# Efficient qubit phase estimation using adaptive measurements

Marco A. Rodríguez-García<EMAIL_ADDRESS>Instituto de Investigaciones
en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de
México, Ciudad Universitaria, Ciudad de México 04510, Mexico  Isaac Pérez
Castillo<EMAIL_ADDRESS>Departamento de Física, Universidad Autónoma
Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de México 09340,
Mexico  P. Barberis-Blostein<EMAIL_ADDRESS>Instituto de Investigaciones
en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de
México, Ciudad Universitaria, Ciudad de México 04510, Mexico

###### Abstract

Estimating correctly the quantum phase of a physical system is a central
problem in quantum parameter estimation theory due to its wide range of
applications from quantum metrology to cryptography. Ideally, the optimal
quantum estimator is given by the so-called quantum Cramér-Rao bound, so any
measurement strategy aims to obtain estimations as close as possible to it.
However, more often than not, the current state-of-the-art methods to estimate
quantum phases fail to reach this bound as they rely on maximum likelihood
estimators of non-identifiable likelihood functions. In this work we
thoroughly review various schemes for estimating the phase of a qubit,
identifying the underlying problem which prohibits these methods to reach the
quantum Cramér-Rao bound, and propose a new adaptive scheme based on covariant
measurements to circumvent this problem. Our findings are carefully checked by
Monte Carlo simulations, showing that the method we propose is both
mathematically and experimentally more realistic and more efficient than the
methods currently available.

##  1 Introduction

The aim of statistical estimation theory in classical systems is to estimate a
probability distribution based on a series of observations. More precisely,
given a statistical model
$\mathcal{P}=\left\\{p(x\mid\theta)\mid\theta\in\Theta\subset\mathbb{R}^{n}\right\\}$
and a series of observations $\\{X_{1},\ldots,X_{N}\\}$ generated by a
particular distribution density $p(x\mid\theta)$, the aim is to design an
estimator $\hat{\theta}(X_{1},,\ldots,X_{N})$ which is as close as possible to
the parameter $\theta$, measured in terms of a cost function, usually the Mean
Square Error (MSE), between $\hat{\theta}$ and $\theta$. A fairly remarkable
result, the celebrated Cramér–Rao bound (CRB), states that the variance of an
unbiased estimator, is at least as high as the inverse of the Fisher
information of $p(x\mid\theta)$. The estimators that saturate this bound are
called efficient. An example of an efficient estimator is the maximum
likelihood, that saturates the CRB when the number of measurements tends to
infinity.

This classical result has, however, some subtleties, namely: for the maximum
likelihood estimator to be consistent (i.e. the estimator converges to the
parameter when the number of outcomes tends to infinity), the likelihood
function must be identifiable (i.e. the likelihood has a unique global
maximum), $\Theta$ must be a compact set, and all probability distributions in
the statistical model must have the same support. Statistical models
satisfying the previous conditions are called regular models.

While the actual role that the process of measurement plays in extracting
information from a classicaly system is not particularly relevant, the
situation is completely different in a quantum system, making the process of
parameter estimation mathematically more involved. Suppose $\rho$ is an
unknown density operator acting on a Hilbert space $\mathcal{H}$, that
represents the state of the system. Let us assume that $\rho$ belongs to a
particular subset of parametrized states

$\mathcal{S}=\left\\{\rho_{\theta};\theta\in\Theta\subset\mathbb{R}^{n}\right\\}\,.$
(1)

The parametrized states $\rho_{\theta}$ are the result of the evolution of an
initial probe state $\rho$ by a trace-preserving dynamical process dependent
upon a parameter $\theta$. Note that different initial conditions give rise to
different subsets of parametrized states.

Quantum parameter estimation aims to produce the best estimation for $\theta$
under the premise that the system is prepared in a state of the parametric
family $\mathcal{S}$. Notice that, unlike its classical counterpart, quantum
estimation consists of two parts: the choice of the measurements to perform on
the system, and then the processing of their outcomes through an estimator
$\hat{\theta}$. Note that measured data is a set of random outcomes with a
probability distribution depending on the parameter, and an estimator is a
random variable whose outcomes estimates the parameter. Thus, the pair
measurement-estimator forms a strategy of estimation in the quantum case one
can play around with seeking for an optimal estimation strategy. Indeed, each
choice of measurement defines a classical statistical model, whose lowest
possible CRB maximises the Fisher information. Thus the maximum of all
possible Fisher informations, over the space of measurements allowed by the
postulates of Quantum Mechanics, is called the Quantum Fisher Information
(QFI), and its inverse, the Quantum Cramér-Rao Bound (QCRB). In other words,
the QCRB is the minimum over all possible MSE of any possible estimation
strategy allowed by Quantum Mechanics. If the Fisher information for a
particular choice of measurement $M$ is equal to the QFI, this measurement is
called optimal.
One way to find an optimal measurement is to use the set of operators
$\left\\{M(j)\right\\}$ to represent the measurement, where each $M(j)$
corresponds to a projector onto each of the eigenspaces of the symmetric
logarithmic derivative (SLD) $\mathcal{L}(\theta)$ [5], defined by

$\frac{\partial\rho_{\theta}}{\partial\theta}=\frac{1}{2}\left(\mathcal{L}(\theta)\rho_{\theta}+\rho_{\theta}\mathcal{L}(\theta)\right)\,.$
(2)

It turns out that, in general, this measurement is so-called locally optimal.
This means that the optimal measurement depends on the actual value of the
parameter we want to estimate and the classical Fisher information given by
that measurement is a local maximum in the parameter space that characterizes
the measurement. Thus estimating the parameter using a locally optimal
measurement is rather impractical. Two approaches have been developed to
overcome this problem. The first one is based on adaptive estimation schemes
which updates a guess for a locally optimal measurement [13, 1]. The second
method seeks a set of initial conditions that do not depend on the unknown
parameter [7, 33]. Both methods are based on the maximum likelihood estimator
(MLE) which implies that in order to obtain it, the aforementioned set of
regularity conditions must be met [11, 20, 6]. We will see that the MLE
derived from optimal measurements may fail to satisfy these conditions and
thus these two approaches fail to attain the QCRB.

The problem of non-identifiable likelihood functions often appears when
estimating the quantum phase of a system, a particular problem with a wide
range of applications from quantum metrology to cryptography, as many problems
can be recast into a quantum phase estimation [10, 24, 27]. In this framework,
when trying to use locally optimal measurements [14, 12, 31, 19, 33], it may
happen that adaptive schemes fail to reach the QCRB [28] due to the fact that
regularity conditions are not met [13]. A paradigmatic example of non-
identifiablity is provided when trying to perform qubit phase estimation which
are, in general, unable to reach the QCRB [7, 1].
The main goal of the present work is to introduce a new adaptive scheme for
quantum phase estimation which saturates the QCRB independently of the initial
condition that generates $\mathcal{S}$. We will first show that the Adaptive
Quantum State Estimation method (AQSE), a general method for parameter
estimation [35], does not achieve the QCRB when applied to a qubit phase
estimation. The reason why AQSE does not converges is that the statistical
model, built with the measurement that maximizes the Fisher information (the
locally optimal measurement), is not identifiable. We will further show that
the non-identifiability problem is solved by taking a sample of measures from
a covariant measurement. The covariant measurement is identifiable and is
chosen to minimize the MSE for one measure. With the covariant sample, we can
then construct a confidence interval where the underlying statistical model is
now regular. Then we will apply the AQSE method inside the confidence
interval.
The paper is organized as follows. We start by giving a brief review of
quantum estimation theory with special emphasis in covariant estimation
techniques for periodic parameter estimation (section 2). Then, we review
different estimation strategies for the phase of a qubit and discuss their
weakness and strengths, in particular we discuss the effects of non-
identifiable probabilistic models in the estimation error (section 3). Once we
understand the problems with different estimation strategies, we present a
two-step estimation scheme. Numerical simulations suggest that this scheme is
able to reach the QCRB (section 4). We end the paper with a summary of the
main results and our conclusions (section 5).

##  2 Quantum parameter estimation

Recall that, given a set of independent observations
$\\{x_{1},\ldots,x_{N}\\}$ from a random variable $X$ with probability density
$p(x\mid\theta)$, the likelihood function is defined as

$L(\theta\mid x_{1},...,x_{N})=\prod_{i=1}^{N}p(x_{i}\mid\theta).$ (3)

From here, we can derive the maximum likelihood estimator for $\theta$, whose
mean square error obeys the CRB [20]

$\text{MSE}(\hat{\theta})\geq\frac{1}{NF(\theta;X)}\,,$ (4)

where the MSE is defined by

$\text{MSE}(\hat{\theta})=E_{\theta}\left[\left(\hat{\theta}-\theta\right)^{2}\right]\,,$
(5)

and $F(\theta;X)$ is the Fisher information:

$\begin{split}F(\theta;X)=E_{\theta}\left[\left(\frac{\partial}{\partial\theta}\log
p(x\mid\theta)\right)^{2}\right].\end{split}$ (6)

Intuitively, the Fisher information quantifies how much information carries a
sample about the unknown parameter. The ultimate aim in classical parameter
estimation is to find the estimator that achieves the Cramér-Rao bound. If the
statistical model $\left\\{p(x\mid\theta)\mid\theta\in\Theta\right\\}$ for a
random variable $X$ is regular the maximum likelihood estimator produced by a
sequence of $N$ independent and identically distributed of $X$ can attain the
Cramér-Rao bound asymptotically for large $N$ [20, 11].
In the quantum case, we must first describe the process of measuring the
system. This is better achieved by using the concept of Positive Operator-
Valued Measures (POVMs). A POVM with outcomes on a set $\mathcal{X}$ is a
family of bounded positive operators

$M=\left\\{M(B);B\text{ is a Borel set in }\mathcal{X}\right\\}$

acting over the Hilbert space of the system $\mathcal{H}$ such that
$M(\mathcal{X})=I$. When the state of the system is given by the density
operator $\rho_{\theta}$, a POVM $M$ specifies the conditional probability of
obtaining the event $B$ by Born’s rule

$P\left(B\mid\theta\right)=\text{Tr}\left[M(B)\rho_{\theta}\right].$ (7)

Thus, in quantum parameter estimation, the Fisher information, given by Eq.
(6), is a function of the POVMs and, henceforth, we will denote it as
$F(\theta;M)$. This implies that, according to Eq. (4), maximising
$F(\theta;M)$ over the set of POVMs gives the lowest CRB. This results into
the so-called Quantum Fisher Information (QFI), which we will denote as
$F_{Q}(\theta)$, and its lowest bound is so-called the Quantum Cramér-Rao
Bound (QCRB) [15, 18, 16].
For the particular case of a scalar parameter estimation, the QFI has the
following form [15, 16, 5]

$F_{Q}(\theta)=\text{Tr}\left[\rho_{\theta}\mathcal{L}(\theta)^{2}\right],$
(8)

where $\mathcal{L}(\theta)$ is the symmetric logarithmic derivative, also
called quantum score, defined by Eq. (2). When the system is in a pure state
$\rho_{\theta}=\left|\psi_{\theta}\right\rangle\left\langle\psi_{\theta}\right|$,
the quantum score is easy to calculate [29], obtaining
$\mathcal{L}(\theta)=2\frac{\partial\rho_{\theta}}{\partial\theta}$, so that
the QFI becomes:

$F_{Q}(\theta)=4\leavevmode\nobreak\
\text{Tr}\left[\left(\frac{\partial}{\partial\theta}\left(\left|\psi_{\theta}\right\rangle\left\langle\psi_{\theta}\right|\right)\right)^{2}\left|\psi_{\theta}\right\rangle\left\langle\psi_{\theta}\right|\right].$
(9)

Notice that the set of POVMs $\left\\{M_{\theta}(j)\right\\}$ constructed as
the projections of the quantum score $\mathcal{L}(\theta)$ do depend on the
parameter $\theta$ one is trying to infer [5]. A natural way around this is to
introduce adaptive schemes to estimate $\theta$. One approach relies on
adaptive quantum estimation schemes based on locally optimal POVMs that could,
in principle, asymptotically construct the optimal POVM without knowing the
parameter beforehand [25, 13, 2, 19]. Nevertheless, a set of precise mild
regularity conditions for each statistical model involved in the method are
required. For instance, in the adaptive quantum state estimation (AQSE) method
[25, 13], it is necessary to assume regular statistical models for every
measurement to guarantee a consistent and efficient estimator. However, for
estimation problems on which the quantum parameter is periodic, as it is the
case for phase estimation, the likelihood functions produced by locally
optimal POVMs are not identifiable [7, 3]. Consequently, there is no
mathematical reason that ensures the saturation of the CRB. The second
approach searches specific initial conditions, for which the optimal POVM does
not depend on the unknown value of $\theta$ [33, 7]. Hence, in principle,
performing an extensive independent sequence of this POVM, along with the
maximum likelihood estimator, it is possible to achieve the quantum Cramér-Rao
bound. Nonetheless, for periodic parameter estimation, this method often
produces non-identifiable likelihood functions, and as a consequence, the CRB
is not attained.
We will show now that, for the case of quantum phase estimation, we can solve
the problem of non-identifiable likelihood functions by using the formalism of
covariant estimation, which considers the symmetries of the system and can
saturate the QCRB under specific initial conditions [16].

###  2.1 Covariant estimation

The following discussion is based on [16]. In quantum parameter estimation,
each $\rho_{\theta}\in\mathcal{S}$ obtains its parametric dependency through a
physical transformation $U_{\theta}$. Usually, the set of
$\left\\{U_{\theta}\right\\}_{\theta\in\Theta}$ forms a group, which induces
an action over $\mathcal{S}$. When the family $\mathcal{S}$ is invariant under
the conjugation of $U_{\theta}$, for any $\theta\in\Theta$, we say that the
problem of estimation involves a symmetry. The problem of quantum parameter
estimation involving symmetries is called covariant. In the following, we
summarise the approach of covariant estimation.
Let $G$ be a locally compact Lie group acting transitively over the parametric
set $\Theta$. Thus for any base point $\theta_{0}\in\Theta$, we have

$\Theta=\theta_{0}\cdot G\cong G/K\,,$ (10)

where $K$ is the stabilizer subgroup of $\theta_{0}$. According to Wigner’s
theorem [34], the Hilbert space $\mathcal{H}$ of the system has a unitary
$G$-representation $U=\left\\{U_{g}\right\\}_{g\in G}$. Hence, the probe
states can be transformed by the conjugation

$\rho\mapsto U_{g}\rho U_{g}^{\dagger}\,,$ (11)

where $U_{g}$ is an element of the $G$-unitary representation of
$\mathcal{H}$. Specifically, we say that we have a quantum covariant
estimation problem whenever

$\left\\{U_{g}\rho_{\theta}U_{g}^{\dagger}\mid\theta\in\Theta\right\\}=\mathcal{S}\quad\forall
g\in G\,.$ (12)

In this framework, a POVM $M$ on the system $\mathcal{H}$ taking values in
$\Theta$ is called _covariant_ with respect the $G$-unitary representation
$\left\\{U_{g}\right\\}$, if for every event $B$ and $g\in G$, it satisfies
that

$M(Bg^{-1})=U_{g}^{\dagger}M(B)U_{g},$ (13)

where $Bg^{-1}=\left\\{\theta\mid\theta=g\theta^{\prime},\theta^{\prime}\in
B\right\\}$. Note that this condition is equivalent to

$P\left(B\mid\rho\right)=P\left(gB\mid U_{g}\rho U_{g}^{\dagger}\right)\,.$
(14)

We will denote as $\mathcal{M}\left(\Theta\right)$ the set of POVMs with
outcomes in $\Theta$. Similarly, we will denote as as
$\mathcal{M}\left(\Theta,U\right)$ the set of the covariant POVMs with respect
the $G$-unitary representation.
The class of covariant POVMs takes advantage of group symmetries.
Specifically, when the outcomes of the measurement are considered as the
estimates, and the cost function $c(\hat{\theta},\theta)$ under consideration
is $G$-invariant, that is
$c(\hat{\theta},\theta)=c(g\hat{\theta},\theta)\leavevmode\nobreak\ \forall
g\in G$, one has that

$\begin{split}E_{g\theta,M}\left[c(\hat{\theta},g\theta)\right]&=\int_{\Theta}c(\hat{\theta},g\theta)Tr\left[M(d\hat{\theta})\rho_{g\theta}\right]=\int_{\Theta}c(g\hat{\theta^{\prime}},g\theta)Tr\left[M(d\hat{\theta^{\prime}})\rho_{\theta}\right]\\\
&=E_{\theta,M}\left[c(\hat{\theta},\theta)\right],\,\forall\,M\in\mathcal{M}\left(\Theta,U\right)\text{
and }\hat{\theta}^{\prime}:=g^{-1}\hat{\theta}.\end{split}$ (15)

As a result, we can find optimal measurements on
$\mathcal{M}\left(\Theta,U\right)$ in the Bayesian approach, by taking the
average of $c(\theta,\hat{\theta})$ over a prior probability measure. The
following theorem ensures this assertion.

######  Theorem 1 ([18]).

When $\Theta$ is compact,

$\begin{split}\min_{M\in\mathcal{M}\left(\Theta\right)}\int_{\Theta}E_{\theta,M}\left[c(\hat{\theta},\theta)\right]\mu(d\theta)=\min_{M\in\mathcal{M}\left(\Theta,U\right)}\int_{\Theta}E_{\theta,M}\left[c(\hat{\theta},\theta)\right]\mu_{\Theta}(d\theta).\end{split}$
(16)

where $\mu_{\Theta}$ is the Haar measure over $\Theta$.

The previous result restricts the search for optimal measurements to the class
$\mathcal{M}\left(\Theta,U\right)$. This fact is particular useful as,
according to the following theorem, the covariant POVMs can be characterised
as follows.

######  Theorem 2 ([16]).

Let $P_{0}$ be a Hermitian positive operator on $\mathcal{H}$, commuting with
the operators $\left\\{U_{g}\right\\}_{g\in K}$ and satisfying

$\int_{\Theta}U_{g(\theta)}P_{0}U_{g(\theta)}^{\dagger}\mu_{\Theta}(d\theta)=I,$
(17)

where $g(\theta)\in G$ is any representative element of the equivalence class
$\theta\in\Theta$. Then, a POVM $M(d\theta)$ with outcomes in $\Theta\cong
G/K$ is covariant if and only if has the form

$M(d\theta)=U_{g(\theta)}P_{0}U_{g(\theta)}^{\dagger}\mu_{\Theta}(d\theta).$
(18)

A particular case, which is relevant for quantum phase estimation, is when the
stability group is the trivial one, that is when $K=\\{e\\}$ with $e$ the
identity element. Here, we have that $\Theta\cong G$, that is, the parametric
space $\Theta$ is on one-to-one correspondence to the elements of the group
$G$. This implies, in turn, that
$E_{\theta,M}\left[c(\hat{\theta},\theta)\right]$ is constant for all
$\theta\in\Theta$ and, as consequence of theorem (1), the optimization of the
expectation of the invariant cost function can be restricted to the set of
covariant POVMs. Notice that for quantum phase estimation, there is a little
caveat: here the one-parameter symmetry group $G$ isomorphic to
$\Theta=[0,2\pi)$ but this is not a compact set. In practice, we can consider
it to be compact, assuming that the unknown parameter is an interior point of
$\Theta$. Thus, we can analyze the CRB for any covariant POVM.
We proceed to discuss in more detail how these ideas apply to quantum phase
estimation for qubit states.

##  3 Phase estimation strategies in a qubit

Let $\mathcal{S}=\left\\{\rho_{\theta};\,\theta\in\Theta=[0,2\pi)\right\\}$ be
a parametric family of density operators, on a $2$-dimensional Hilbert space
$\mathcal{H}$. In this case, each state $\rho_{\theta}\in\mathcal{S}$
represents the state of a qubit. Here, the states $\rho_{\theta}$ obtain their
parametric dependency applying an arbitrary unitary transformation
$U_{\theta}$ over a probe state $\rho$ on $\mathcal{H}$, that is

$\rho_{\theta}=U_{\theta}\rho U_{\theta}^{\dagger}\,.$ (19)

An arbitrary unitary transformation (up to a global phase) on a
$2$-dimensional Hilbert space can be written using the generators of the Lie
algebra su$(2)$ and has the following form [23, 26]

$U_{\theta}=e^{-i\theta\vec{n}\cdot\frac{\vec{\sigma}}{2}}\,,$ (20)

where $\vec{n}=n_{x}\hat{x}+n_{y}\hat{y}+n_{z}\hat{z}\in\mathbb{R}^{3}$ is a
unit vector and
$\vec{\sigma}=\sigma_{1}\hat{x}+\sigma_{2}\hat{y}+\sigma_{3}\hat{z}$, with
$\sigma_{i}$, for $i=1,2,3$, the Pauli matrices. The estimation problem in
this case consists in finding the best strategy to estimate the phase
$\theta\in\Theta$ of the exponential operator appearing in Eq. (20).
To analyse the QCRB in this case it is better to work using the Bloch sphere
representation for a qubit state. The probe state can be written as

$\rho=\frac{1}{2}\left(I+\vec{a}\cdot\vec{\sigma}\right)\,,\quad\quad\left\lVert\vec{a}\right\rVert\leq
1\,,$ (21)

and the transformation $U_{\theta}$ can be seen as the rotation of the probe
Bloch vector $\vec{a}$ by the angle $\theta$ around the axis $\vec{n}$. We
will denote the transformed vector as $\vec{a}(\theta)$, so the transformed
qubit state $\rho_{\theta}$ is solely determined by $\vec{a}(\theta)$. One can
show that, for this problem, the quantum score takes the following form [7]:

$\mathcal{L}(\theta)=\left(\vec{n}\times\vec{a}(\theta)\right)\cdot\vec{\sigma}\,,$
(22)

where

$\vec{a}(\theta)=\cos(\theta)\vec{a}+\left(\sin(\theta)(\vec{n}\times\vec{a})\right)+\left(2(\vec{n}\cdot\vec{a})\sin^{2}\left(\frac{\theta}{2}\right)\right)\vec{n}\,.$
(23)

This implies that, according to Eq. (8), the quantum Fisher information
becomes

$F_{Q}=\left\lVert\vec{a}\right\rVert^{2}-(\vec{a}\cdot\vec{n})^{2}\,.$ (24)

Thus, if the probe qubit is a pure state (i.e.,
$\left\lVert\vec{a}\right\rVert=1$) and the Bloch vector $\vec{a}$ is
orthogonal to the rotation axis $\vec{n}$, then Eq. (24) achieves their
maximal value $F_{Q}^{\text{max}}=1$. We will refer to this as the optimal
initial conditions. Let us proceed to discuss the different estimation
strategies for the phase $\theta$.

###  3.1 Locally Optimal POVMs

From the expression of the quantum score, given by Eq. (22), it is
straightforward obtain that its projection operators are given by:

$M_{g}(0)=\frac{1}{2}\left(I+\frac{\left(\vec{n}\times\vec{a}(g)\right)}{\left\lVert\vec{n}\times\vec{a}(g)\right\rVert}\cdot\vec{\sigma}\right)\,,\quad\quad
M_{g}(1)=\frac{1}{2}\left(I-\frac{\left(\vec{n}\times\vec{a}(g)\right)}{\left\lVert\vec{n}\times\vec{a}(g)\right\rVert}\cdot\vec{\sigma}\right)\,,\quad\quad
g\in\Theta\,.$ (25)

Hence, the family of POVMs is $M_{g}=\left\\{M_{g}(0),M_{g}(1)\right\\}$, with
outcomes in the set $\mathcal{X}=\left\\{0,1\right\\}$, yield the following
classical Fisher information

$F(\theta;g)=\frac{F_{Q}\cos^{2}(\theta-g)}{1-F_{Q}\sin^{2}(\theta-g)}\,.$
(26)

From Eq. (26) it is fairly obvious to realize that $F(\theta,g)=F_{Q}(\theta)$
whenever $g=\theta$ for any $\theta\in\Theta$. That is, the POVM
$M_{\theta}=\left\\{M_{\theta}(0),M_{\theta}(1)\right\\}$ is locally optimal.
Note, however, that if $\vec{a}\perp\vec{n}$ then

$F(\theta;g)=\frac{\cos^{2}(\theta-g)}{1-\sin^{2}(\theta-g)}=1=F_{Q}^{\text{max}}.$
(27)

In other words, when the probe state is prepared in the optimal initial
conditions, the POVM $M_{g}$ is optimal, independent of $\theta\in\Theta$.
Therefore, any POVM $M_{g}$ is optimal.

At this point, one may think that a POVM $M_{g}$ produces an optimal
estimation strategy in the sense that if we were to perform a sequence of $N$
independent measurements using the maximum likelihood estimator, it should be
possible to saturate the QCRB in the asymptotic regime. This is, however,
false: the POVM $M$ yields a non-identifiable likelihood function, and thus
the maximum likelihood estimator is not consistent. Indeed, when a sequence of
$N$ identically quantum systems with Hilbert space $\mathcal{H}$ is prepared
in the same state $\rho_{\theta}$, the composite system is described by the
$N$-tensor product
$\underbrace{\mathcal{H}\otimes\cdots\otimes\mathcal{H}}_{N\text{-times}}$ and
its state is described by an $N$-fold tensor product state
$\underbrace{\rho_{\theta}\otimes\cdots\otimes\rho_{\theta}}_{N-\text{
times}}$. Then, the likelihood function produced by the outcomes
$\vec{x}=\left(x_{1},...,x_{N}\right)\in\mathcal{X}^{N}$ from a sequence of
$N$ independent POVMs $M(g)$ is

$\begin{split}L(\theta\mid\vec{x};g)&=\text{Tr}\left[\left(M_{g}(x_{1})\otimes\cdots\otimes
M_{g}(x_{N})\right)\rho_{\theta}\otimes\cdots\otimes\rho_{\theta}\right]=\prod_{i=1}^{N}\text{Tr}\left[M_{g}(x_{i})\rho_{\theta}\right]=\prod_{i=1}^{N}p(x_{i}\mid\theta;g)\\\
&=p(0\mid\theta;g)^{m}p(1\mid\theta;g)^{N-m}\,,\end{split}$ (28)

where $m$ is the number of $0$’s in $\vec{x}$, and

$p\left(x\mid\theta;g\right)=\begin{cases}\text{Tr}\left[M_{g}(1)\rho_{\theta}\right]=\frac{1}{2}\left[1+\sin\left(\theta-g\right)\sqrt{F_{Q}}\right]\,,\quad
x=1\,,\\\
\text{Tr}\left[M_{g}(0)\rho_{\theta}\right]=\frac{1}{2}\left[1-\sin\left(\theta-g\right)\sqrt{F_{Q}}\right]\,,\quad
x=0\,.\end{cases}$ (29)

By looking for the MLE of $\theta$, denoted $\hat{\theta}_{\text{MLE}}$, we
obtain:

$\hat{\theta}_{\text{MLE}}=\text{arg
max}_{\theta\in\Theta}L(\theta\mid\vec{x};g)=\text{arcsin
}\left(\frac{1}{\sqrt{F_{Q}}}\left[1-\frac{2m}{N}\right]\right)+g\,,$ (30)

which returns two values in $\Theta=\left[0,2\pi\right)$, indicating the the
likelihood is non-identifiable. Thus, the Fisher information alone is not
enough to characterize the error of this estimation strategy. The previous
discussion also can be found in [7].

Now we consider the case of POVMs with $k\geq 3$ outcomes. Any element of a
POVM $M=\left\\{M(i)\right\\}$ with outcomes in a set
$\mathcal{X}=\left\\{1,2,...,k\right\\}$ can be written as
$M(i)=f_{0}^{(i)}I+\vec{f}^{(i)}\cdot\vec{\sigma}$, where
$\sum_{i=1}^{k}f_{0}^{(i)}=1$ and $\sum_{i=1}^{k}\vec{f}^{(i)}=\vec{0}$. When
a qubit is in a state $\rho_{\theta}$, the probability of measuring the
outcome $i$ is

$p(i\mid\theta)=\text{Tr}\left[\rho_{\theta}M(i)\right]=f_{0}^{(i)}+\vec{a}(\theta)\cdot\vec{f}^{(i)}.$
(31)

A necessary and sufficient condition for identifiability is that, for any
parameters $\theta_{1},\theta_{2}\in\Theta$, the set of equations
$p(i\mid\theta_{1})=p(i\mid\theta_{2})$, $i=1,...,k$ admits only one solution
in the parametric space $\Theta$ [21].

The problem of identifiability could be avoided by considering POVMs with
several outcomes because it is easier to satisfy the requirements for
identifiability given above. However, a POVM that produces an identifiable
likelihood is not necessarily optimal. Outside the optimal initial conditions,
there is no measurement $M$ that does not depend on $\theta$ and that
saturates the QCRB [1]. In [7] it is shown a family of identifiable POVMs with
more than $2$ outcomes, however, except for the optimal initial condition,
this family is not locally optimal. The results in [1, 7] do not forbid the
existence of an identifiable and locally optimal POVM with more than $2$
outcomes, but we could not find one. Note also that for a small number of
measurements, the locally optimal POVM does not necessarily saturate the QCRB.
Nevertheless it is possible to built a POVM that, for one measurement,
produces an identifiable model that minimizes the mean square error. We will
review now how to build this POVM by using the covariant approach [16]. Once
we show how this POVM is built, we proceed with our proposal to achieve the
QCRB for qubit phase estimation.

###  3.2 Covariant phase estimation for qubits

The following discussion is based on [17], where $M_{*}$ is deduced for any
shift parameter.

As we have previously mentioned, phase estimation estimation is covariant with
trivial stabilizer group since the group $G=\left[0,2\pi\right)$ equipped with
addition modulo $2\pi$ acts over itself. Then, $\Theta$ is isomorphic to $G$.
Moreover, $U=\left\\{U_{\theta}\right\\}_{\theta\in\left[0,2\pi\right)}$ is a
$G$-unitary representation of $\mathcal{H}$. Using Eq. (18) the covariant POVM
$M$ to estimate $\theta$ has the form

$M(d\,\hat{\theta})=e^{-i\hat{\theta}\vec{n}\cdot\frac{\vec{\sigma}}{2}}P_{o}e^{i\hat{\theta}\vec{n}\cdot\frac{\vec{\sigma}}{2}}\frac{d\hat{\theta}}{2\pi}\,,$
(32)

where $P_{o}$ is a positive operator satisfying Eq. (17).
The generator of the unitary representation is the operator
$J:=\vec{n}\cdot\frac{\vec{\sigma}}{2}$, which has a spectral decomposition

$\begin{split}&J=\frac{1}{2}\left|\tfrac{1}{2}\right\rangle\left\langle\tfrac{1}{2}\right|-\frac{1}{2}\left|-\tfrac{1}{2}\right\rangle\left\langle-\tfrac{1}{2}\right|\,,\end{split}$
(33)

where have used that
$\left|\pm\tfrac{1}{2}\right\rangle\left\langle\pm\tfrac{1}{2}\right|=\frac{1}{2}\left(I\pm\vec{n}\cdot\vec{\sigma}\right)$.
In the basis
$\left\\{\left|\tfrac{1}{2}\right\rangle,\left|-\tfrac{1}{2}\right\rangle\right\\}$,
one can express $M(d\,\hat{\theta})$ as

$M(d\,\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\sum_{m,n\in\left\\{-\frac{1}{2},\frac{1}{2}\right\\}}e^{i\hat{\theta}\left(n-m\right)}p_{nm}\left|m\right\rangle\left\langle
n\right|,$ (34)

where $p_{mn}=\left\langle m|P_{o}|n\right\rangle$. Thus, any covariant POVM
is characterized by the real numbers $0\leq p_{nm}\leq 1$. Then, by Eq. (7)

$\begin{split}&P(\hat{\theta}\in
B|\theta)=\int_{B}\frac{d\hat{\theta}}{2\pi}\sum_{m,n\in\left\\{-\frac{1}{2},\frac{1}{2}\right\\}}e^{i(n-m)(\hat{\theta}-\theta)}p_{mn}\left\langle
n\right|\rho\left|m\right\rangle.\end{split}$ (35)

Note that the measure (35) is a $2\pi$-periodic probability distribution, so
it is natural to consider the moments of the random variable $e^{i\phi}$
instead. The first moment for a circular distribution $p(\phi)$ is defined as

$\begin{split}E\left[e^{i\phi}\right]=\int_{0}^{2\pi}e^{i\phi}p(\phi)d\,\phi\,,\end{split}$
(36)

so from here we can estimate the phase as
$\overline{\phi}=\text{Arg}\left(E\left[e^{i\phi}\right]\right)$.
To quantify the correct dispersion of the estimates we use the Holevo variance
[16]

$V^{H}_{M}(\hat{\theta})=\mu^{-2}-1\,,$ (37)

where $M\in\mathcal{M}\left(\Theta,U\right)$ and $\mu=\left\rvert
E\left[e^{i\hat{\theta}}\right]\right\lvert$. If one has a biased estimator,
then $\mu=E\left[\cos(\hat{\theta}-\theta)\right]$ [3]. A circular estimator
$\hat{\theta}$ is unbiased if $e^{i\hat{\theta}}\propto
E\left[e^{i\hat{\theta}}\right]$. For narrowly peaked and symmetric
distributions around $\theta$,
$V^{H}_{M}(\hat{\theta})\sim\text{MSE}_{M}(\hat{\theta})$. As a result,
Holevo’s variance is lower bounded by

$V^{H}_{M}(\hat{\theta})\geq\frac{1}{F_{Q}(\theta)}\,.$

As the quantum Fisher information reaches its largest values for the families
of pure states, let us restrict our attention only to pure quantum probes.
Moreover, when the probe is a pure state, one can find the covariant POVM that
minimizes the Holevo variance using the spectrum of $J$ for any shift
parameter [16, 17]. Here, we adapt the proof of [16] to the problem of phase
estimation.

######  Theorem 3.

Let $\rho_{\theta}=U_{\theta}\rho U_{\theta}^{\dagger}$, with
$\rho=\left|\psi\right\rangle\left\langle\psi\right|$ and

$M_{*}(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\sum_{m,n\in
Spec(J)}e^{i\hat{\theta}\left(n-m\right)}\cfrac{P_{m}\rho
P_{n}}{\sqrt{\text{Tr}[P_{m}\rho]\text{Tr}[P_{n}\rho]}},$ (38)

where $P_{m}=\left|m\right\rangle\left\langle m\right|$ is the associated
projector to the eigenvalue $m$ of $J$. Then, for any
$M\in\mathcal{M}\left(\Theta,U\right)$,

$V_{M}^{H}(\hat{\theta})\geq V_{M*}^{H}(\hat{\theta})\,.$ (39)

###### Proof.

In Dirac notation, $\rho=\left|\psi\right\rangle\left\langle\psi\right|$ so
that Eq. (38) takes the following form

$M_{*}(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\sum_{m,n\in
Spec(J)}e^{i\hat{\theta}\left(n-m\right)}p_{mn}^{*}\left|m\right\rangle\left\langle
n\right|,$ (40)

with $p_{mn}^{*}=\frac{\left\langle m|\psi\right\rangle}{|\left\langle
m|\psi\right\rangle|}\frac{\left\langle\psi|n\right\rangle}{|\left\langle\psi|n\right\rangle|}$.
On other hand, from Eq. (34), we have

$E_{M,\theta}[e^{ik\hat{\theta}}]=\sum_{\begin{subarray}{c}m,n:\\\
|m-n|=k\end{subarray}}p_{mn}\left\langle\psi|m\right\rangle\left\langle
n|\psi\right\rangle e^{-i(n-m)\theta}.$ (41)

Since $|p_{mn}|\leq 1$,

$\begin{split}E_{M,\theta}[e^{ik\hat{\theta}}]&\leq\sum_{\begin{subarray}{c}m,n:\\\
|m-n|=k\end{subarray}}\left\langle\psi|m\right\rangle\left\langle
n|\psi\right\rangle e^{-i(n-m)\theta}\leq\sum_{\begin{subarray}{c}m,n:\\\
|m-n|=k\end{subarray}}|\left\langle\psi|m\right\rangle||\left\langle
n|\psi\right\rangle|e^{-i(n-m)\theta}.\end{split}$ (42)

where the equality is achieved if and only if $p_{mn}=p_{mn}^{*}$. Setting
$k=1$ yields the assertion $\square$. ∎

Besides, if $\left\langle m|\psi\right\rangle$ is a constant for all
$m\in\text{Spec}(J)$, the measurement $M_{*}$ has information about $\theta$
equal to the quantum Fisher information [17]. For qubits, the previous
condition is equivalent to have an initial condition $\vec{a}\cdot\vec{n}=0$.

In this context, we now prove that the measurement $M_{*}$ maximizes the
Fisher information over the set of covariant POVMs.

######  Theorem 4.

Let $\rho_{\theta}=e^{-i\theta\vec{n}\cdot\frac{\vec{\sigma}}{2}}\rho
e^{-i\theta\vec{n}\cdot\frac{\vec{\sigma}}{2}}$, with

$\begin{split}\rho=\left|\psi\right\rangle\left\langle\psi\right|=\frac{1}{2}\left[I+\vec{a}\cdot\hat{n}\right]\,,\quad\quad\left\lVert\vec{a}\right\rVert=1.\end{split}$

Then, for any $M\in\mathcal{M}\left(\Theta,U\right)$,

$F(\theta;M)\leq F(\theta;M_{*})\,,$ (43)

where $M_{*}(d\hat{\theta})$ is defined according to Eq. (38).

###### Proof.

Writing explicitly Eq. (35) we obtain

$p(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\left[1+2\text{Re}\left(e^{-(\hat{\theta}-\theta)}\cdot
C\right)\right],$ (44)

where
$C=p_{\footnotesize{\frac{1}{2}-\frac{1}{2}}}\left\langle-\tfrac{1}{2}|\psi\right\rangle\left\langle\psi|\tfrac{1}{2}\right\rangle$
is a complex number with module $|C|$ and phase $\varphi$. Thus,

$p(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\left[1+2|C|\cos\left((\hat{\theta}-\theta)+\varphi\right)\right].$
(45)

Therefore the Fisher information,

$\begin{split}F(M)&=\frac{1}{2\pi}\int_{\Theta}d\hat{\theta}\frac{4|C|^{2}\sin^{2}\left((\hat{\theta}-\theta)+\varphi\right)}{1+2|C|\cos\left((\hat{\theta}-\theta)+\varphi\right)}=\frac{4|C|^{2}}{2\pi}\int_{\Theta}dx\frac{\sin^{2}\left(x\right)}{1+2|C|\cos\left(x\right)}.\end{split}$
(46)

The value of the expression given by Eq. (46) increases monotonically as $|C|$
increases. As
$\lvert\left\langle\pm\tfrac{1}{2}|\psi\right\rangle\rvert=\sqrt{\tfrac{1}{2}\left(1\pm(\vec{a}\cdot\vec{n})\right)}$,
then $|C|=|\leq\frac{1}{2}\sqrt{F_{Q}}$, and the maximum value for $F(M)$ is
attained by $M_{*}$ $\square$. ∎

To find an explicit epxression of $p(d\hat{\theta}|\theta)$, we first notice
that

$\begin{split}\left|\pm\tfrac{1}{2}\right\rangle\left\langle\pm\tfrac{1}{2}|\psi\right\rangle\left\langle\psi|\mp\tfrac{1}{2}\right\rangle\left\langle\mp\tfrac{1}{2}\right|=\frac{1}{4}\left[2(\vec{a}\cdot\vec{\sigma})+i(\vec{a}\times\vec{n})\cdot\vec{\sigma}-(\vec{n}\cdot\vec{\sigma})(\vec{n}\cdot\vec{a})\right]\,,\end{split}$
(47)

so that Eq. (38) can be rewritten as follows

$M_{*}(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\left[I+\frac{1}{\sqrt{F_{Q}}}\left[\left(\vec{a}\cdot\vec{\sigma}-(\vec{n}\cdot\vec{\sigma})(\vec{n}\cdot\vec{a})\right)\cos(\hat{\theta})-\sin(\hat{\theta})(\vec{a}\times\hat{n})\cdot\vec{\sigma}\right]\right]\,.$
(48)

Then, the measurement $M_{*}$ yields the probability density function

$p(d\hat{\theta}\mid\theta)=\frac{d\hat{\theta}}{2\pi}\left[1+\sqrt{F_{Q}}\cos(\hat{\theta}-\theta)\right]\,.$
(49)

As $M_{\star}$ is a POVM with an infinity number of outcomes, it is reasonable
to get an identifiable statistical model independent of $\theta$. To
illustrate this fact, we perform a numerical simulation, generating random
numbers with distribution (49). A particular likelihood function produced by
(49) is shown in Fig. 1.

Figure 1: Plot of the likelihood function corresponding to the probability
density Eq. (49) for the measure $M_{\star}$ and $N=64$ probes. We have
considered the value of the actual parameter to be $\theta=2$ and used initial
conditions $\vec{a}\cdot\vec{n}=0$. For clarity we have rescaled the figure so
that the maximum equals to one.

Moreover, according to Eq. (49), the Fisher information given $M_{*}$ reads:

$F(\theta;M_{*})=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{F_{Q}\sin^{2}(\hat{\theta}-\theta)}{1+\sqrt{F_{Q}}\cos(\hat{\theta}-\theta)}d\hat{\theta}=1-\sqrt{1-F_{Q}}\,.$
(50)

When $\vec{a}\cdot\vec{n}=0$, $F_{Q}=1$ and therefore
$F(\theta;M_{*})=F^{\text{max}}_{Q}=1$. This is not surprising since, under
this optimal initial condition,
$\lvert\left\langle\tfrac{1}{2}|\psi\right\rangle\rvert$ =
$\lvert\left\langle-\tfrac{1}{2}|\psi\right\rangle\rvert$. Consequently,
assuming this optimal setup, if the maximum likelihood is used for $N$
independent copies of the system, one can attain the QCRB asymptotically for
large $N$. Nevertheless, under the optimal initial condition, there are other
covariant POVMs that can attain the QCRB. To see this, let us characterize all
covariant measurements for quantum phase estimation of qubits.

Every positive bounded operator $P_{o}$ can be expressed in the Bloch
representation, that is, $P_{o}=d_{0}+\vec{d}\cdot\vec{\sigma}$, where
$(d_{0},\vec{d})$ are real. When evaluating Eq. (17), it turns out that
$P_{o}$ forms a valid covariant POVM if $d_{0}=1$ and $\vec{d}$ is orthogonal
to $\vec{n}$. Therefore any covariant POVM has the form

$\begin{split}M(d\hat{\theta})&=\frac{d\hat{\theta}}{2\pi}\left[I+\vec{d}\cdot\vec{\sigma}\cos(\hat{\theta})-\sin(\hat{\theta})(\vec{d}\times\hat{n})\cdot\vec{\sigma}\right]\,.\end{split}$
(51)

When a qubit is in the state $\rho_{\theta}$, the outcomes for the POVM
$M(d\hat{\theta})$ follows the probability density

$\begin{split}p(d\hat{\theta}|\theta)&=\frac{d\hat{\theta}}{2\pi}\left[1+\left(\vec{a}\cdot\vec{d}\right)\cos(\hat{\theta}-\theta)-\vec{a}\cdot(\vec{d}\times\vec{n})\sin(\hat{\theta}-\theta)\right]\,.\end{split}$
(52)

From this result, one can calculate the Fisher information associated with the
covariant POVM $M$. In particular, the case
$\vec{d}=\pm(\vec{n}\times\vec{a})$ is of interest, since the classical Fisher
information reads

$F\left(\theta;\left.M\right\rvert_{\vec{d}=\pm(\vec{n}\times\vec{a})}\right)=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{F_{Q}^{2}\cos^{2}(\hat{\theta}-\theta)}{1\pm
F_{Q}\sin(\hat{\theta}-\theta)}d\hat{\theta}\,.$ (53)

As far as we are aware of, equations (51)-(53) are new results. Note that, in
general, $F(\theta,M_{*})\geq
F\left(\theta;\left.M\right\rvert_{\vec{d}=\pm(\vec{n}\times\vec{a})}\right)$,
but for the particular case $\vec{a}\cdot\vec{n}=0$, the POVM
$\left.M\right\rvert_{\vec{d}=\pm(\vec{n}\times\vec{a})}$ is able to reach the
maximum of the Fisher information, $F^{\text{max}}_{Q}=1$. Hence, the POVM
given by Eq. (51) is equivalent to the measurement $M_{*}$.

Let us briefly summarise the results we have found thus far. Firstly, we have
shown that estimation strategies based on locally optimal POVM with two
outcomes cannot achieve the QCRB since the corresponding likelihood is non-
identifiable. POVMs with more than two outcomes may be identifiable but we
could not find the POVMs that are locally optimal (globally optimal POVMs do
not exist [1]). Secondly, we have seen that for continuous and periodic
outcomes the covariant POVM solves the identifiability problem and minimizes
the MSE for one measurement, however, it can only achieve the QCRB (in the
limit of a large number of measurements) with optimal initial conditions
$\vec{a}\cdot\vec{n}=0$. Hence, to the best of our knowledge, it does not
exist a set of independent and identical measurements that achieve the QCRB.

It turns out that, under the optimal set of initial conditions, the covariant
POVM $M_{*}$ has been previously investigated and it is usually called
canonical phase measurement [17, 3, 22] which, according to Eq. (40), takes
the following form:

$M_{*}^{can}(d\hat{\theta}):=\frac{d\hat{\theta}}{2\pi}\sum_{m,n\in
Spec(J)}e^{i\hat{\theta}\left(n-m\right)}\left|m\right\rangle\left\langle
n\right|\,.$ (54)

Although the optimal covariant POVM is hard to realize experimentally, it can
be well approximated by POVMs with large number of elements or adaptive
measurements [7, 30, 22]. In particular, the canonical phase measurement can
be implemented using adaptive measurements with quantum feedback [22], and the
POVM $M_{*}$ can be well approximated using a POVM with $k\geq 8$ that
discretizes Eq. (49).

###  3.3 Entangled measurement

It is actually possible to achieve the QCRB for a probe not orthogonal to the
rotation axis if we relax the condition of identical and independent set of
measures. Let us indeed see it is possible to attain the QCRB for entangled
measurements for any initial condition. Following [17], we estimate
$\theta\in[0,2\pi)$ for the family of states $\rho^{\otimes N}_{\theta}$ in
the Hilbert space $\mathcal{H}^{\otimes N}$ with $\rho_{\theta}=U_{\theta}\rho
U_{\theta}^{\dagger}$ and $\rho$ being a probe pure qubit. Note that
$\rho^{\otimes N}_{\theta}$ can be written as $\rho^{\otimes
N}_{\theta}=e^{-i\theta J^{(N)}}\rho^{\otimes N}e^{i\theta J^{(N)}}$, where

$J^{(N)}=\left(\hat{n}\cdot\frac{\vec{\sigma}}{2}\right)\otimes\cdots\otimes
I+\cdots+I\otimes\cdots\otimes\left(\hat{n}\cdot\frac{\vec{\sigma}}{2}\right)\,,$
(55)

that is, the family of states $\rho^{\otimes N}_{\theta}$ in
$\mathcal{H}^{\otimes N}$ is covariant with respect to the unitary
representation $e^{-i\theta J^{(N)}}$. Therefore, as in the case of one probe,
the POVM that minimizes the Holevo variance is expressed in terms of the
spectrum of $J^{(N)}$. Therefore, the measurement $M_{*}^{(N)}$ that minimizes
the Holevo variance has the expression

$\begin{split}&M_{*}^{(N)}(d\hat{\theta})=\frac{d\hat{\theta}}{2\pi}\left[\sum_{\begin{subarray}{c}\lambda,\lambda^{\prime}\in\\\
Spec\left(J^{(N)}\right)\end{subarray}}e^{i\hat{\theta}(\lambda^{\prime}-\lambda)}\frac{P_{\lambda}\rho^{\otimes
N}P_{\lambda^{\prime}}}{\sqrt{Tr\left[P_{\lambda}\rho^{\otimes
N}\right]Tr\left[P_{\lambda^{\prime}}\rho^{\otimes
N}\right]}}\right],\end{split}$ (56)

where, $P_{\lambda}$ is the projection operator of $J^{(N)}$ associated to the
eigenvalue $\lambda$.
The difference here, in contrast to the single measurement case, is that here
the spectrum of $J^{(N)}$ is degenerate. Thus

$P_{*}=\int_{\Theta}M(d\hat{\theta})\neq I,$

and therefore $M_{*}^{(n)}$ is not a resolution of identity. However, this is
not a real problem because it can be extended to a POVM by adding $I-P_{*}$ in
the orthogonal complement to the subspace
$\mathcal{H}_{*}:=P_{*}(\mathcal{H}^{\otimes N})$. So, the optimal POVM is

$M_{*}^{(N)}:=P_{*}\oplus I-P_{*}.$

In particular, if $N=1$, $M_{*}^{(N)}=M_{*}$.
As stated in [17], $M_{*}^{(N)}$ can achieve the QCRB for any probe state in
the asymptotic limit. Besides, the scaling of the error is proportional to the
square in the number of probes (Heisenberg scaling). Unfortunately,
$M_{*}^{(N)}$ is an entangled measurement. Hence, if one wants to construct an
entangled POVM, it is necessary to have $N$ copies of the probe and perform
quantum operations over all of them, making the implementation of this type of
POVMs an experimental challenge.

###  3.4 Adaptive state quantum estimation (AQSE)

As we mentioned in the introduction, unlike its classical counterpart, quantum
parameter estimation yields estimators which depend on the parameter one wants
to estimate. A way to tackle this problem is using an adaptive quantum
estimations scheme [13, 25, 35]. It works as follows: suppose one has a set of
optimal POVMs $\left\\{M_{g}\right\\}_{g\in\Theta}$. One begins with an
arbitrary initial guess $g_{0}$. Then one applies the optimal measurement at
$g_{0}$, $M_{g_{0}}$. Assuming the data $x_{1}$ is observed, one applies the
MLE to the likelihood function $L_{1}(\theta\mid
x_{1};g_{0})=p(x_{1}\mid\theta;g_{0})$ to obtain an estimate
$\hat{\theta}_{1}(x_{1})=g_{1}$. This process is then repeated iterative. For
$n\geq 2$, one applies the POVM $M_{g_{n-1}}$, with
$g_{n-1}=\hat{\theta}_{n-1}(x_{1},...,x_{n-1}))$, obtaining the outcome
$x_{n}$. Here $\hat{\theta}_{n-1}$ is the estimation from the previous step
using the outcomes $x_{1},...,x_{n-1}$. The likelihood function for the $n$th
step is

$L_{n}(\theta\mid
x_{1},...,x_{n-1};g_{n-1})=\prod_{i=1}^{n}p(x_{i}\mid\theta;g_{i-1})\,,$ (57)

where $x_{i}$ is the data observed at step $i$. Applying the MLE one obtains
the $n$th guess $g_{n}=\hat{\theta}_{n}\left(x_{1},...,x_{n}\right)$. Even
though this method relaxes the condition of identical measurements, this does
not guarantee that the resulting classical statistical model is regular thus
impacting its performance [8].
To illustrate this let us revisit the problem of non-identifiability for the
POVM $M_{g}$ previously discussed. To fix ideas we take $\theta=2$ and
consider $N=64$ measurements. In panel A of Fig 2 we show the result of the
likelihood function for $g=1.5$ using Eq. (29). The likelihood is indeed non-
identifiable and presents two global maxima in the interval $[0,2\pi)$. Panel
B in the same figure shows the results of using AQSE starting with the initial
value $g_{0}=1.5$. In this case we can see that there is only one global
maxima but at the incorrect value of $\theta$. This is the result of having an
original non-identifiable model, whose wrong maximum has been enhanced due the
particular random trajectory in the parameter space generated by AQSE. Thus,
even though AQSE is able to lift a possible degeneracy of the global maxima
there is no a priori way to control on how to enhance the correct maximum.

Figure 2: Plot of the likelihood functions for the two-outcome POVM $M_{g}$
given by Eq. (25) and considering $\theta=2$ and $N=64$ probes. Panel (A)
shows the likelihood function given by formula (28). Here we have considered
$g=1.5$ and the number of ones was 32 out of the possible $N=64$. For clarity
the figure has been rescaled so that the maxima equal to unity. Panel (B)
shows the log-likelihood function for the AQSE scheme given by Eq. (57) using
$64$ adaptive steps. The existence of trajectories in the parameter space
enhancing the incorrect maximum explains why the AQSE does not reach the QCRB.
In both cases, the initial condition has been taken to be
$\vec{a}\cdot\vec{n}=0.5$.

With a modest amount of foresight, it seems fairly natural to expect that if
we knew in which interval the parameter lies and restricted the likelihood
over that interval -so as to ensure that the restricted likelihood is
identifiable- then the adaptive scheme will converge to the QCRB fast. To
analyse this we have compared the behaviour of AQSE in two different
scenarios: unrestricted and restricted likelihoods. In both cases we have
taken $\theta=\pi$. For the unrestricted scheme the parameter space is
$\Theta=[0,2\pi)$, while for the restricted one we take
$\theta\in\Theta=[\frac{\pi}{2},\frac{3\pi}{2})$, so that the likelihood is
identifiable. In Fig. 3 we show the result of these two cases for two extremal
initial conditions $\vec{a}\cdot\vec{n}=0.5$ (panel A) and
$\vec{a}\cdot\vec{n}=0$ (panel B), corresponding to the lowest and largest
value of $F_{Q}$, respectively. In both cases the numerical method was done
using a bootstrap simulations with $10000$ repetitions.
From these results we can observe that in the unrestricted case the lack of
identifiability directly affects the scaling of the Holevo variance as a
function of the number of adaptive steps. Furthermore, numerical results show
that AQSE does not saturate the QCRB. On the other hand, in the restricted
case, we find that the MLE saturates the QCRB around $128$ measurements for
both initial conditions. Thus we conclude that a possible deficient
performance of AQSE is due to the non-identifiability for $\theta$ in the
likelihood functions.

Figure 3: Result of Holevo’s variance as a function of the number of probes
for different estimation strategies and for two different initial conditions:
$\vec{a}\cdot\vec{n}=0.5$, corresponding to the smallest quantum Fisher
information (shown in panel A) and $\vec{a}\cdot\vec{n}=0$ (shown in panel B).
In all cases we used $10000$ sequences for each number of probes. As we can
see the restricted AQSE performs much better than the AQSE. We have also
included the strategy based on the optimal covariant POVM $M_{*}$ as a
benchmark. Notice that for both initial conditions the AQSE does not reach the
QCRB, while for optimal initial condition the sequence of covariant POVMs is
able to attain the QCRB (see panel B).

Finally, in Fig. 3, we have also compared the performance of AQSE with the
optimal covariant inference. In this case, the sequence of $M_{*}$
measurements has a better scaling for a small number of measurements (less
than $4$). Then, we can also conclude that the optimal covariant measurement
is the best independent strategy in a regime of small $N$. This is expected
because the covariant measurement is built to minimize the MSE for one
measurement.
In the next section, we propose an estimation scheme that uses the covariant
estimation and the AQSE method to avoid the problem of non-identifiable
likelihood functions and thus is able to saturate the QCRB for any initial
condition.

##  4 Adaptive estimation scheme with confidence intervals

From the previous discussion it seems clear that, conditioned of knowing in
which interval the actual phase lies in and restricting the likelihood to that
interval so it becomes identifiable, the AQSE method converges efficiently to
the QCRB. With this in mind, we propose the following two-step scheme which
allows us to initially guess the interval, thus making the iterative scheme a
regular problem.
Our scheme consists of two main steps. In the first step one uses a sequence
of independent optimal covariant measurements to produce a confidence interval
(CI) using the maximum likelihood estimate $\hat{\theta}_{\text{MLE}}$. The CI
gives a range of plausible values where the unknown parameter most likely lies
in. In the second step, one applies the AQSE method restricted to the
confidence interval. Furthermore, one is able to update the center of the CI
by the MLE of each previous step. The idea of using CIs to improve the error
in estimations is exemplified, in a different context, in [4].

Let us assume that we have $N$ copies of the state system $\rho$, so that each
measurement is performed consecutively over each copy. Let us denote the
result of each measurement as $x_{i}$. In the first part of our scheme we
apply $N_{1}$ covariant measurements $\mathcal{M}_{1}$ given by Eq. (40), that
is

$\mathcal{M}_{1}:=M_{*}(dx_{k})\,,$ (58)

and then we construct the MLE

$\hat{\theta}_{\text{MLE}}=\text{arg
max}_{\theta\in\Theta}\prod_{i=1}^{N_{1}}p\left(dx_{i}\mid\theta\right)=\text{arg
max}_{\theta\in\Theta}\prod_{i=1}^{N_{1}}\frac{dx_{i}}{2\pi}\left(1+\sqrt{F_{Q}}\cos(x_{i}-\theta)\right)$
(59)

to obtain the estimation
$\hat{\theta}_{\text{MLE}}\left(x_{1},...,x_{N_{1}}\right)$. The CI is then
given by [9]:

${\rm
CI}(\hat{\theta}_{\text{MLE}}(\vec{x}))=\left(\hat{\theta}_{\text{MLE}}(\vec{x})-c\cdot
F^{-\frac{1}{2}},\,\hat{\theta}_{\text{MLE}}(\vec{x})+c\cdot
F^{-\frac{1}{2}}\right)\,,$ (60)

where $c$ is the appropriate critical value in the standard normal
distribution (e.g. $1.96$ for $95\%$ of confidence or $2.58$ for $99\%$ of
confidence) and $F$ is the Fisher information of $M_{*}$ given by Eq. (50). To
determine the minimum sample size $N_{1}$ in order to get a confidence
interval of size $2E$, we use the CRB to write the lower bound

$N_{1}\geq\frac{c^{2}}{FE^{2}}\,.$ (61)

In the second step of our scheme we then use the remainder of the copies, that
is $N_{2}=N-N_{1}$, to perform measurements using the POVM

$\mathcal{M}_{2}:=M_{\hat{\theta}_{\text{MLE}}\left(x_{1},...,x_{k}\right)}\,.$
(62)

All in all, our two-step method is given by:

$\hat{\theta}_{\text{MLE}}=\text{arg
max}_{\theta\in\Theta}\prod_{i=1}^{N_{1}}p(dx_{i}\mid\theta)\prod_{k=N_{1}}^{N_{2}}p(x_{k+1}\mid\theta;\hat{\theta}_{\text{MLE}}(x_{1},...,x_{k}))\,,$
(63)

with
$\Theta=\mathrm{CI}\left(\hat{\theta}_{\text{MLE}}\left(x_{1},...,x_{k}\right)\right)$.
Eq. (63) is the main result of this work.
Before discussing some numerical results of our newly introduced method it is
important to discuss the possible sources of errors associated to it. Suppose
that we set a confidence level $0\leq C_{l}\leq 1$ for a given marginal error
$E$. Then we can write that

$V^{H}\left(\hat{\theta}_{\text{MLE}}\right)=C_{l}\Delta_{1}+\left(1-C_{l}\right)\Delta_{2}\,,$
(64)

where the two types of errors $\Delta_{1}$ and $\Delta_{2}$ correspond to the
error associated to the AQSE method when the CI correctly includes the value
of the actual parameters, and when it does not, respectively. Let us call the
latter intervals bad CIs. Further, let us take $E<\frac{\pi}{2}$, so that
likelihood functions used in the AQSE method are identifiable. Thus, the error
$\Delta_{1}$ when the CI includes the value of the parameter is lowered
bounded by

$\Delta_{1}\geq\frac{1}{F_{Q}N_{2}+N_{1}F(\theta;M_{*})}\,.$ (65)

To characterize $\Delta_{2}$, first we note that there are two types of bad
CIs. To see this, let $\theta_{A}$ be the parameter’s value and
$\theta_{B}=\theta_{A}+\pi$. Then, the likelihood function produced by AQSE
has two local maxima, one around $\theta_{A}$, and the other around
$\theta_{B}$. The first type of bad CIs corresponds to those that include the
second maximum around $\theta_{B}$. If one applies AQSE restricted to these
CIs then $\theta_{\text{MLE}}$ tends to $\theta_{B}$. The second type of bad
CIs appear when the interval does not include $\theta_{B}$. In this case, the
maximum likelihood estimation produced by AQSE can tend to the point in the
interval closest to either $\theta_{A}$ or to $\theta_{B}$. When the
estimation is closest to $\theta_{A}$, by updating the interval’s center with
the posterior estimates, one can, in principle, obtain a confidence interval
that captures the parameter’s value. This explain why we obtain better results
updating the center of the confidence interval.

To obtain a bad interval of type one, it is necessary that most of the data
obtained from the covariant inference be numbers close to $\theta_{B}$ (the
likelihood function has a global maximum around $\theta_{B}$). From Eq. (49),
for a covariant sample of size $N$, it follows that the probability of
obtaining a likelihood function with the global maximum around $\theta_{B}$ is

$\left[\int_{\theta_{B}-\epsilon}^{\theta_{B}+\epsilon}p\left(d\hat{\theta}\mid\theta\right)\right]^{N}=\left[\int_{\theta_{B}-\epsilon}^{\theta_{B}+\epsilon}\frac{d\hat{\theta}}{2\pi}\left(1+\sqrt{F_{Q}}\cos(\hat{\theta}-\theta)\right)\right]^{N}\,.$
(66)

Thus, for sufficiently large $N$, the above probability can be neglected. In
this way, for a confidence level close to $1$, and a sufficient small marginal
error $E$, one expects that the number of bad confidence intervals disappear
as the number of adaptive steps increases. To illustrate this fact, in Table
1, we show the number of bad intervals as a function of the adaptive steps in
the AQSE part produced by a bootstrap simulation of $10000$ repetitions. Here
we have chosen a marginal error of $\frac{\pi}{4}$, with a confidence level of
$0.99$. In this case, we need to perform $11$ and $22$ covariant measurements
for the initial conditions $\vec{a}\cdot\vec{n}=0$ and
$\vec{a}\cdot\vec{n}=0.5$, respectively, to get the desired CI. For these
values, the number of type one bad confidence intervals is negligible. Setting
$\epsilon=E$ in (66), the probabilities of obtaining them are $2.3\times
10^{-10}$ and $2.3\times 10^{-18}$ for these two initial conditions,
respectively.

AQSE Steps | # bad CIs ($\vec{a}\cdot\vec{n}=0$) | # bad CIs ($\vec{a}\cdot\vec{n}=0.5$)
---|---|---
0 | 111 | 125
4 | 50 | 56
8 | 19 | 20
16 | 8 | 1
32 | 6 | 1
48 | 2 | 0
Table 1: Number of bad confidence intervals as a function of adaptive steps
for the AQSE part where the real value of $\theta$ was $\pi$.

Note that, the distribution for the estimates can be approximated as a mixture
of $3$ normal distributions,
$\mathcal{N}(\theta,\frac{1}{F_{Q}N_{2}+N_{1}F(\theta;M_{*})})$,
$\mathcal{N}(\theta+E,\frac{1}{F(\theta,M_{*},M)})$, and
$\mathcal{N}(\theta-E,\frac{1}{F(\theta,M_{*},M)})$, where $M$ is the sequence
of AQSE and $E$ is the marginal error. This implies that the lower bound for
the Holevo variance, when the center of the CI is not updated, is given by

$\begin{split}V^{H}\left(\hat{\theta}_{\text{MLE}}\right)&\geq\frac{1}{F_{Q}N_{2}+N_{1}F(\theta;M_{*})}+(1-C_{l})E^{2}\,.\end{split}$
(67)

The second term on the right-hand side of Eq. (67) dominates when
$N_{2}\to\infty$. This has a very simple interpretation: if the parameter is
outside the confidence interval, making more steps in the AQSE method does not
diminish the error. Unfortunately, we are not able to provide a lower bound to
the Holevo variance in the case for which the center of the CI is also
updated. Nevertheless, as we will see, by setting a confidence level close to
$1$, a sufficiently small marginal error $E$, and by updating the center of
the CIs, the Holevo variance approximates the QCRB in a small number of steps.
To assess the performance of our method we have performed a Monte Carlo
simulations by setting $E=\frac{\pi}{4}$ and $C_{l}=.99$. Moreover, we have
compared our numerical results with the ideal case ($C_{l}=1$) and an
estimation strategy for which the center of the CIs is not updated. The
results are summarised in Figures 4 and 5 for initial conditions
$\vec{a}\cdot\vec{n}=0$ and $\vec{a}\cdot\vec{n}=0.5$.
In Fig. 4 we have considered the initial condition $\vec{a}\cdot\vec{n}=0$, so
that $F(\theta;M_{*})=1$, and fixed $N_{1}=11$. The vertical dashed line at
$N_{1}=11$ separates the point between covariant and AQSE inferences. Notice
that in the covariant part, for $N\leq 11$, both strategies corresponding to
either updating or not the CIs centers yields the same result. However, for
$N>11$, if the center of the CI is not updated, so that according to Eq. (67)
it second term dominates, the estimation error grows quickly as the number of
measurements increases. If, however, the center of the CIs is updated then the
estimation error tends to the QCRB. Similarly in Fig. 5, for which we have
chosen $N_{1}=22$, $\vec{a}\cdot\vec{n}=0.5$ and $F(\theta;M_{*})=0.75$, we
can observe the same qualitative behaviour, showing that our estimation
strategy tends to the QCRB as the number of steps increases for any set of
initial conditions.

Figure 4: Plot showing Holevo’s variance as a number of probes for our scheme.
The optimal strategy (orange rhomboid markers), in which we update the centers
of the CIs, tends to the QCRB as the number of probes increases. The latter
bound corresponds to take a confidence level of $100\%$. We also show the
alternative strategy for which the centers of the CIs are not updated (yellow
pentagon markers). This result is lowered bounded by the solid black line,
whose formula is given by Eq. (67). We have chosen the initial condition
$\vec{a}\cdot\vec{n}=0$. Here the vertical blue line at $N=11$ marks the point
separating the covariant inference performance from the AQSE inference to our
two-step scheme. Figure 5: Plot showing Holevo’s variance as a function of the
number of probes $N$ for our two-step adaptive scheme. The optimal strategy
(orange rhomboid markers), in which we update the centers of the CIs, tends to
the QCRB as the number of probes increases. This result is lowered bounded by
the dashed black line, which is the QCRB corrected using Eq. (67) for a
confidence level of $100\%$. We also show the alternative strategy for which
the centers of the CIs are not updated (yellow pentagon markers). This result
is lowered bounded by the solid black line, whose formula is given by Eq.
(67). In this case, we have chosen the initial condition
$\vec{a}\cdot\vec{n}=0.5$. Here the vertical blue line at $N=22$ marks the
point separating the covariant inference performance from the AQSE inference
to our two-step scheme. Finally, the horizontal dashed red line corresponds to
the Cramér-Rao bound when only covariant measurements are used instead.

We conclude by showing in Fig. 6 the performance of all the methods we have
discussed throughout this work considering the initial conditions
$\vec{a}\cdot\vec{n}=0.5$ and $\vec{a}\cdot\vec{n}=0$, which correspond to the
smallest and largest quantum Fisher information, respectively. Obviously the
optimal performance for the different estimation strategies is independent of
$\theta$, since the Holevo variance is invariant under translation modulo
$2\pi$ over the parametric space. For all the strategies, the behavior is
qualitatively the same for any initial condition. As expected, the entangled
strategy is the best one and reaches the QCRB. The second best strategy is the
restricted ASQE, but this is unrealistic as it assumes that one knows
beforehand an interval where the MLE is regular and includes the parameter.
Our method, which we emphasize is the more realistic both mathematically and
experimentally, is the third best strategy, as it reduces the error of the
estimates improving the performance of AQSE and approximating the estimate to
the QCRB.

Figure 6: Holevo variance vs the number of probes for different strategies of
estimation. In order to calculate the Holevo variance for the schemes based on
the maximum likelihood estimator. A sequence of $10000$ measurements were
simulated in each point. The curve for the entangled measurement was
analytically calculated. The curve for the proposed scheme shows the
performance with a $99\%$-confidence intervals. The restricted AQSE method
assume that the parametric space is an interval of length $\pi$ that includes
the real value of $\theta$. The best strategy is the entangled measurement,
followed by the restricted AQSE. These two strategies are unrealistic to
implement. The third best strategy is our proposal. The y-axis is in log
scale.

##  5 Summary and Conclusions

In this work we have thoroughly examined several strategies for quantum phase
estimation. Their common denominator relies on maximizing likelihood density
functions which are generally non-identifiable and/or non-optimal. This
implies that the QCRB cannot be attained. We have developed a two-step
strategy that circumvents this problem. Our method relies first on covariant
measurements to identify a confidence interval within which the actual
parameter is most likely to be, and then to apply an adaptive technique
restricted to that interval. When compared with the current existing methods,
ours is mathematically more robust, as it reaches the QCRB for any initial
condition, and experimentally is more realistic, since neither an entangled
measurement nor a priori information of the parameter’s value are needed.
Finally, we believe that our scheme can be generalized to any system so long
as one can use a set of measurements to construct confidence intervals.

However, based on the results presented here and the current body of work, a
number of questions remain open. First of all, it is not clear to us whether
there exists a better strategy for which the subset of covariant measurements
is intertwined with the subset of adaptive ones so as to identify the CI much
faster. Secondly, the generalization to this work to multiparameter estimation
does not seem straightforward, since although it is indeed possible to
construct the covariant POVMs [16] for this case, little is known about non-
identifiability for multiparameter likelihood functions. These, and other
issues, will be addressed in forthcoming works.

##  6 Acknowledgments

We thank Laboratorio Universitario de Cómputo de Alto Rendimiento (LUCAR) of
IIMAS-UNAM for their service on information processing. This research was
supported by the Grant No. UNAM-DGAPA-PAPIIT IG100518 and IG101421, and
Doctoral scholarship CONACYT 334231. I.P.C. acknowledges funding support from
the London Mathematical Laboratory where he is an External Fellow.

##  Appendix A Code implementation

Our numerical results have been implemented using R language. An R library
with the various methods discussed here can be publicly found in the
repository [32]. To reproduce the numerical points for covariant strategy,
AQSE and restricted-AQSE presented in Fig. 6, use the function:

    Hvar_Scheme(theta_real, par_space, n_boost, num_prob, n,a, strategy),

set the value of $\theta$, `theta_real` from $0$ to $2\pi$, set `par_space` as
the vector `c(0, 2*pi)`, set the size of bootstrap `n_boost=1000`, vary the
desired number of probes `num_prob`, set the initial unitary vectors for the
probe and the axis of rotation `n` `a`, and vary the index `strategy` from $1$
to $3$. The index $1$ represents the estimation strategy for independent
covariant POVMs, $2$ represents the AQSE strategy, $3$ represents the
restricted-AQSE strategy. To reproduce the points for our proposed estimation
scheme use the function

    ECI_Hvar(theta_real, par_space, n_boost, num_prob, n,a, C_lev, Margin_Err),

where you have to set the confidence level `C_lev` and marginal error
`Margin_Err` to the desired values. Finally, to calculate the curve for the
entangled estimation strategy use the function

    Ent_Hvar(theta_real, par_space, n,a, num_prob).

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