Quantum measurement optimization by decomposition of measurements into extremals

TL;DR

This paper introduces an efficient numerical method for optimizing quantum measurements by decomposing them into extremals, significantly improving parameter estimation strategies in quantum metrology.

Contribution

The paper presents a novel numerical approach that leverages the convex structure of quantum measurements to optimize parameter estimation, outperforming previous methods.

Findings

Strong numerical advantage for small target errors

Effective in qubit and harmonic oscillator systems

Improves quantum measurement strategies

Abstract

Using the convex structure of positive operator value measurements and of several quantities used in quantum metrology, such as quantum Fisher information or the quantum Van Trees information, we present an efficient numerical method to find the best strategy allowed by quantum mechanics to estimate a parameter. This method explores extremal measurements thus providing a significant advantage over previously used methods. We exemplify the method for different cost functions in a qubit and in a harmonic oscillator and find a strong numerical advantage when the desired target error is sufficiently small.

Equations

$$p(\mathbf{y}|\theta )$$

$$⟨\hat{\theta }(\mathbf{y})⟩=\int \text{rmd}\mathbf{y}p(\mathbf{y}|\theta )\hat{\theta }(\mathbf{y})=\theta .$$

$${\varsigma }^2\equiv \int (\hat{\theta }(\mathbf{y})-\theta {)}^{2}p(\mathbf{y}|\theta )\text{rmd}\mathbf{y}.$$

$$F(\theta )\equiv \int {\left(\frac{\partial \ln p(\mathbf{y}|\theta )}{\partial \theta }\right)}^{2}p(\mathbf{y}|\theta )\text{rmd}\mathbf{y}.$$

$${\varsigma }^{2}F(\theta )⩾1.$$

$${\Xi }^2=\int (\hat{\theta }(\mathbf{y})-\theta {)}^{2}P(\mathbf{y},\theta )\text{rmd}\theta \text{rmd}\mathbf{y},$$

$${\Xi }^{2}Z⩾1,$$

$$Z=\int F(\theta )\lambda (\theta )\text{rmd}\theta +\int {\left(\frac{\partial \ln \lambda (\theta )}{\partial \theta }\right)}^2\lambda (\theta )\text{rmd}\theta .$$

$$\int \hat{E}(\xi )\text{rmd}\xi =1,$$

$$p(\xi |\theta )=Tr(\rho (\theta )E(\xi )).$$

$$F_Q(\theta )=\underset{\{\hat{E}(\xi )\}}{\max }\int {\left(\frac{\partial \ln p(\xi |\theta )}{\partial \theta }\right)}^{2}p(\xi |\theta )\text{rmd}\xi .$$

$${\varsigma }^2{F}_Q(\theta )⩾1,$$

$${\varsigma }^2{\tilde{Z}}_Q⩾1,$$

$${\tilde{Z}}_Q=\underset{\{\hat{E}(\xi )\}}{\max }\left(\int {\left(\frac{\partial \ln P(\mathbf{y},\theta )}{\partial \theta }\right)}^{2}P(\mathbf{y},\theta )\text{rmd}\theta \text{rmd}\mathbf{y}\right).$$

$$\frac{1}{2}\int \frac{(p_1^{\prime }{)}^2}{p_1}\text{rmd}\mathbf{x}+\frac{1}{2}$$

$$⟨i|Q_m|j⟩=U_{im;j1}$$

$$Aa=b,$$

$$\begin{array}[]{l@{\quad} r c r }\mathrm{find}&x&&\\ \mathrm{subject\leavevmode\nobreak\ to}&Ax&=&b,\quad x\geq 0.\end{array}$$

$$a=p{x}^{\mathrm{ext}}+(1-p)x^{\prime },$$

$$p=\underset{i}{\min }\frac{a_i}{x_i^{\mathrm{ext}}},$$

$$\text{Q}=p{\text{Q}}^{\mathrm{ext}}+(1-p){\text{Q}}^{\prime }.$$

$$|\psi(\theta,\eta)\rangle=\left(\begin{array}[]{c}e^{-i\theta/2}\cos(\eta/2)\\ e^{i\theta/2}\sin(\eta/2)\end{array}\right),$$

$$\mathbf{P}=\{\rho (\xi ),\mathcal{I}-\rho (\xi ){\}}_{\xi }$$

$$\rho (\xi )=|\psi (\xi ,\eta )⟩⟨\psi (\xi ,\eta )|\}.$$

$$p_{\xi }(1|\theta )$$

$$p_{\xi }(2|\theta )$$

$$F^{(\xi ,\eta )}(\theta )=\frac{{\sin }^2(\eta )}{1+{\cos }^2(\eta ){\tan }^2((\xi -\theta )/2)},$$

$$F_Q(\theta )=\underset{\xi }{\max }{F}^{(\xi ,\eta )}(\theta )={\sin }^2(\eta ).$$

$${\Delta }_{\mathrm{RSM}}$$

$${\Delta }_{\mathrm{RESM}}$$

$${\tilde{Z}}_Q^{\mathbf{P}}=\underset{\xi }{\max }\int \text{rmd}\theta p(\theta )F^{(\xi )}(\theta ).$$

Full text

1. 1 Introduction

2. 2 Mathematical tools

3. 2.1 Crámer-Rao inequality

4. 2.2 Bayesian Cramér-Rao inequality

5. 2.3 Quantum Cramér-Rao inequality

6. 2.4 Bayesian quantum Crámer-Rao inequality

7. 3 Numerical calculations

8. 3.1 Convexity

9. 3.2 The algorithm

10. 4 Examples

11. 4.1 Qubit

12. 4.2 Phase estimation

13. 4.3 Coherent plus thermal state

14. 4.4 A note for reproducing results

15. 5 Conclusions

16. 6 Acknowledgements

1 Introduction

The goal of quantum metrology is to find limits in the precision of parameter estimation of quantum systems [1, 2, 3, 4, 5]. Recent research includes theoretical and experimental advances [6, 7]. Knowing these limits allows to know if a given measurement strategy minimizes the estimation errors. The strategy that minimizes the error will be called the optimal measurement strategy. Given a quantum state that depends on a set of parameters, the optimal measurement strategy for estimating these parameters can be used to estimate the quantum state. These strategies has applications in quantum technologies, for example they can can be used together with quantum control for quantum state manipulation [8].

To estimate a parameter of a physical setup one acquires data through measurements and the estimation of the parameter is obtained by applying a function, known as the estimator, to the data. Data has a random component. The probability distribution of measurement outcomes can be modelled using a statistical model of the experiment: the probability distribution of outcomes conditioned to the value of the parameter. Given a mathematical model of a system using quantum mechanics, the statistical model is obtained once it is decided which operator is going to be measured. Notice that we can include classical noise in the quantum mechanical description. Going from the data to the estimation of the parameters that minimizes the error is not trivial.

Given a cost function that quantifies the error, the optimal measurement strategy consist of the quantum measurement and estimator that extremizes it. But, finding the extreme of a cost function over all possible quantum measurements and estimators is not simple. When a Cramér-Rao type inequality exists, the problem can be reduced to find the extreme of another cost function over all the quantum measurements. This simplifies the problem because it is not longer necessary to maximize over the space of estimators.

It is possible, though very costly, to numerically find the maximum over all quantum measurements of cost functions. The straightforward way is to randomly sample the space of all positive operator value measures (POVMs), evaluate the cost function on this sample, and keep the maximum value obtained. This method, that we will call the random sampling method (RSM), is very inefficient because the POVM space is large.

In this paper we show how using the algorithm proposed by Sentís et al. [9] we can numerically find the maximum over all quantum measurements of some cost functions, that are useful in quantum metrology, in a way that is orders of magnitude faster than using the RSM. This paper presents a direct application of this algorithm. We used it as a method to extract extremal POVMs from a general POVM. It is easy to produce efficiently a general POVM from a random unitary matrix, however, it is not trivial to produce randomly extremal POVMs, this is where the algorithm plays an essential role.

We will call our method the random extreme sampling method (RESM). The techniques here presented can be used, for example, to find numerically the quantum van Trees information  [10] or the quantum Fisher information in the case that the initial state is not pure. Together with the value of the cost function maximum, the quantum measurement that maximizes it is obtained.

We start presenting the mathematical tools, including the Crámer-Rao inequality and its quantum extensions in section 2. In the following section we present the RESM in detail (section 3). We finish the bulk of this article comparing the performance of RESM against RSM in a two-level scenario, and benchmarking the accuracy of the method against other ansatz for a harmonic oscillator in two different physical situations. We close the article with some concluding remarks in section 5.

2 Mathematical tools

We discuss how to get the best parameter estimation given a statistical model. Then we discuss how to apply this ideas for a quantum system.

2.1 Crámer-Rao inequality

Now we introduce some basic quantities needed to develop further discussion. Let

[TABLE]

be the distribution probability of the outcomes $\mathbf{y}$, of the random variable $\mathbf{Y}$, conditioned to a fixed value of the real parameter $\theta$. We assume that each $\mathbf{y}$ is a set of real numbers of fixed finite size. This is the statistical model. The function $\hat{\theta}(\mathbf{y})$ is called the estimator and gives and estimation of $\theta$. The estimator is unbiased when is in average correct,

[TABLE]

The uncertainty of the estimator is given by the mean squared error

[TABLE]

We say that the measurement strategy is optimal if the estimator minimizes the uncertainty. Finally, let us define the Fisher information

[TABLE]

If the estimator is unbiased (i.e. if (2) holds), using the Cauchy-Schwartz inequality, one arrives to the Cramér-Rao inequality [11, 12]:

[TABLE]

Note that $\varsigma^{2}$ depends on the choice of the specific estimator ($\hat{\theta}(\mathbf{y})$), whereas the Fisher information depends only on the probability density function of the random variable. From the Cramér-Rao inequality, we see that the inverse of the Fisher information bounds from below the mean squared error independently of the estimator we use; the larger the Fisher information is the smaller the error bound. The best measurement strategy saturates the Cramér-Rao inequality. Fisher showed that in the limit where the number of measurements goes to infinity, the maximum likelihood estimator saturates this inequality [13].

2.2 Bayesian Cramér-Rao inequality

In general each time a measurement is done, the parameter is different. Let us give two examples. The first is a length that changes due to thermal fluctuations. The second, a parameter (such as the phase of a phase shifter) characterizing an element of an ensemble of non-identical objects. Both situations can be modelled assuming that the parameter to be measured is a random variable (with its characteristic distribution). A third case, in which a random variable approach is suitable, is when the parameter is fixed, but we have some partial knowledge contained in an a priori distribution. The Bayesian Cramér-Rao inequality can be used to decide what is the best estimator in this situation.

Lets assume now that each time we make an experiment the parameter we want to estimate is different. We model the parameter as the random variable $\Theta$, with outcomes $\theta$, and probability distribution $\lambda(\theta)$. The outcomes of the experiment are modelled as the random variable $\mathbf{Y}$, with outcomes $\mathbf{y}$, and probability distribution $p(\mathbf{y}|\theta)$. The experiment is modelled in the following way: first we take a value, $\theta_{1}$, from the random variable $\Theta$; the outcome of the experiment is $\mathbf{y}_{1}$ which is taken from the random variable $\mathbf{Y}$ with distribution probability $p(\mathbf{y}|\theta_{1})$; using $\mathbf{y}_{1}$ we can estimate $\theta_{1}$ by $\hat{\theta}(\mathbf{y}_{1})$, After repeating the experiment $n$ times, we have $n$ estimations $\hat{\theta}_{i}$ with $i=1\dots n$. The error of experiment $i$ is $(\hat{\theta}_{i}-\theta_{i})$. The mean squared error is, after performing the experiment $n$ times, $(1/n)\sum_{i=1}^{n}(\hat{\theta}(\mathbf{y}_{i})-\theta_{i})^{2}$. This is the cost function we want to minimize over all the estimators $\hat{\theta}$. In the limit $n\rightarrow\infty$ it can be written as

[TABLE]

where $P(\mathbf{y},\theta)=p(\mathbf{y}|\theta)\lambda(\theta)$. It can be shown that the error is bound from below by the Cramér-Rao type inequality[14],

[TABLE]

where the generalized Fisher information, $Z$, can be written as

[TABLE]

The first term of the sum is the expectation value of the Fisher information, the second term is the Fisher information of the probability distribution of the possible values of the parameter. Note that we already know something about the parameter; this a priory knowledge is given by $\lambda(\theta)$. As can be seen from the previous equation, the generalized Fisher information is larger than the Fisher information of the knowledge we already have of the parameter. This has a simple interpretation: we can use $\lambda(\theta)$ to estimate the parameter and measuring the system necessarily diminishes the error in the estimation of the parameter. The best strategy for measuring the outcomes of a random variable is given by the estimator, $\hat{\theta}$, that saturates this inequality.

In this context, we found useful [15], a review of bayesian inference in physics.

2.3 Quantum Cramér-Rao inequality

We want to find the best measurement strategy to estimate a parameter, $\theta$, that appears in the Hamiltonian of a quantum system. In order to estimate the parameter we proceed as follows: we start with an initial state and let the system evolve some time, after which the state of the system is $\rho(\theta)$. One then measures some observable of the system; we use the result to estimate $\theta$. Fixing the Hamiltonian, time and initial state, we want to know if the strategy we are using minimizes the error in the parameter estimation.

Measurements in quantum mechanics are described by the positive operator valued measure (POVM), which we briefly recall in order to fix the notation. If $\{E(\xi)\}$ is a POVM parametrized by the real parameter $\xi$, for each value of $\xi$, $E(\xi)$ is a self-adjoint operator on the system Hilbert space, they satisfy

[TABLE]

and the probability of measuring the result $\xi$ is

[TABLE]

Notice that $\xi$ can also belong to a finite set (or a combination of several discrete and continuous indices), if the number of possible outcomes is finite. The expressions throughout this article generalize replacing $\int\rmd\xi$ by $\sum_{\xi}$.

Fixing the POVM, and thinking of (10) as the distribution probability of the outcomes [as in (1)], one can use the tools introduced in section 2.1. In particular, we can calculate the Fisher information and use the Cramér-Rao inequality to know if a given estimator is optimal. Note that there exist a dependence of the Fisher information on the POVM we choose. In order to have the lowest bound for the error we maximize the Fisher information over all the possible measurements [16]

[TABLE]

The quantity $F_{Q}(\theta)$ is known as the quantum Fisher information and through the Cramér-Rao inequality,

[TABLE]

tells us the minimal possible error for the best measurement strategy for estimating a parameter appearing in the Hamiltonian of a quantum system. Equation (12) holds since Cramér-Rao inequality is valid for every POVM, therefore it is valid for the one in which the maximum Fisher information is attained. The POVM that maximizes $F_{Q}$ is the one that should be used to get the smallest error in the parameter estimation [1]; we call this POVM the optimal POVM. If the quantum state is pure there are analytical formulas for finding $F_{Q}$, otherwise no general formulae are know and one must rely in numerical methods.

2.4 Bayesian quantum Crámer-Rao inequality

Assume now that each time we prepare the quantum system we are going to meaure, the parameter $\theta$ we want to estimate is taken from a random variable $\Theta$ with probability distribution $\lambda(\theta)$.

The POVM, $E_{\rm max}(\xi)$, that maximizes the generalized Fisher information, $\int\left(\frac{\partial\ln P(\mathbf{y},\theta)}{\partial\theta}\right)^{2}P(\mathbf{y},\theta)\rmd\theta\rmd\mathbf{y}$, together with the appropiate estimator, saturates the Cramér-Rao type inequality

[TABLE]

where

[TABLE]

We call $\widetilde{Z}_{Q}$ the quantum Van Trees information.

If we want to minimize the error in the parameter estimation, and we codify what we know about the parameter in the probability distribution $\lambda(\theta)$, we have to implement the quantum measurement given by $E_{\rm max}(\xi)$ [10].

3 Numerical calculations

The calculation of cost functions as $F_{Q}$ or $\widetilde{Z}_{Q}$ is not easy, as it implies an optimization over all POVMs. In this section we present an efficient numerical procedure to calculate the maxima, over all POVMS, of convex cost functions.

3.1 Convexity

The quantum Van Trees information is convex; this follows directly noticing that set of POVMs [17, 18] and the Fisher information are convex. Fisher information can be rewritten as $F=\int(p^{\prime})^{2}/p\rmd\mathbf{x}$, where the prime indicates derivative with respect to $\theta$. It then follows that

[TABLE]

provided that $p_{1,2}\geq 0$ [19]. For a combination with other weights, continuity and a recursive procedure imply convexity of $F$. But then, the Van Trees information is also convex, as the integral of convex functions is also convex.

Since the maximum of the convex cost functions lies on the extremal points of all POVMs, we only need to search in this subset simplifying greatly the optimization task. A way to sample randomly such a set is presented in the following paragraphs.

3.2 The algorithm

The outline of the algorithm is as follows: We produce a random POVM, and decompose it in extremals. We then evaluate the cost function using all its extremal POVMs and choose the one which reaches the highest value. We repeat the procedure several times and keep the optimal POVM. We provide an implementation in [20].

Random POVMs

To produce a random POVM, we use backwards the purification algorithm [21] which transforms a general POVM into a usual projective measurement in an enlarged space. We start from a unitary matrix $U$ acting on a Hilbert space resulting of the product of the original Hilbert space, and an ancilla space of dimension equal to the number of outcomes of our POVM. This matrix is chosen according to the Haar measure in the enlarged Hilbert space [22]. To generate a member from such ensemble we build a matrix $A$ with Gaussian complex numbers (with equal standard deviation and zero mean). The desired matrix, $U$, is one that diagonalizes $A+A^{\dagger}$. Each of the operators $Q_{m}$ of the POVM we want to generate are defined via its matrix elements as

[TABLE]

where we are using tensor index notation for the space in which $U$ acts, the first corresponding to the original space, and the second to the additional ancilla space. Since for all POVMs one can build a unitary transformation in the extended space such that (16) holds [21], sampling all unitaries in the extended space guaranties sampling all POVMs with the corresponding number of outcomes.

Conversion to a rank-1 POVM

To proceed further, we need a rank-1 POVM, so we must transform the aforementioned POVM accordingly. Recall that a rank-1 POVM is one whose elements are all rank-1 operators. For all operators of the POVM that are not rank-1, we apply standard procedures to decompose them, for example the spectral decomposition for normal operators. Notice that the number of elements can change after this step. Let us call $n$ the number of outcomes of the rank-1 POVM obtain.

Obtaining an element of the decomposition

Let us define $a_{i}=\tr Q_{i}$, and $A_{ij}=a_{j}^{-1}\tr(Q_{j}G_{i})$ with $\{G_{j}\}$ an orthonormal traceless base for hermitian matrices of the appropriate dimension. In our case, we used the Gell-Mann matrices. We also define $A_{d^{2},j}=1$, so that the completeness condition over POVMS reads

[TABLE]

if we define the $d^{2}$-dimensional vector $b=(0,\cdots,0,d)$. We now propose the linear program

[TABLE]

Even though $x=a$ is a solution to the problem, the usual numerical algorithms provide an extremal point, which defines an extremal POVM [18, 9]. Notice that if an element $x_{i}$ of the solution is 0, it means that we do not include the operator in the POVM. Let this extremal solution be $x^{\rm ext}$.

To obtain the extremal POVM we start by defining $x^{\prime}$ via

[TABLE]

with $p$ a scalar. Requiring that $x^{\prime}\geq 0$ can be enforced letting

[TABLE]

which in turn implies that $p$ is a probability and that for some $i$, $x^{\prime}_{i}=0$. If we define $\textbf{Q}^{\rm ext}=x^{\rm ext}\textbf{Q}$ and $\textbf{Q}^{\prime}=x^{\prime}\textbf{Q}$, we can write

[TABLE]

Indeed, Q is an extremal POVM [9], and since one of the elements of $x^{\prime}$ is null, $\textbf{Q}^{\prime}$ is a $n-1$ output POVM for which we can iterate the algorithm until a single output POVM is obtained.

Notice that with this algorithm, all POVMs with a given number of outputs can in principle be sampled.

4 Examples

In this section we apply the method described in section 3 to estimate the quantum Fisher information and the quantum Van Trees information, $\widetilde{Z}_{Q}$. We observe a big advantage in terms of numerical effort using this method compared with finding the maximum using randomly chosen POVMs.

4.1 Qubit

We consider a spin $1/2$ particle in a superposition pure state (see [23]),

[TABLE]

where $\theta\in[0,2\pi)$ is the phase between the two basis states and $\eta\in[0,\pi]$ is a known parameter that characterizes the weight of each element of the superposition. The problem is the following: we want to find the best estimate strategy for the phase $\theta$ when $\eta$ is known.

Through the rest of the article we shall consider the following sets of POVMs:

[TABLE]

so each POVM is parametrized by $\xi$ and has two elements that corresponds to the outcomes $1$ and $2$. In this subsection we shall consider the particular case

[TABLE]

Using Eq. (10) we obtain that the probability of measuring outcome $1$ or $2$ for the POVM $\xi$ is given by

[TABLE]

The Fisher information for this probability distribution is

[TABLE]

which is a function of the parameter we want to estimate, i.e. $\theta$. Notice that there is a dependence on the initial state, via $\eta$, and on the POVM used, via $\xi$. We make this dependence on the POVM explicit via a superscript. When the state is pure, the maximum quantum Fisher information can be analytically calculated [16]; in this example the quantum Fisher information is the maximum of $F^{(\xi)}(\theta)$ with respect to $\xi$:

[TABLE]

In order to know if the RESM is useful, we apply the RSM and RESM methods and compare their results with the exact result Eq. (25). We define the errors

[TABLE]

where $F_{\rm RSM}$ and $F_{\rm RESM}$ are the Fisher information numerically calculated using the RSM and RESM respectively. In Fig. 1(left) we plot running time vs error, for the two methods. It is clear from the plot that RESM is better and the longer the program runs the better the results using RESM compared with RSM. For this example, we obtain an error two orders of magnitude smaller running the program the same time.

Now we consider that $\theta$ is a random variable with probability distribution $p(\theta)$; limits to the error in the estimation of its outcomes are given by the Cramér-Rao type inequality Eq. (13). First we consider the maximization of the generalized Fisher information over the family of POVMs, $\mathbf{P}(\xi)$, given by (21) and (22)

[TABLE]

Because we are using a subset of all the POVMS $\widetilde{Z}_{Q}^{\mathbf{P}}\leq\widetilde{Z}_{Q}$, nevertheless this approach allow us to get an analytic approximation for the quantum van Trees information. We assume that $\theta$ has a uniform distribution in $[0,2\pi)$, i.e. $p(\theta)=1/2\pi$ in (26). For a uniform superposition ($\eta=\pi/2$), the Fisher information becomes independent of $\xi$; in fact $\widetilde{Z}_{Q}^{\mathbf{P}}=\widetilde{Z}_{Q}=1$, see (24). This implies that any POVM from the family $\mathbf{P(\xi)}$ maximizes the Fisher information. In general we obtain

[TABLE]

so we can assert that if only POVMs of the family $\mathbf{P(\xi)}$ are allowed, the best estimation is in the case where $\eta=\pi/2$.

Now we apply RESM to calculate $\widetilde{Z}_{Q}$ and compare them with $\widetilde{Z}_{Q}^{\mathbf{P}}(\eta)$, see figure 1(right). The maximum of $\widetilde{Z}_{Q}$ is obtained when $\eta=\pi/2$. That means that the lowest error in the phase estimation is obtained when the weights of the superposition are the same. The figure suggest that $\widetilde{Z}_{Q}^{\mathbf{P}}=\widetilde{Z}_{Q}$.

4.2 Phase estimation

We want to estimate the phase difference $\theta$ between two paths that light can follow, see [10] for a similar calculation. We probe the system with a coherent state, such that one path yields the state $|\alpha\rangle$ (with $\alpha$ a complex number) and the other

[TABLE]

where $\hat{n}$ is the number operator.

Assume that the object that creates the phase difference is subject to fluctuations such that the phase difference between the two paths is different each time the experiment is done. One can model this assuming that $\theta$ is a random variable. We consider a Gaussian distribution centered at $\pi$, with standard deviation $\pi/4$ and trimmed at the edges ([math], $2\pi$). Using RESM we calculated the quantum van Trees information for different values of $|\alpha|$. The results are depicted in 2 (left) as red dots. The line is obtained using (21) with $\rho(\xi)=|\phi(\xi)\rangle\langle\phi(\xi)|$ as an ansatz. The figure suggest that the family of POVMS proposed is a good ansatz.

4.3 Coherent plus thermal state

As a final example, we consider estimating a parameter, chosen from a given distribution, encoded in a non-pure state. In general, there are no analytical expressions for the quantum Fisher information for this case. We calculate $\widetilde{Z}_{Q}$ in order to bound the error in estimating the parameter. We build upon the last example, considering a mixture of (28) and a thermal state. Let

[TABLE]

with

[TABLE]

be the state in which the parameter ($\theta$) is encoded. For the right side of fig. (2), we used a Gamma distribution of the form,

[TABLE]

with $\alpha=4$ and $\beta=1.5$.

In figure 2 (right) we show the numerical calculations of $\widetilde{Z}_{Q}$ using the algorithm RESM. We compare it with the case of an initial coherent pure state and with the ansatz composed of the two outcome POVM (21) with $\rho(\xi)=|\phi(\xi)\rangle\langle\phi(\xi)|$, see (29). As expected, $\widetilde{Z}_{Q}$ is larger for a pure state: a coherent pure state is better for estimating a phase than the mixture (29).

4.4 A note for reproducing results

The code implementation can be obtained in the repository [20]. To reproduce the results presented in section 4.1, set the flag -o to Qubit and vary the flag --EtaAngle from 0 to $\pi$. For the results in sections 4.2 and 4.3, set the flag -o to CohPlusTher and to CohPlusTherGamma respectively. We also set the temperature with -T 0.001, the mixing constant --MixConstant 0.5, the number of times to sample the space with -s 150, the dimension of the Hilbert space to describe the system with --HilbertDim 7 and the number of outcomes of the POVM with --Outcomedim 10. For pure state, as in section 4.2, set --MixConstant 1. The squared norm of $\alpha$ is set with the option --MeanPhotonNumb, which can be varied to reproduce the plots. The whole data set can be obtained with the command make all.

5 Conclusions

The random extreme sampling method (RESM) can be used to find efficiently the maximum of a cost function over all possible quantum measurements. Particularly it is useful to find limits in the precision of parameter estimation, through the cost function known as the quantum Fisher information, when the state to be measured is a mixed state. It can also be used to find the optimal measurement strategy by a given convex cost function by finding the POVM that maximizes it, at a considerable lower computational cost.

6 Acknowledgements

Support by PASPA-DGAPA, UNAM, projects CONACyT 285754 and UNAM-PAPIIT IG100518, IN-107414.