we consider the case \(d=2\). We want to compare the performance of the minimal settings in \(\mathcal {T}_{3,4}\) with those settings that employ more input states and observables. We shall do so by picking a test channel, finding a minimal setting that performs well, and comparing its performance with a nonminimal setting in \(\mathcal {T}_{M,L},\,M>3,L\ge 4\) that performs well in this set while the total number \(N_T\) of trials is fixed.

Set \(N=N_T\setminus (LM)\) and choose a randomly generated collection \(\{\mathrm{T}_m\}_{m=1}^{100},\,\mathrm{T}_m\in \mathcal {T}_{M,L}\).

Perform 50 experiments for each \(\mathrm{T}_m\). At the \(l\)-th experiment, we have a sample data \(\{f_{jk}^m(l)\}\) corresponding to \(\chi \) and \(\mathrm{T}_m\). From \(\{f_{jk}^m(l)\}\) compute the estimate \(\hat{\chi }_{m}(l)\) using the ML method and the corresponding error norm \(e_m(l)=\Vert \hat{\chi }_m(l)-\chi \Vert / \Vert \chi \Vert \).

When the experiments corresponding to \(\mathrm{T}_m\) are completed, compute the mean error norm \(\mu _m=\frac{1}{50}\sum _{l=1}^{50}e_m(l) \).

- When we have \(\mu _m\) for \(m=1 \ldots 100\), compute$$\begin{aligned} \bar{\mu }_{L,M}=\min _{m\in \{1,\ldots ,50\}} \mu _m. \end{aligned}$$

## 5 Conclusions

convex methods are more reliable than the inverse method (at least) when the number of measurement is small

experimental settings which are richer than the minimal one do not improve the quality of the estimate. The critical parameter, as long as accuracy of the estimation is concerned, is the total size of the data set.

## Footnotes

- 1.
These topics could also be studied from an abstract, frame-theoretical viewpoint [14]: however, in order to maintain contact with well-established notations and concepts in quantum information theory, we choose a more direct approach.

- 2.
If the optimization is constrained to \(\mathcal{A}_+\cap \mathcal{I},\) we are guaranteed that \(f_{jk}\) will tend to be positive for a sufficiently large numbers of trials.

## Notes

### Acknowledgments

The authors would like to thank Alberto Dall’Arche, Andrea Tomaello, Prof. Paolo Villoresi and Dr. Giuseppe Vallone for stimulating discussions on the topics of this paper. Work partially supported by the QFuture research grant of the University of Padova, and by the Department of Information Engineering research project “QUINTET.”

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