# Determination of the asymptotic limits of adaptive photon counting
measurements for coherent-state optical phase estimation

M. A. Rodríguez-García  Instituto de Investigaciones en Matemáticas Aplicadas
y en Sistemas, Universidad Nacional Autónoma de México, Ciudad Universitaria,
Ciudad de. México 04510, Mexico  M. T. DiMario  Center for Quantum Information
and Control, Department of Physics and Astronomy, University of New Mexico,
Albuquerque, New Mexico 87131  Joint Quantum Institute, National Institute of
Standards and Technology and the University of Maryland, College Park,
Maryland 20742  P. Barberis-Blostein  Instituto de Investigaciones en
Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México,
Ciudad Universitaria, Ciudad de. México 04510, Mexico  F. E. Becerra
<EMAIL_ADDRESS>Center for Quantum Information and Control, Department of
Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131

###### Abstract

Physical realizations of the canonical phase measurement for the optical phase
are unknown. Single-shot phase estimation, which aims to determine the phase
of an optical field in a single shot, is critical in quantum information
processing and metrology. Here we present a family of strategies for single-
shot phase estimation of coherent states based on adaptive non-Gaussian,
photon counting, measurements with coherent displacements that maximize
information gain as the measurement progresses, which have higher
sensitivities over the best known adaptive Gaussian strategies. To gain
understanding about their fundamental characteristics and demonstrate their
superior performance, we develop a comprehensive statistical analysis based on
Bayesian optimal design of experiments, which provides a natural description
of these non-Gaussian strategies. This mathematical framework, together with
numerical analysis and Monte Carlo methods, allows us to determine the
asymptotic limits in sensitivity of strategies based on photon counting
designed to maximize information gain, which up to now had been a challenging
problem. Moreover, we show that these non-Gaussian phase estimation strategies
have the same functional form as the canonical phase measurement in the
asymptotic limit differing only by a scaling factor, thus providing the
highest sensitivity among physically-realizable measurements for single-shot
phase estimation of coherent states known to date. This work shines light into
the potential of optimized non-Gaussian measurements based on photon counting
for optical quantum metrology and phase estimation.

##  I Introduction

Optical phase estimation is ubiquitous in many fundamental and practical
problems ranging from quantum state preparation [1, 2, 3], sensing [4],
communications [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], and information processing
[15]. In a photonic metrology problem [16, 17, 18, 19], an optical probe state
interacts with a physical system to interrogate its properties. This
interaction maps parameters of the system to the state of the optical probe,
where an optimal readout can be performed [17, 18, 15, 19]. When the physical
property of the system is mapped onto the phase of the optical probe, the
optimal quantum measurement is the canonical phase measurement [20], which
consists of projections onto phase eigenstates [21]. However, while
theoretically the canonical phase measurement exists, the physical realization
of projections onto phase eigenstates are not physically known [22]. Thus in
practical estimation problems in quantum metrology one seeks to determine the
limits in precision of physically realizable measurements, and the degree with
which they approach to the fundamental quantum limit in sensitivity [23, 24,
25, 26, 27, 17].

Physically realizable measurements of the optical phase have been widely
investigated [20, 28, 21] for diverse metrological problems with quantum and
classical fields [29, 30, 31, 32, 33] including sensing small deviations from
a known phase [29, 30, 31, 34, 32, 33] and estimation with repeated sampling
[35, 36] and measurements [30, 37, 38, 39, 40, 41]. Beyond these specific
estimation problems, measurements of the phase of a single optical mode in a
single-shot are central for quantum state preparation and detection [42, 43],
waveform estimation and sensing [44, 45, 46, 47], and quantum information
processing [48, 49, 50]. The standard measurement for optical phase estimation
is the heterodyne measurement [21], which samples both quadratures of the
electromagnetic field simultaneously from which the phase can be estimated.
However, the achievable sensitivity of the heterodyne measurement [21] is far
below the ultimate measurement sensitivity allowed by physics, given by the
canonical phase measurement [51, 21]. Adaptive measurement techniques based on
homodyne detection, a Gaussian measurement, can be used to align the phase
quadrature of the optical field with the measurement setting where the
homodyne measurement provides maximum sensitivity [32]. Adaptive homodyne has
been theoretically shown to surpass the heterodyne limit and get closer to the
canonical phase measurement for optical phase estimation of coherent states
[20, 21], providing the most sensitive Gaussian measurement of the optical
phase so far [21].

In a complementary measurement paradigm, quantum measurements of coherent
states based on photon counting, displacement operations, and feedback [8, 2,
9, 10, 11, 12, 13, 6, 52, 14] have enabled state discrimination below the
Gaussian sensitivity limits and approaching the Helstrom bound [5]. Recently,
some of the authors proposed and demonstrated a non-Gaussian measurement
strategy for single-shot phase estimation of coherent states, able to surpass
the heterodyne limit and approach the sensitivity limit of a canonical phase
measurement in the presence of loss and noise of real systems [2]. These
measurement strategies are based on realizing coherent displacements of the
input field and monitoring the output field with photon number resolving
detection. The information from the detection outcomes is then used to
implement real time feedback of displacement operations optimized to maximize
the measurement sensitivity of the phase of the input state. This estimation
strategy is the most sensitive single-shot non-Gaussian measurement of a
completely unknown phase encoded in optical coherent states so far [2]. In
this strategy the displacement operation optimization is realized by
maximizing either the information gain in subsequent adaptive steps of the
measurement or the sharpness of the posterior phase distribution. While these
cost functions are functionally different, both perform similarly and get
close to the ultimate sensitivity allowed by physics, the Crámer-Rao lower
bound (CRLB), in the limit of many photons and many adaptive measurements.
While the work in Ref. [2] demonstrated the potential of non-Gaussian
measurements for single-shot phase estimation, the superiority over adaptive
homodyne detection was not proven. A deeper understanding of the properties of
convergence and ultimate limits of the estimators produced by non-Gaussian
measurements is still missing. This is an open problem due to the complexity
associated with the analysis of these adaptive non-Gaussian strategies.

In this article we investigate a family of adaptive non-Gaussian strategies
based on photon counting for single-shot optical phase estimation, and assess
their performance compared to Gaussian measurements and the canonical phase
measurement. To analyze these non-Gaussian strategies, we use the mathematical
framework of Bayesian optimal design of experiments, which provides a natural
description of non-Gaussian adaptive strategies, allowing us to investigate
their fundamental characteristics and determine the limits in sensitivity,
which up to now has been a challenging problem. Our work provides a
comprehensive statistical analysis of adaptive non-Gaussian measurements for
parameter estimation, and the requirements to approach optimal bounds in the
asymptotic limit. We show that strategies based on photon counting and
feedback for single-shot phase estimation of coherent states provide superior
sensitivity over the best known adaptive Gaussian strategies, having the same
functional form as the canonical phase measurement in the asymptotic limit,
differing only by a scaling factor. This work provides a deep insight into the
potential of optimized non-Gaussian measurements for quantum communication,
metrology, sensing, and information processing.

##  II  Results

###  II.1 Holevo Variance of Non-Gaussian Estimation Strategies

The non-Gaussian phase estimation strategies investigated here are based on
photon counting, displacement operations, and feedback, and are optimized by
maximizing a specific cost function. These strategies maximize either the
estimation precision (by minimizing the Holevo variance [51]), or the
information gain about the unknown parameter based on entropy measures
including mutual information, the Kullback-Lieber divergence, and the
conditional entropy [53, 54]. We note that these cost functions produce non-
identifiable likelihood functions that do not allow to correctly estimate a
cyclic parameter, such as the phase [55]. To address this problem, these non-
Gaussian strategies use the Fisher information to optimize the displacement
operations, which are the dynamical control variable in the strategy, to
guarantee that these cost functions provide identifiable likelihood functions,
and to enable optical phase estimation with near optimal performance.

In the problem of single-shot phase estimation with coherent states, an
electromagnetic field in a coherent state
$\rho_{0}=\left|\alpha\right\rangle\left\langle\alpha\right|$ interacts with a
physical system and experiences a unitary transformation
$e^{\mathrm{i}\phi\hat{n}}$, where $\hat{n}$ is the number operator. The phase
$\phi$ induced in the coherent state carries information about the system,
which can be extracted by a measurement of the output state
$\rho(\phi)=e^{\mathrm{i}\phi\hat{n}}\rho_{0}e^{-\mathrm{i}\phi\hat{n}}=\left|e^{\mathrm{i}\phi}\alpha\right\rangle\left\langle
e^{-\mathrm{i}\phi}\alpha\right|\,$.

Measurements onto $\rho(\phi)$, together with an estimator $\widehat{\phi}$ on
the measurement outcomes, provide an estimate of $\phi$, and a measurement
strategy aims to obtain the best estimation of the physical parameter. The
efficiency of such a strategy is characterized by the estimation variance as a
function of the number of photons in the coherent state. The most efficient
strategy provides a variance at the highest convergence rate towards zero as
the number of photons increase.

The standard measurement paradigm for phase estimation of Gaussian states is
the heterodyne measurement (a Gaussian measurement), with an estimator
variance of
$\text{Var}\left[\widehat{\phi}_{\text{Het}}\right]=1/2\lvert\alpha\rvert^{2}$.
Within the paradigm of Gaussian measurements, adaptive homodyne strategies
optimized to minimize the Holevo variance have achieved the best performance
among Gaussian measurements for single-shot phase estimation of coherent
states [20]. The best adaptive Gaussian measurement reported to date, termed
the Adaptive Mark-II (MKII), achieves a Holevo variance in the limit of large
number of photons of:

$\text{Var}\left[\widehat{\phi}_{\text{MKII}}\right]\approx\frac{1}{4\lvert\alpha\rvert^{2}}+\frac{1}{8\lvert\alpha\rvert^{3}}\,.$
(1)

While this optimized Gaussian strategy surpasses the heterodyne limit, it has
an error of order $1/|\alpha|^{3}$ above the Cramér-Rao Lower Bound
($1/4\lvert\alpha\rvert^{2}$), and does not reach the performance of the
canonical phase measurement [21]:

$\text{Var}\left[\widehat{\phi}_{\text{CPM}}\right]\approx\frac{1}{4\lvert\alpha\rvert^{2}}+\frac{5}{32\lvert\alpha\rvert^{4}}\,.$
(2)

In this work, we numerically show that non-Gaussian strategies for single-shot
phase estimation based on photon counting, optimized displacement operations,
and real-time feedback achieve an estimator variance smaller than Gaussian
strategies with an asymptotic scaling in the limit of high mean photon numbers
of:

$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]\approx\frac{0.250\pm
0.001}{\lvert\alpha\rvert^{2}}+\frac{0.520\pm
0.010}{\lvert\alpha\rvert^{4}}\,.$ (3)

Figure 1A summarizes the main result comparing the three asymptotic variances
for the canonical phase measurement (solid blue), MKII (solid red), and non-
Gaussian strategies (solid green and points). Figure 1B, shows the excess
Holevo variance compared to the canonical phase measurement
($\text{Var}[\cdot]-\text{Var}[\hat{\phi}_{\text{CPM}}]$) for Heterodyne
(purple), MKII (red), and non-Gaussian strategies (green).

Fig. 1:  Asymptotic variances for different phase measurements. a: Holevo
variance in the limit of large mean photon (MPN) $\lvert{\alpha}\rvert^{2}$
for the canonical phase measurement in Eq. (2), the most sensitive Gaussian
measurement known to date (MKII) [21] in Eq. (1), and the non-Gaussian
strategies in Eq. (3). The points show numerically values for the non-Gaussian
strategies in a region where the analytical expression is not valid. Error
bars represent a 1-$\sigma$ standard deviation. b: Excess phase variance
($\text{Var}[\cdot]$ \- $\text{Var}[\widehat{\phi}_{\text{CPM}}]$) for
Heterodyne, MKII, and non-Gaussian strategies.

These non-Gaussian strategies implement a series of adaptive steps with
displacement operations optimized to maximize information gain, while ensuring
efficient phase estimators in the asymptotic limit. These strategies surpass
the best known Gaussian strategy in Eq. (1), and have the same functional form
as the canonical phase measurement in the asymptotic limit, differing only by
a scaling factor, thus providing the highest sensitivity among physically-
realizable measurements for single-shot phase estimation of coherent states
known to date.

Achieving this performance with non-Gaussian estimation strategies, however,
requires a deep understanding of the measurement process. To gain this
understanding, we use the mathematical framework of Bayesian optimal
experimental design, which provides a natural description of adaptive non-
Gaussian measurements. This allows us to optimize these strategies for single-
shot phase estimation with a Holevo variance given by Eq. (3).

###  II.2  Bayesian optimal design of experiments

Phase estimation of coherent states based on photon counting with adaptive
coherent displacement operations can be defined in the context of Bayesian
optimal design of experiments. Optimal design of experiments allows for
improving statistical inferences about quantities of interest by appropriately
selecting the values of control variables known as designs [54, 56, 57]. In
this framework, it is assumed that the experimental data $y$ (the measurement
outcomes) can be modeled as an element of the set

$\mathcal{P}_{\Phi}=\left\\{p(y\mid\mathbf{d},\mathbf{\phi}),\,\,\,\mathbf{d}\in\mathcal{D}\,\,\,\mathbf{\phi}\in\Phi\right\\},$
(4)

where $\mathbf{d}$ is a parameter called design chosen from some set
$\mathcal{D}$ called design space, $\bm{\phi}\in\Phi$ is an unknown parameter
to be estimated, and the data $y$ comes from a sample space
$\mathcal{Y}\subseteq\mathbb{R}$. In this paradigm, the experimenter has full
control over the designs $\mathbf{d}$ and the ability to adjust them prior to
making a measurement. This allows for optimizing such measurement for
estimating the unknown parameter $\bm{\phi}$. Bayesian optimal design of
experiments goes beyond standard methods for parameter estimation based on
Bayesian statistical inference [58, 59, 60, 61, 62], by providing the suitable
mathematical framework to ensure optimal designs to find efficient estimators
for a general parameter space [63, 64, 65, 66].

In the Bayesian approach of optimal design [54, 56], the initial lack of
knowledge about $\bm{\phi}$ is modeled as a prior probability distribution
$p(\bm{\phi})$. The measurement aims to reduce the uncertainty of $\bm{\phi}$
as much as possible by the use of Bayes’ theorem over the prior distribution.
In an estimation problem, the optimal choice for the designs
$\mathbf{d}\in\mathcal{D}$ maximize the expected value of a cost function
$U(\bm{d},\bm{\phi},y)$ with respect to the possible outcomes of $y$ and
$\bm{\phi}$:

$\begin{split}\mathbf{d}^{\text{opt}}&=\arg\max_{\mathbf{d}\in\mathcal{D}}\text{E}\left[U(\mathbf{d},\bm{\phi},y)\right]\\\
&=\arg\max_{\mathbf{d}\in\mathcal{D}}\int_{\mathcal{Y}}\int_{\Phi}U(\mathbf{d},\bm{\phi},y)p(\bm{\phi}\mid\mathbf{d},y)d\bm{\phi}p(y\mid\mathbf{d})dy\\\
&=\arg\max_{\mathbf{d}\in\mathcal{D}}\int_{\mathcal{Y}}\int_{\Phi}U(\mathbf{d},\bm{\phi},y)p\left(\bm{\phi},y\mid\mathbf{d}\right)d\bm{\phi}dy\,.\end{split}$
(5)

A standard approach in optimal design of experiments considers choosing
$\mathbf{d}^{\text{opt}}$ at the beginning of the experiment an then sample
data from $p(y\mid\mathbf{d}^{\text{opt}},\bm{\phi})$ for all subsequent
trials. An alternative approach considers dynamically updating
$\mathbf{d}^{\text{opt}}$ on each trial, as more data is collected. The
advantage of this approach is that adaptive estimation strategies are never
less efficient than the non-adaptive ones [67].

The implementation of Bayesian optimal design of experiments requires a cost
function, such as the Kullback-Lieber divergence between the prior and
posterior distributions [68]:

$\begin{split}U(\mathbf{d},y)&=D_{\text{KL}}\left[p(\bm{\phi}\mid\mathbf{d},y)\mid\mid
p(\bm{\phi})\right]\\\
&=\int_{\Phi}p(\bm{\phi}\mid\mathbf{d},y)\log\left[\frac{p(\bm{\phi}\mid\mathbf{d},y)}{p(\bm{\phi})}\right]d\bm{\phi}.\end{split}$
(6)

The Kullback-Lieber divergence provides a distance between probability
distribution $p(\bm{\phi}\mid\mathbf{d},y)$ and $p(\bm{\phi})$ [53]. If
$p(\bm{\phi}\mid\mathbf{d},y)$ is equal to $p(\bm{\phi})$ then
$U(\mathbf{d},y)=0$ and there is not any gain of information about $\bm{\phi}$
by measuring with design $\mathbf{d}$ and outcome $y$.

Another cost function considered in optimal design is the conditional entropy
between the plausible values of $\bm{\phi}$ and $y$

$\begin{split}U(\mathbf{d})&=-H(\bm{\phi}\mid Y)\\\
&=-\sum_{y\in\mathcal{Y}}p(y)\int_{\Phi}p(\bm{\phi}\mid\mathbf{d},y)\log\left[p(\bm{\phi}\mid\mathbf{d},y)\right]d\bm{\phi},\end{split}$
(7)

which is a measure of how much information is needed to describe the outcomes
of the random variable $\bm{\phi}$ given that the value of another random
variable $Y$ is known [53]. However, we note that cost functions Eq. (6) and
Eq. (7) can be related to the mutual information [53]:

$\begin{split}U(\mathbf{d})=I(\bm{\phi}\mid
Y)&=\text{E}_{y}\left[D_{\text{KL}}\left[p(\bm{\phi}\mid\mathbf{d},y)\mid\mid
p(\bm{\phi})\right]\right]\\\ &=H(p(\bm{\phi}))-H(\bm{\phi}\mid
Y).\end{split}$ (8)

As a result, designs $d$ maximizing any of these cost functions in Eq. (6),
Eq. (7), or Eq. (8) are equivalent [54, 69, 56]. Moreover, in the asymptotic
limit, maximizing these cost functions is equivalent to minimizing the
determinant of the covariance matrix (D-optimality design criteria), that is,
in the asymptotic limit, optimizing these cost functions is equivalent to
optimizing any member within the family of D-optimal designs [54, 68].

Our goal is to apply the theory of Bayesian optimal design of experiments to
the problem of phase estimation of coherent states with photon counting and
adaptive coherent displacement operations. The adaptive non-Gaussian
estimation strategy consists of several parts: i) in the first adaptive step,
it uses an specific cost function and the prior information to choose the
design; ii) then, it performs a measurement; iii) based on the measurement
outcome, it uses Bayes’ theorem to update the probability distribution; iv)
and lastly, it uses a recursive approach, where this posterior probability
distribution becomes the prior of the subsequent measurement step i). The
estimation of $\bm{\phi}$ at each adaptive step is obtained from the maximum
posterior estimator (MAP) of the posterior probability distribution. This
approach requires that the MAP converge to the true value of the parameter
when the number of measurements increases.

In the adaptive mathematical framework of optimal experimental design,
Paninski [67] proved that under a set of regular modelling conditions and in
the case when $\phi\in\mathbb{R}$, cost functions based on mutual information
can allow for designs that lead to asymptotically consistent and efficient MAP
estimators with variance

$\sigma^{2}_{\text{INFO}}=\left(\underset{C\in
co\left(F(\phi;\mathbf{d})\right)}{\operatorname{arg}\,\operatorname{max}}\;\left\lvert
C\right\rvert\right)^{-1}.$ (9)

Here $co\left(F(\phi;\mathbf{d})\right)$ denotes the convex closure of the set
of Fisher information functions $F(\phi;\mathbf{d})$. In other words, the
estimations produced by $\widehat{\phi}$ converge to $\phi$ (consistency), and
the distribution of $\widehat{\phi}$ tends to a normal distribution with mean
$\phi$ and variance given by Eq. (9) when the number of adaptive steps tends
to infinity (efficiency).

Formally, the regularity conditions introduced in [67] can be stated as
follows:

  1. 1.

The parameter space $\Phi$ is a compact metric space.

  2. 2.

The log likelihood, $\log\left(p(y\mid\phi,\mathbf{d})\right)$ is uniformly
Lipschitz in $\phi$ with respect to some dominating measure on $\mathcal{Y}$.

  3. 3.

The likelihood function is identifiable for $\phi$, that is, the likelihood
function has a unique global maximum.

  4. 4.

The prior distribution, assigns a positive probability to any neighborhood of
the real value of $\phi$.

  5. 5.

The Fisher information functions $F(\phi;\mathbf{d})$ are well defined for any
$\phi\in\Phi$ and $\mathbf{d}\in\mathcal{D}$.

  6. 6.

The maximum of the convex closure of the set of Fisher information functions
$\left\\{F(\phi,\mathbf{d})\mid\mathbf{d}\in\mathcal{D},\,\phi\in\Phi\right\\}$
must be positive-definite, i.e., $\max_{C\in
co\left(F(\phi;\mathbf{d})\right)}\left\lvert C\right\rvert>0$.

We note that in the case of estimation of a scalar parameter, the maximization
of mutual information is equivalent to the minimization of the mean square
error (MSE) [67]:

$\begin{split}\rm{MSE}(\widehat{\phi})&=\rm{E}\left[\right(\widehat{\phi}-\phi\left){}^{2}\right]\\\
&=\text{Var}\left(\widehat{\phi}\right)+\left(\rm{E}\left[\widehat{\phi}\right]-\phi\right)^{2}.\end{split}$
(10)

This shows that the MSE is a trade-off between the estimator’s variance and
its bias. As a result, since the phase is a scalar quantity, in the asymptotic
limit both the mutual information Eq. (8) and the mean square error Eq. (10)
are appropriate cost functions to find optimal estimation strategies. Even
more, for unbiased estimators (such as the MAP estimator on the asymptotic
limit), the MSE corresponds to the estimator variance, then the maximization
of mutual information is equivalent to the minimization of estimator variance.
In practice, however, an estimation strategy would use a cost function that
can be calculated efficiently and with a high rate of convergence.

Phase estimation in coherent states.
Optimal phase estimation of coherent states of light aims to obtain the best
estimate, from the outcomes of a physical measurement, of an unknown phase
$\phi\in[0,2\pi)$ encoded in a coherent state
$\rho(\phi)=\left|e^{i\phi}\alpha\right\rangle\left\langle
e^{-i\phi}\alpha\right|\,$. The most general description of a physical
measurement is given by a POVM, a positive operator-valued measure. A
measurement $M$ with a discrete set of outcomes $\left\\{m\,|\,m\in
S\subseteq\mathbb{Z}\right\\}$, can be represented as a POVM
$M=\left\\{M(m)\,|\,m\in S\right\\}$, where the operators $M(m)$ are positive
bounded $M(m)>0$ and resolve the identity $\sum_{m}M(m)=I$, ${\forall m\in S}$
[70]. By the Born rule, the probability for $m$ conditioned to $\phi$ is

$\text{ Tr}\left[M(m)\rho(\phi)\right]=p\left(m\mid\phi\right)\,.$ (11)

According to information theory, if an estimator $\widehat{\phi}$ of ${\phi}$
is constructed using a sample from a POVM $M$, the limit in the accuracy of
$\widehat{\phi}$ is given by the Crámer-Rao Bound [71, 72]

$\text{Var}_{\phi}\left[\widehat{\phi}\right]\geq\frac{1}{F_{M}(\phi)}.$ (12)

Here $F_{M}(\phi)$ is the Fisher information of $M$ about $\phi$, which
quantifies how much information about $\phi$ is carried in a sample from $M$:

$F_{M}(\phi)=\text{E}_{\phi}\left[\left(\frac{\partial}{\partial\phi}\left[\log\left(p\left(m\mid\phi\right)\right)\right]\right)^{2}\right].$
(13)

Since the Fisher information in Eq. (13) depends on the POVM $M$, the
maximization of the Fisher information over all POVMs provides the lowest
possible Cramér-Rao bound. This maximum Fisher information over all POVMs is
known as the quantum Fisher information $F_{\text{Q}}$ (QFI), and the lowest
possible bound in the accuracy of $\widehat{\phi}$ is known as the quantum
Cramér-Rao bound (QCRB) [5, 73, 72]. In the case of phase estimation of
coherent states $F_{\text{Q}}=4\lvert\alpha\rvert^{2}$, and the QCRB is:

$\text{Var}_{\phi}\left[\widehat{\phi}\right]\geq\frac{1}{4\lvert\alpha\rvert^{2}}.$
(14)

In the limit of large photon number $\lvert\alpha\rvert\gg 1$, the QCRB is
saturated by the canonical phase measurement (the optimal phase measurement),
which is described by the POVM [51]:

$M(\widehat{\phi})=\frac{1}{2\pi}\sum_{n,m=0}^{\infty}\left|n\right\rangle\left\langle
m\right|e^{\mathrm{i}\left(n-m\right)\widehat{\phi}},$ (15)

where $\left|n\right\rangle$ is an eigenstate of the number operator
$\hat{n}$. The operator $M(\widehat{\phi})$ is an element of the canonical
phase measurement whose outcome is a number
$\widehat{\phi}\in\left[0,2\pi\right)$, which can be used as an estimation for
$\phi$.

The optimality of the canonical phase measurement indicates that its Holevo
variance

$V_{\text{CPM}}=\left\lvert
e^{-\alpha^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{n!\sqrt{n+1}}\right\rvert^{2}-1\,,$
(16)

is the fundamental bound of estimation precision [51, 20]. Although there are
proposals that attempt to implement this POVM, they have not been able to
reach the fundamental bound Eq. (16), or Eq. (2). For instance, in [74] it was
possible to obtain the canonical measurement distribution as the marginal of a
joint measurement in phase space producing a worse performance in the context
of phase estimation. Moreover, the canonical phase measurement was implemented
for the case of one-photon wave packet using quantum feedback [22]. However,
for the case of higher dimensional states, such as coherent states, this
problem remains open.

While there is not a satisfactory known method to implement the canonical
phase measurement, Gaussian strategies serve as a standard of physically
realizable measurement techniques for phase estimation of coherent states. The
natural benchmark in the Gaussian strategies is the heterodyne detection,
whose variance is lower bounded by
$\text{Var}\left[\widehat{\phi}_{\text{Het}}\right]=1/2\lvert\alpha\rvert^{2}$
[21]. Several adaptive Gaussian schemes have been shown to exceed the lower
bound for heterodyne detection. The most efficient Gaussian measurement
reported to date, termed the Adaptive Mark-II (MkII) strategy [20], has a
variance in the limit of $\lvert\alpha\rvert\gg 1$ given by Eq. (1).
Nonetheless, these adaptive Gaussian strategies cannot reach the performance
for the canonical phase measurement in Eq. (2) [21].

The proposed non-Gaussian strategies for single-shot phase estimation of
coherent states are based on optimized adaptive measurements with photon
number resolving detection. These non-Gaussian strategies are able to
outperform the best known Gaussian strategies and closely follow the
performance of the canonical phase measurement in the limit of large photon
number.

###  II.3 Adaptive non-Gaussian phase estimation

The proposed non-Gaussian adaptive estimation strategies based on adaptive
photon counting [2] aim to estimate the phase $\phi\in\Phi=[0,2\pi)$ of a
coherent state
$\rho(\phi)=\left|e^{\mathrm{i}\phi}\alpha\right\rangle\left\langle
e^{-\mathrm{i}\phi}\alpha\right|\,$ with mean photon number
$\text{E}\left[\hat{n}\right]=\lvert{\alpha}\rvert^{2}$ using a finite number
of adaptive measurement steps, and based on the prior information $p(\phi)$
about $\phi$. In every adaptive step $l=1,2,\cdots,L$, the input coherent
state with energy $\lvert{\alpha}\rvert^{2}/L$ interferes with a local
oscillator field, which implements a displacement operation
$\hat{D}\left(\beta\right)\left|\alpha\right\rangle=\left|\alpha+\beta\right\rangle,\,\beta\in\mathbb{C},$
with phase and amplitude chosen by some rule, in general depending on previous
measurement outcomes. This is followed by a photon number detection
measurement with a given photon number resolution (PNR) $m$ of the detectors
[6], $m\in\mathbb{N}$. In practice, since the energy in each adaptive step is
$\lvert{\alpha}\rvert^{2}/L$, the strategy will only require moderate PNR
resolution ($m<10$) as $L$ increases.

In the first adaptive measurement $l=1$ [2], the strategy makes a random guess
hypothesis $\beta_{0}\in\mathbb{C}$ about the optimal $\beta$, and applies the
POVM

$\displaystyle\left\\{\hat{D}(\beta_{0})\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta_{0})\right\\}_{n=0}^{m-1}\cup$
$\displaystyle\left\\{\mathbb{I}-\sum_{n=0}^{m-1}\hat{D}(\beta_{0})\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta_{0})\right\\}$ (17)

over the state $\left|e^{\mathrm{i}\phi}\alpha/\sqrt{L}\right\rangle$. In the
POVM in Eq. (II.3), the sum over Fock states $\left|n\right\rangle\left\langle
n\right|$ models the photon detection on the displaced state with a detector
with PNR($m$) [6]. The corresponding Wigner function describing a photon-
number resolving detector shows non-Gaussian features with negative values.
For this reason, these adaptive estimation techniques are called “non-
Gaussian”, despite that the estimator produced is asymptotically normal (this
result will be proved in the remainder of this section).

Given the detection outcome $n_{1}$ in $l=1$, the posterior probability
distribution becomes

$p\left(\phi\mid n_{1};\beta_{0}\right)\propto\mathcal{L}(\phi\mid
n_{1};\beta_{0})p(\phi),$ (18)

where $\mathcal{L}\left(\phi\mid n_{1};\beta_{0}\right)$ is the likelihood
function given by

$\begin{split}\mathcal{L}\left(\phi\mid
n_{1};\beta_{0}\right)&=p\left(n_{1}\mid\phi;\beta_{0}\right)\\\
&=\operatorname{Tr}\left[\hat{D}(\beta_{0})\left|n_{1}\right\rangle\left\langle
n_{1}\right|\hat{D}^{\dagger}(\beta_{0})\rho(\phi)\right]\,.\end{split}$ (19)

The phase estimate $\phi_{1}$ in this adaptive step corresponds to the MAP
estimator $\widehat{\phi}_{\text{MAP}}$,
$\phi_{1}=\widehat{\phi}_{\text{MAP}}(n_{1})$, with the posterior probability
distribution in Eq. (18). Using the posterior phase distribution in Eq. (18)
as the prior for the next adaptive step $l=2$, the strategy optimizes a cost
function $U(\beta)$ to obtain the next value of $\beta$ denoted as
$\beta_{1}$, and implements the POVM in Eq. (II.3) with $\beta_{1}$. The
Bayesian updating procedure is repeated at each step $l\geq 2$. After $l$
adaptive measurements the posterior probability distribution becomes

$\begin{split}p(\phi\mid\mathbf{n},\bm{\beta})&=p(\phi\mid
n_{l},...,n_{1},\beta_{l-1},...,\beta_{0})\\\
&\propto\prod_{i=1}^{l}p(n_{i}\mid\phi,\beta_{i-1})p(\phi).\end{split}$ (20)

Here $n_{i}$ is the observed photon detection at step $i$. Using the MAP on
this phase distribution, we obtain the $lth$ estimation $\widehat{\phi}_{l}$.
The procedure is repeated until the last measurement step $L$.

This parameter estimation strategy is a particular case of Bayesian optimal
design of experiments, where the parameters $\beta\in\mathbb{C}$ are the
designs, and which are optimized to estimate a phase $\phi\in[0,2\pi)$. Since
the optimal value for $\beta$ on each adaptive step depends on all previous
detection results, the cost function to be optimized is a function of the
posterior distribution in Eq. (20). A suitable choice of cost function, such
as the mutual information or the estimator variance, can provide a sequence of
estimations $\widehat{\phi}_{n}$ that approaches the true value of $\phi$ [2].

In the case of estimation of cyclic parameters, such as phase estimation, the
posterior distribution in Eq. (20) is $2\pi$ periodic, and the moments of
$\widehat{\phi}$ cannot be calculated as in the linear case [51]. In such
situations, the first moment of the cyclic random variable $X$ is defined as
$\rm{E}\left[e^{\mathrm{i}X}\right]$, and the dispersion of an estimator
$\widehat{\phi}$ is calculated using the Holevo Variance [51]:

$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]=\frac{1}{\left\lvert\rm{E}\left[e^{\mathrm{i}\widehat{\phi}}\right]\right\rvert^{2}}-1\,,$
(21)

which is the analogous to the mean square error. The minimization of the
uncertainty about $\phi$ (positive square root of Eq. (21)), requires
maximization of
$S(\beta,m)=\left\lvert\rm{E}\left[e^{\mathrm{i}\widehat{\phi}}\right]\right\rvert$,
known as the sharpness of the distribution. Then, the suitable cost function
for the adaptive protocol is the average sharpness:

$\bar{S}(\beta,m)=\sum_{n=0}^{m}p(n)\left\lvert\int_{\Phi}e^{\mathrm{i}\phi}p\left(\phi\mid
n,\beta\right)d\phi\right\rvert.$ (22)

Identifiability of Likelihood.
To guarantee a consistent asymptotic estimator the optimized estimation
strategies require to satisfy the regularity conditions 1-6 described in Sec.
II.2. For phase estimation, the regularity conditions $1\text{ and }2$ are
satisfied given that $\phi$ is an interior point of $\Phi=[0,2\pi)$. Moreover,
given that the probability

$\begin{split}&p(n\mid\phi;\beta)=\text{Tr}\left[\left|\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}\right\rangle\left\langle\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}\right|\hat{D}(\beta)\left|n\right\rangle\left\langle
n\right|\hat{D}^{\dagger}(\beta)\right]\\\
&=\frac{\exp\left(-\lvert\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}-\beta\rvert^{2}\right)\lvert\frac{\alpha
e^{\mathrm{i}\phi}}{\sqrt{L}}-\beta\rvert^{2n}}{n!},\,\alpha\in\mathbb{R}_{+},\,\beta\in\mathbb{C}\end{split}$
(23)

is well defined and two times differentiable, the conditions $4$, $5$, and $6$
are directly satisfied. However, the condition $3$ is not satisfied in
general. If one chooses $\beta$ as the value that maximizes the mutual
information (8) or the average sharpness (22), the resulting likelihood
function can have in general two maxima, that is, a non-identifiable
likelihood function [30]. In that case it is not possible to guarantee the
existence of a consistent estimator.

To address the challenge of working with non-identifiable likelihood
functions, we use designs with a fixed relation between the phase of $\beta$
and the amplitude $|\beta|$, given by $\beta=f(\theta)e^{\mathrm{i}\theta}$,
with $\theta\in[0,2\pi)$ and $f(\theta)$ a real function. These experimental
designs result in a cost function $U$ that is a function of $\theta$.

To see how this method solves the problem of non-identifiability, we observe
that the probability $p(n\mid\phi;\beta)$ in Eq. (23) is Poisson distributed,

$\begin{split}p(n\mid\phi;\beta=\lvert\beta\rvert
e^{\mathrm{i}\theta})&=\frac{e^{-\lambda}\cdot\lambda^{n}}{n!},\end{split}$
(24)

with
$\lambda=\lvert\alpha\rvert^{2}/L+\lvert\beta\rvert^{2}-2\lvert\alpha\rvert\lvert\beta\rvert\cos\left(\phi-\theta\right)/\sqrt{L}$.
Then, for $L$ adaptive steps with results $\mathbf{n}=(n_{1},...,n_{L})$, the
likelihood function is the product of $L$ probability functions of the form of
Eq. (24):

$\begin{split}\mathcal{L}(\mathbf{n},\phi)=\prod_{i=1}^{L}p(n_{i}\mid\phi;\beta_{i}=\lvert\beta\rvert
e^{\mathrm{i}\theta_{i}})&=\prod_{i=1}^{L}\frac{e^{-\lambda_{i}}\cdot\lambda_{i}^{n_{i}}}{n_{i}!}\,.\end{split}$
(25)

Here, each $\theta_{i}$ depends on the cost function and previous POVM
outcomes. The choice of the experimental designs with $|\beta|=f(\theta)$ can
force the adaptive strategy to change $\theta_{i}$ in each step. In this case,
the likelihood function in Eq. (25) becomes identifiable because the product
of probability functions with different $\theta_{i}$ produces a likelihood
with a global maximum. To see this, note that if $\theta_{i}=\theta$ is fixed,
even if the parameter $|\beta|$ is different in each adaptive step of the
protocol, the likelihood function from a sequence of independent random
variables with probability function given by Eq. (24) has two maxima over
$\Phi$. One of the maxima will always be around $\phi$ and the other will
depend on the value of $\theta$. On the other hand, if the strategy allows
$\theta$ to change at each adaptive step, the second maximum is suppressed,
since the functions whose product constitutes the likelihood Eq. (25) will
each have a second maxima at different positions
$\theta_{i}\neq\theta_{j}\,(i\neq j)$. As a result, with the experimental
designs $\beta=f(\theta)\exp(\mathrm{i}\theta)$, the likelihood functions
satisfy all the regularity conditions described in Sec. II.2.

Bayesian optimal design of $|\beta|$.
Any viable estimation strategy should aim to achieve the QCRB (14). Therefore,
natural choice for $\lvert{\beta}\rvert$ is the one that maximizes the Fisher
information. For a discrete random variable, the Fisher information is given
by [72, 75]:

$F(\phi)=\sum_{n=0}^{\infty}p(n\mid\phi)\left(\frac{\partial}{\partial\phi}\log\left(p(n\mid\phi)\right)\right)^{2}.$
(26)

The Fisher information for a particular design $\beta=\lvert{\beta}\rvert
e^{\mathrm{i}\theta}$ for Poisson distributions Eq. (24) results in

$F(\phi;\beta)=\frac{4\lvert\alpha\rvert^{2}\lvert\beta\sin^{2}(\phi-\theta)\rvert^{2}}{\lvert\alpha\rvert^{2}+L\lvert\beta\rvert^{2}-2\lvert\alpha\rvert\lvert\beta\rvert\cos(\phi-\theta)\sqrt{L}}.$
(27)

Optimizing over $|\beta|$, the Fisher information becomes:

$F(\phi,\beta_{\text{opt}})=4\lvert\alpha\rvert^{2}/L,$ (28)

where

$\beta_{\text{opt}}=\frac{\lvert\alpha\rvert}{\sqrt{L}\cos(\phi-\theta)}$ (29)

is the value of $|\beta|$ that maximizes the Fisher information.
Unfortunately, since $\beta_{\text{opt}}$ depends on $\phi$, its
implementation is not practical, because it would be required to already know
a priori the very same parameter that one wants to estimate. To address this
problem, the optimal Bayesian design $\beta_{\text{opt}}$ can be estimated as:

$\widehat{\beta}_{\text{opt}}=\frac{\lvert\alpha\rvert}{\sqrt{L}\cos(\widehat{\phi}-\theta)},$
(30)

where $\widehat{\phi}$ is the MAP estimator. As a result, the non-Gaussian
estimation strategy has now only one design, the phase $\theta$. With
$\beta=\widehat{\beta}_{\text{opt}}$, the Fisher information becomes:

$F(\Delta,\widehat{\beta}_{\text{opt}})=\frac{4\,\sin^{2}(\Delta)\,\alpha^{2}}{L(\cos^{2}\left(\delta+\Delta\right)-2\,\cos(\Delta)\,\cos\left(\delta+\Delta\right)+1)}\,,$
(31)

where $\delta=\widehat{\phi}-\phi$ and $\Delta=\phi-\theta$.

Note that $F(\Delta,\widehat{\beta}_{\text{opt}})$ becomes a random variable
with outcomes depending on $\widehat{\phi}$ through
$\widehat{\beta}_{\text{opt}}$. Moreover, for a random initial design
$\theta_{1}$, a cost function given by the expected sharpness or the mutual
information with $\widehat{\beta}_{\text{opt}}$ in Eq. (30), the likelihood
function in Eq. (25) becomes identifiable. As a result, the non-Gaussian
strategy then leads to consistent and efficient MAP estimators [67].

Asymptotic behavior.
In general $\delta=\widehat{\phi}-\phi\neq 0$, and the Fisher information has
a strong dependence on $\theta$. For example, if $\theta\to\phi$ then
$F(\Delta=0,\widehat{\beta}_{\text{opt}})\to 0$, and negligible information is
gained when the system is measured. Therefore, the optimal value of $\theta$
should result in the value of $\Delta$ that maximizes the expected value of
$F(\Delta,\widehat{\beta}_{\text{opt}})$. By observing that the expected value
$\rm{E}[\delta^{2n+1}]=0$, with $n\in\mathbb{N}$, we see that the optimal
value of $\Delta$ is $\pi/2$. As a result, an efficient adaptive strategy
would make $\Delta_{L}=\phi-\theta_{L}$ tend to $\pi/2$ as $L$ increases.
However, in the limit of $\Delta\to\pi/2$ and $\delta\to 0$,
$|\widehat{\beta}_{\text{opt}}|\to\infty$ (see Eq. (30)), and we expect that
when the strategy is implemented $\Delta<\pi/2$. These findings are consistent
with our numerical simulations of the non-Gaussian adaptive strategy, where we
observe that for $L\gg 1$, $\Delta\to\pi/2-\epsilon$, with $\epsilon$ a small
positive real number, and $|\beta|$ stays finite around a value that the PNR
detector in the strategy can resolve. Note that $\hat{\beta}_{\text{opt}}$,
which maximizes the Fisher information given $\theta$, does not necessarily
maximizes the cost function $U(\beta)$ for $\beta\in\mathbb{C}$. However
restricting $\beta$ to the set that maximizes the Fisher information makes the
phase of $\beta$ change in each step forcing the likelihood to be
identifiable. Moreover, when $\hat{\phi}\to\phi$, $\hat{\beta}_{\text{opt}}$
tends to the value that maximizes $U$ [2].

Given that the Fisher Information is additive, for any $L$ measurements made
using the optimal design, $\beta_{\text{opt}}$ in Eq. (29), the total Fisher
information for this design corresponds to the quantum Fisher information
$4|\alpha|^{2}$. However, since $\beta_{\text{opt}}$ is not known, it is not
possible to choose a design $\beta$ for which its Fisher information equals
the quantum Fisher information. This is already implied by the fact that the
canonical phase measurement does not saturate the Cramér-Rao bound. To show
this, we observe that for and estimator very close to the true value
$\delta=\widehat{\phi}-\phi\ll 1$ for $L$ adaptive measurements, the Fisher
information (Eq. (31)) is

$F(\phi,\widehat{\beta}_{\text{opt}})\approx 4\,|\alpha|^{2}(1-\delta^{2})\,.$
(32)

On the other hand, the best possible estimator of $\delta$ in each step
satisfies $E\left[\delta^{2}\right]\geq 1/4|\alpha|^{2}$, so that

$F(\phi,\widehat{\beta}_{\text{opt}})\lesssim 4\,|\alpha|^{2}-1\,,$ (33)

independently of $L$. We conclude that the adaptive non-Gaussian strategies do
not saturate the Cramér-Rao bound for finite $|\alpha|$. Nevertheless, these
adaptive estimation schemes outperform the most sensitive Gaussian strategy
known to date, and show a similar asymptotic scaling as the canonical phase
measurement.

###  II.4 Performance of Non-Gaussian Adaptive Strategies

We numerically investigate the performance of the estimator produced by non-
Gaussian adaptive strategies for phase estimation for different number of
adaptive steps $L$, photon number resolution PNR($m$), and average photon
number $|\alpha|^{2}$. To assess the performance of these strategies, we
compare our results with the best known Gaussian measurement for phase
estimation, termed Mark II (MKII) strategy [21], and with the canonical phase
measurement. As discussed in Sec. II.2, the performance of non-Gaussian
adaptive strategies using cost functions including the Kullback-Lieber
divergence Eq. (6), conditional entropy Eq. (7), mutual information Eq. (8),
and expected sharpness Eq. (22) are equivalent in the asymptotic limit. In our
numerical simulations, however, we use the expected sharpness in Eq. (22) as
the cost function for the optimization of the strategy, because it
substantially reduces the number of operations for this optimization and the
computational overhead.

In our numerical analysis, we use Monte Carlo simulations and generate
sufficient numerical data samples to reduce statistical uncertainties. Fig. 2A
shows the Holevo variance for the non-Gaussian adaptive strategy, as a
function of the number of adaptive steps $L$ for $|\alpha|^{2}=1$, for
different PNR: $m=1,3,6$. We observe that the adaptive non-Gaussian scheme
with PNR($1$) and $L\geq 100$ (green dots with error bars) outperforms the
most sensitive Gaussian strategy known to date, the MKII, (red dashed line).
Moreover, strategies with PNR($3$) (yellow dots with error bars) and PNR($6$)
(light blue dots with error bars) outperform the MKII strategy with only
$L\approx 30$ and $L\approx 20$, respectively, achieving a smaller Holevo
variance with fewer adaptive measurements compared to PNR(1). We have observed
similar behavior for non-Gaussian strategies optimizing different cost
functions, such as the mutual information.

To investigate the asymptotic behavior of the adaptive non-Gaussian strategy,
we assume an exponential dependence for the Holevo variance with $L$ (solid
lines in Fig. 2A):

$y(L,\alpha)=\text{A}e^{-\text{B}\cdot L}+\text{C}.$ (34)

The exponential model is a few parameter model that allows us to quantify
asymptotic trends when the datasets have rapidly decaying tails. This is a
widely used model for studying the asymptotic scaling of estimators as a
function of resources in diverse metrological problems [64, 76, 77].

We fit the numerical data from Monte Carlo simulations to Eq. (34) to estimate
the constants $A$, $B$, and $C$. Our results show that the asymptotic constant
$C=0.751\pm 0.002$ for PNR(1), $C=0.719\pm 0.004$ for PNR(3), and $C=0.714\pm
0.003$ for PNR(6). We observe that these values are smaller than the asymptote
of the MKII Gaussian strategy, but larger than the one for the canonical phase
measurement ($0.673$, blue dashed line). We note that while $C$ for PNR(3) is
larger than for PNR(6), they are statistically equal due to their
uncertainties. This prevents us from drawing any conclusions for larger values
of $m$.

Fig. 2: Asymptotic limit for the estimator of Holevo variance and optimal
design. a: Holevo variance for $\alpha=1$ and PNR $m=1,3,6$ as a function of
adaptive steps $L$. Note that the non-Gaussian strategy surpasses the MKII
strategy (red dashed line at $y=0.767$) with $L>100$, $L>30$ and $L>20$ for
PNR($1$), PNR($3$) and PNR($6$), respectively. The lines are obtained fitting
the exponential model Eq. (34). Estimates for these strategies result in
$(A,B,C,\mathrm{RSE})=(0.11\pm 0.02,0.059\pm 0.014,0.7145\pm 0.003,0.00058)$
for PNR($6$), $(A,B,C,\mathrm{RSE})=(0.22\pm 0.04,0.082\pm 0.0157,0.719\pm
0.004,0.00076)$ for PNR($3$), $(A,B,C,\mathrm{RSE})=(0.41\pm 0.05,B=0.117\pm
0.013,0.7517\pm 0.0027,0.00052)$ for PNR($1$). The shaded region represents
one standard deviation. b: Optimal design as a function of $L$. As $L$
increases the optimal design tends to $\pi/2$. Numerical data are obtained
with $10000$ Monte Carlo sequences. Error bars represent a 1-$\sigma$ standard
deviation.

Figure 2b shows the design $\Delta=\widehat{\phi}-\theta$, the phase of the
displacement field, as a function of $L$. We observe that for non-Gaussian
adaptive strategies with increasing PNR, $\Delta$ tends to $\pi/2$ for large
$L$ ($L=200$). This observation is consistent with the theoretical framework
of optimal design of experiments (Sec. II.3), which states that
$\Delta_{optimal}=\pi/2$ for $L\to\infty$. Moreover, as PNR increases, the
strategies show a faster convergence to the asymptotic value of the Holevo
variance, which translates in a smaller error in the estimation (see Fig. 2a.)
These observations further support our theoretical model of non-Gaussian
strategies for phase estimation.

Fig. 3:  Estimator variance produced by the non-Gaussian strategy. Holevo
variance of adaptive Non-Gaussian strategies as a function of
$\lvert\alpha\rvert^{2}$, normalized to QCRB, for different adaptive
measurement steps (solid lines), together with the canonical phase measurement
(blue dashed line). For $L=100$ the Holevo variance differs from the QCRB by
$3\%$ exemplifying the tendency to the ultimate precision limit in the regime
of large number of photons in the asymptotic limit. Simulations consist of
$10000$ Monte Carlo simulation sequences with PNR $m=3$.

We investigate the asymptotic performance of the estimator variance produced
by the non-Gaussian strategy for large $|\alpha|^{2}$. Figure 3 shows the
Holevo variance for the non-Gaussian adaptive strategy as a function of
$|\alpha|^{2}$ for different $L$ normalized to the QCRB. We observe that for
large $|\alpha|^{2}$ (with $L=100$), the adaptive non-Gaussian strategy tends
to the CRLB.

To build a model for the performance of this strategy for large
$|\alpha|^{2}$, we propose three candidate models for the Holevo variance for
$L\gg 1$:

$\displaystyle\widehat{y}_{1}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{2}}{\lvert\alpha\rvert^{3}}+\frac{A_{3}}{\lvert\alpha\rvert^{4}},$
(35a) $\displaystyle\widehat{y}_{2}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{2}}{\lvert\alpha\rvert^{3}},$
(35b) $\displaystyle\widehat{y}_{3}$ $\displaystyle=$
$\displaystyle\frac{A_{1}}{\lvert\alpha\rvert^{2}}+\frac{A_{3}}{\lvert\alpha\rvert^{4}}\,,$
(35c)

to describe our numerical observations in Fig. 3 based on the Monte Carlo
simulations. The model that best describes our observations allows us to
determine, with a certain degree of confidence, the performance of non-
Gaussian adaptive strategies, and compare them with the best Gaussian
strategy, Eq. (1), and the canonical phase measurement, Eq. (2).

We discriminate among plausible candidate models using the technique of
backward elimination [78]. We start with the candidate $\widehat{y}_{1}$, Eq.
(35a), and test the deletion of each variable $A_{i}$ using the $p$-value of a
hypothesis testing procedure ($H_{0}:\,A_{i}=0$, $H_{A}:\,A_{i}\neq 0$). Given
that $p>0.1$ for $A_{2}$ and $p<0.001$ for $A_{1}$ and $A_{3}$, we conclude,
with confidence larger than $99\%$ , that the model reflecting the behavior of
the data presented in Fig. 3 is $\widehat{y}_{3}$, Eq. (35c).

To find $A_{1}$ and $A_{3}$ in the limit $L\to\infty$ in model
$\widehat{y}_{3}$, we fit the Holevo variance as a function of $|\alpha|^{2}$
for $\lvert{\alpha}\rvert^{2}>5$ to the model $\widehat{y}_{3}$ for different
values of $L$. Given a number of adaptive steps $L$, each fitting provides a
set of coefficients $\left\\{A_{1},A_{3}\right\\}$. Hence, to obtain the trend
of each coefficient $A_{1}$ and $A_{2}$ as $L$ increases, we fit them to an
exponential model of the form $A_{i}=D_{i}\exp\left(-E_{i}L\right)+F_{i}$.
Fig. 4 shows the coefficients $A_{1}$ and $A_{3}$ as a function of $L$.
Adjusting this exponential model and observing that in the limit $L\gg 1$
$A_{i}\rightarrow F_{i}$, we obtain $A_{1}=0.250\pm 0.001$ and $A_{3}=0.52\pm
0.01$. Therefore, we conclude with a $99\%$ confidence level that the Holevo
variance for the adaptive non-Gaussian strategy in the limit $L\to\infty$ for
large $|\alpha|$ is:

$\text{Var}_{\text{H}}\left[\widehat{\phi}\right]\approx\frac{0.250\pm
0.001}{\lvert\alpha\rvert^{2}}+\frac{0.520\pm
0.010}{\lvert\alpha\rvert^{4}}\,.$ (36)

This equation is the main result of this work, also shown in Eq. (3). We
observe that the Holevo variance for the adaptive non-Gaussian strategy has a
similar dependance with $|\alpha|$ as the canonical phase measurement,
differing only in the scaling of the correction term of order
$1/|\alpha|^{4}$, see Eq. (2). Moreover, we note that the best Gaussian
strategy known to date, the MKII adaptive homodyne, has a correction term of
order $1/|\alpha|^{3}$ [20], see Eq. (1). Then, for large $|\alpha|$, the non-
Gaussian estimation strategies show a much better scaling in the Holevo
variance than the best known Gaussian strategy, and closely follows the
canonical phase measurement. Furthermore, these non-Gaussian phase estimation
strategies can be implemented with current technologies [2], and our work
demonstrates their superior performance over all the physically realizable
strategies for single-shot phase estimation of coherent states reported to
date.

Fig. 4:  Coefficients of model $\widehat{y}_{3}$ in Eq. (35c) for the Holevo
Variance $\text{Var}_{\text{H}}[\widehat{\phi}]$, as a function of $L$. Note
that in the limit of $L\to\infty$ the coefficient $A_{1}$ for the term
$1/\lvert\alpha\rvert^{2}$ tends to $0.25$, and $A_{3}$ of term
$1/\lvert\alpha\rvert^{4}$ tends to $0.52$. Curves are fits to an exponential
model $A_{i}=D_{i}\exp(-E_{i}L)+F_{i}$ with coefficients
$(D,E,F,\mathrm{RSE})=(0.065\pm 0.013,0.060\pm 0.010,0.250\pm 0.001,1.87\times
10^{-7})$ for the left panel, and $(D,E,F,\mathrm{RSE})=(0.095\pm
0.015,0.279\pm 0.119,0.522\pm 0.010,1.82\times 10^{-5})$ for the right panel.
The shaded regions represent a 1-$\sigma$ standard deviation.

The optimized non-Gaussian adaptive strategies based on photon counting
analyzed in this work are not the only possible strategies for single-shot
phase estimation, and there could be other strategies based on photon counting
with better performances. In this work, we studied estimation strategies using
cost functions that are consistent with D-optimal designs. However, there may
be other cost functions that could provide a further improvements to non-
Gaussian strategies. Moreover, we note that the relation in Eq. (30) between
the phase and the magnitude of the displacement field was used to obtain
identifiable likelihood functions and ensure efficient estimators in the
asymptotic limit. While we chose the relation in Eq. (30) because it maximizes
the Fisher information, there is no mathematical proof that this choice is
optimal, or that other choices for this relation would not provide higher
sensitivities. Finally, in the presented adaptive non-Gaussian strategies for
phase estimation, the displacement operations were optimized in every adaptive
step at a time, i. e. using local optimizations in the adaptive steps. We
note, however, that local optimal strategies do not necessarily ensure global
optimality [13]. Strategies with global optimizations, where all adaptive
steps are optimized simultaneously, could probably lead to better
performances. However, the computational overhead required for performing
global optimizations beyond $L=10$ adaptive steps prevents us from being able
to investigate global optimized strategies. To overcome these limitations,
future investigations could make use of machine learning methods, such as
neural networks and reinforcement learning [79], to lower the complexity of
these calculations.

##  III  Discussion

Non-Gaussian measurement strategies for phase estimation approaching the
quantum limits in sensitivity, as set by the canonical phase measurement, can
become a tool for enhancing the performance of diverse protocols in quantum
sensing, communications, and information processing. Optical phase estimation
approaching the quantum limit can be used to prepare highly squeezed atomic
spin states based on measurement backaction [42] for quantum sensing and
metrology [80]; enhance phase contrast imaging of biological samples with
optical probes at the few photon level to avoid photodamage and ensure
integrity of the sample; and to enhance fidelities in quantum communications
with phase coherent states that require phase tracking close to the quantum
level between receiver and transmitter with few-photon pulses [81, 82], while
allowing for quantum receivers to decode information encoded in coherent
states below the quantum noise limit [83, 13, 52].

As a direct application for quantum information processing, non-Gaussian
measurements based on photon counting and displacement operators can be used
for full reconstruction of quantum states with on-off detectors [84] and PNR
detectors [85] in a multi-copy state setting by sampling phase space with non-
Gaussian projections. The theoretical methods to assess the performance of
adaptive non-Gaussian measures for phase estimation described in this work
could be used to study methods for adaptive quantum tomography [86, 87] based
on photon counting for high dimensional quantum states, and investigate their
asymptotic advantages over adaptive homodyne tomography [88].

From the practical point of view, our work provides insight into the design of
practical, highly efficient measurement strategies for phase estimation based
on photon counting. It shows that non-Gaussian strategies optimizing any cost
function within the family of D-optimal designs are equally efficient, and
demonstrates the advantages of higher photon number resolution in the
strategies to reduce estimation errors. This understanding allows the
experimenter to chose cost functions that can be efficiently calculated and
optimized to achieve higher convergence rates, while selecting a PNR given the
desired/target error budget in the estimation problem for specific
applications. This knowledge will be critical for the design and development
of future sensors based on photon counting for diverse applications in
communication, phase-contrast imaging, metrology, and information processing.

In conclusion, we investigate a family of non-Gaussian strategies for single-
shot phase estimation of optical coherent states. These strategies are based
on adaptive photon counting measurements with a finite number of adaptive
steps implementing coherent displacement operations, optimized to maximize
information gain as the measurement progresses [2]. We develop a comprehensive
statistical analysis based on Bayesian optimal design of experiments that
provides a natural description of adaptive non-Gaussian strategies. This
theoretical framework gives a fundamental understanding on how to optimize
these strategies to enable efficient estimators with a high degree of
convergence towards the ultimate limit, the Cramér Rao lower bound.

We use numerical simulations to show that optimized adaptive non-Gaussian
strategies producing an asymptotically efficient normal estimator achieve a
much higher sensitivity than the best Gaussian strategy known to date, which
is based on adaptive homodyne [20]. Moreover, we show that the Holevo variance
of the estimator for the adaptive non-Gaussian strategy has a similar
dependence as the canonical phase measurement in the asymptotic limit of large
photon numbers, differing by a scaling factor in the second-order correction
term. Our work complements the work in single-shot phase measurements for
single-photon wave packets in two dimensions [22] using quantum feedback with
Gaussian measurements, and paves the way for the realization of optimized
feedback measurements approaching the canonical phase measurement for higher
dimensional states based on non-Gaussian operations.

DATA AVAILABILITY
The data that support the findings of this study are available from the
authors upon request.

CODE AVAILABILITY
Our numerical results have been implemented using MATLAB and Julia language. A
functions to reproduce the numerical results can be publicly found in the
repository [89]. To obtain a set of estimations produced by the non Gaussian
strategy, use the program: ` Bootstrap_phase_estimation.jl`, setting the
desired values of $L$, Bootstrap size, MPN, in the `Adapt_Steps`, `Boot_Reps`,
and `MPNS` variables.

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 IG101421 and the National Science
Foundation (NSF) grans # PHY-1653670 and PHY-2210447.

AUTHOR CONTRIBUTIONS
F.E.B. and P.B. conceived the idea and supervised the work. M.A.R. and P.B.
conducted the theoretical and statistical study. M.T.D. and M.A.R. contributed
in the code for simulations. All authors contributed to the analysis of the
theoretical and numerical results and contributed to writing the manuscript.

COMPETING INTERESTS
The authors declare that there are no competing interests.

## References

  * [1] Giovannetti, V., Lloyd, S. & Maccone, L.  Quantum-enhanced measurements: Beating the standard quantum limit.  _Science_ 306, 1330–1336 (2004).  URL https://science.sciencemag.org/content/306/5700/1330.
  * [2] DiMario, M. T. & Becerra, F. E.  Single-shot non-gaussian measurements for optical phase estimation.  _Phys. Rev. Lett._ 125, 120505 (2020).  URL https://link.aps.org/doi/10.1103/PhysRevLett.125.120505.
  * [3] Demkowicz-Dobrzański, R., Kołodyński, J. & Guţă, M.  The elusive heisenberg limit in quantum-enhanced metrology.  _Nat. Commun._ 3, 1–8 (2012).
  * [4] Abbott, B. P. e. a.  Observation of gravitational waves from a binary black hole merger.  _Phys. Rev. Lett._ 116, 061102 (2016).  URL https://link.aps.org/doi/10.1103/PhysRevLett.116.061102.
  * [5] Helstrom, C.  _Quantum Detection and Estimation Theory_.  Mathematics in Science and Engineering: a series of monographs and textbooks (Academic Press, 1976).  URL https://books.google.com.mx/books?id=fv9SAAAAMAAJ.
  * [6] Becerra, F. E., Fan, J. & Migdall, A.  Photon number resolution enables quantum receiver for realistic coherent optical communications.  _Nat. Photonics_ 9, 48–53 (2015).
  * [7] Dolinar, S. J.  An optimum receiver for the binary coherent state quantum channel.  Research Laboratory of Electronics, MIT, Quarterly Progress Report No. 111 (1973), p. 115.
  * [8] DiMario, M. T. & Becerra, F. E.  Channel-noise tracking for sub-shot-noise-limited receivers with neural networks.  _Phys. Rev. Res._ 3, 013200 (2021).  URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.013200.
  * [9] DiMario, M. T. & Becerra, F. E.  Phase tracking for sub-shot-noise-limited receivers.  _Phys. Rev. Res._ 2, 023384 (2020).  URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.023384.
  * [10] DiMario, M., Kunz, L., Banaszek, K. & Becerra, F.  Optimized communication strategies with binary coherent states over phase noise channels.  _npj Quantum Inf._ 5, 1–7 (2019).
  * [11] DiMario, M. T. & Becerra, F. E.  Robust measurement for the discrimination of binary coherent states.  _Phys. Rev. Lett._ 121, 023603 (2018).  URL https://link.aps.org/doi/10.1103/PhysRevLett.121.023603.
  * [12] DiMario, M. T., Carrasco, E., Jackson, R. A. & Becerra, F. E.  Implementation of a single-shot receiver for quaternary phase-shift keyed coherent states.  _J. Opt. Soc. Am. B_ 35, 568–574 (2018).  URL http://josab.osa.org/abstract.cfm?URI=josab-35-3-568.
  * [13] Ferdinand, A., DiMario, M. & Becerra, F.  Multi-state discrimination below the quantum noise limit at the single-photon level.  _npj Quantum Inf._ 3, 1–7 (2017).
  * [14] Becerra, F. _et al._ Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination.  _Nat. Photonics_ 7, 147–152 (2013).
  * [15] Slussarenko, S. & Pryde, G. J.  Photonic quantum information processing: A concise review.  _Appl. Phys. Rev._ 6, 041303 (2019).
  * [16] Plenio, M. B.  Logarithmic negativity: A full entanglement monotone that is not convex.  _Phys. Rev. Lett._ 95, 090503 (2005).  URL https://link.aps.org/doi/10.1103/PhysRevLett.95.090503.
  * [17] Polino, E., Valeri, M., Spagnolo, N. & Sciarrino, F.  Photonic quantum metrology.  _AVS Quantum Science_ 2, 024703 (2020).
  * [18] Giovannetti, V., Lloyd, S. & Maccone, L.  Advances in quantum metrology.  _Nat. Photonics_ 5, 222–229 (2011).
  * [19] Flamini, F., Spagnolo, N. & Sciarrino, F.  Photonic quantum information processing: a review.  _Rep. Prog. Phys._ 82, 016001 (2018).  URL https://doi.org/10.1088/1361-6633/aad5b2.
  * [20] Wiseman, H. M.  Adaptive phase measurements of optical modes: Going beyond the marginal $q$ distribution.  _Phys. Rev. Lett._ 75, 4587–4590 (1995).  URL https://link.aps.org/doi/10.1103/PhysRevLett.75.4587.
  * [21] Wiseman, H. M. & Killip, R. B.  Adaptive single-shot phase measurements: The full quantum theory.  _Phys. Rev. A_ 57, 2169–2185 (1998).
  * [22] Martin, L. S., Livingston, W. P., Hacohen-Gourgy, S., Wiseman, H. M. & Siddiqi, I.  Implementation of a canonical phase measurement with quantum feedback.  _Nat. Phys._ 16, 1046–1049 (2020).  URL https://doi.org/10.1038/s41567-020-0939-0.
  * [23] Braunstein, S. L. & Caves, C. M.  Statistical distance and the geometry of quantum states.  _Phys. Rev. Lett._ 72, 3439–3443 (1994).
  * [24] Demkowicz-Dobrzański, R., Jarzyna, M. & Kołodyński, J.  Quantum limits in optical interferometry.  _Prog. Opt._ 60, 345–435 (2015).
  * [25] Genoni, M. G., Olivares, S. & Paris, M. G. A.  Optical phase estimation in the presence of phase diffusion.  _Phys. Rev. Lett._ 106, 153603 (2011).
  * [26] Lee, C., Oh, C., Jeong, H., Rockstuhl, C. & Lee, S.-Y.  Using states with a large photon number variance to increase quantum fisher information in single-mode phase estimation.  _J. Phys. Commun._ 3, 115008 (2019).  URL https://doi.org/10.1088/2399-6528/ab524a.
  * [27] Bradshaw, M., Lam, P. K. & Assad, S. M.  Ultimate precision of joint quadrature parameter estimation with a gaussian probe.  _Phys. Rev. A_ 97, 012106 (2018).
  * [28] Wiseman, H. M. & Killip, R. B.  Adaptive single-shot phase measurements: A semiclassical approach.  _Phys. Rev. A_ 56, 944–957 (1997).
  * [29] Anisimov, P. M. _et al._ Quantum metrology with two-mode squeezed vacuum: Parity detection beats the heisenberg limit.  _Phys. Rev. Lett._ 104, 103602 (2010).
  * [30] Huang, Z., Motes, K. R., Anisimov, P. M., Dowling, J. P. & Berry, D. W.  Adaptive phase estimation with two-mode squeezed vacuum and parity measurement.  _Phys. Rev. A_ 95, 053837 (2017).  URL https://link.aps.org/doi/10.1103/PhysRevA.95.053837.
  * [31] Anderson, B. E. _et al._ Phase sensing beyond the standard quantum limit with a variation on the su(1,1) interferometer.  _Optica_ 4, 752–756 (2017).
  * [32] Izumi, S. _et al._ Optical phase estimation via the coherent state and displaced-photon counting.  _Phys. Rev. A_ 94, 033842 (2016).
  * [33] Slussarenko, S. _et al._ Unconditional violation of the shot noise limit in photonic quantum metrology.  _Nat. Photonics_ 11, 700–703 (2017).
  * [34] Anderson, B. E., Schmittberger, B. L., Gupta, P., Jones, K. M. & Lett, P. D.  Optimal phase measurements with bright- and vacuum-seeded su(1,1) interferometers.  _Phys. Rev. A_ 95, 063843 (2017).
  * [35] Daryanoosh, S., Slussarenko, S., Berry, D. W., Wiseman, H. M. & Pryde, G. J.  Experimental optical phase measurement approaching the exact heisenberg limit.  _Nat. Commun._ 9, 4606 (2018).
  * [36] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J.  Entanglement-free heisenberg-limited phase estimation.  _Nature_ 450, 393–396 (2007).
  * [37] Hentschel, A. & Sanders, B. C.  Machine learning for precise quantum measurement.  _Phys. Rev. Lett._ 104, 063603 (2010).
  * [38] Hou, Z. _et al._ Control-enhanced sequential scheme for general quantum parameter estimation at the heisenberg limit.  _Phys. Rev. Lett._ 123, 040501 (2019).
  * [39] Larson, W. & Saleh, B. E. A.  Supersensitive ancilla-based adaptive quantum phase estimation.  _Phys. Rev. A_ 96, 042110 (2017).
  * [40] Lumino, A. _et al._ Experimental phase estimation enhanced by machine learning.  _Phys. Rev. Applied_ 10, 044033 (2018).
  * [41] Zheng, K., Xu, H., Zhang, A., Ning, X. & Zhang, L.  Ab initio phase estimation at the shot noise limit with on–off measurement.  _Quantum Inf. Process._ 18, 329 (2019).
  * [42] Deutsch, I. H. & Jessen, P. S.  Quantum control and measurement of atomic spins in polarization spectroscopy.  _Opt. Commun._ 283, 681–694 (2010).  URL https://www.sciencedirect.com/science/article/pii/S0030401809010517.  Quo vadis Quantum Optics?
  * [43] Bouchoule, I. & Mølmer, K.  Preparation of spin-squeezed atomic states by optical-phase-shift measurement.  _Phys. Rev. A_ 66, 043811 (2002).
  * [44] Iwasawa, K. _et al._ Quantum-limited mirror-motion estimation.  _Phys. Rev. Lett._ 111, 163602 (2013).
  * [45] Tsang, M.  Quantum metrology with open dynamical systems.  _New Journ. of Phys._ 15, 073005 (2013).
  * [46] Aasi, A. J. A. B. e. a., J.  Enhanced sensitivity of the ligo gravitational wave detector by using squeezed states of light.  _Nat. Photonics_ 7, 613–619 (2013).
  * [47] Nair, R., Yen, B. J., Guha, S., Shapiro, J. H. & Pirandola, S.  Symmetric $m$-ary phase discrimination using quantum-optical probe states.  _Phys. Rev. A_ 86, 022306 (2012).
  * [48] van Loock, P., Lütkenhaus, N., Munro, W. J. & Nemoto, K.  Quantum repeaters using coherent-state communication.  _Phys. Rev. A_ 78, 062319 (2008).
  * [49] Munro, W. J., Nemoto, K. & Spiller, T. P.  Weak nonlinearities: a new route to optical quantum computation.  _New J. of Phys._ 7, 137 (2005).
  * [50] Nemoto, K. & Munro, W. J.  Nearly deterministic linear optical controlled-not gate.  _Phys. Rev. Lett._ 93, 250502 (2004).
  * [51] Holevo, A. S.  _Probabilistic and Statistical Aspects of Quantum Theory; 2nd ed._ Publications of the Scuola Normale Superiore. Monographs (Springer, Dordrecht, 2011).  URL https://cds.cern.ch/record/1414149.
  * [52] Becerra, F., Fan, J. & Migdall, A.  Photon number resolution enables quantum receiver for realistic coherent optical communications.  _Nat. Photonics_ 9, 48–53 (2015).
  * [53] Cover, T. M. & Thomas, J. A.  _Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)_ (Wiley-Interscience, USA, 2006).
  * [54] Chaloner, K. & Verdinelli, I.  Bayesian Experimental Design: A Review.  _Stat. Sci._ 10, 273 – 304 (1995).  URL https://doi.org/10.1214/ss/1177009939.
  * [55] Rodríguez-García, M. A., Castillo, I. P. & Barberis-Blostein, P.  Efficient qubit phase estimation using adaptive measurements.  _Quantum_ 5, 467 (2021).  URL https://doi.org/10.22331/q-2021-06-04-467.
  * [56] Ryan, E., Drovandi, C., McGree, J. & Pettitt, T.  A review of modern computational algorithms for bayesian optimal design.  _Int. Stat. Rev._ 84, 128–154 (2016).  URL https://eprints.qut.edu.au/75000/.
  * [57] Suzuki, J.  Quantum-state estimation problem via optimal design of experiments.  _Int. J. Quantum Inf._ 19, 2040007 (2021).  URL https://doi.org/10.1142/S0219749920400079.  eprint https://doi.org/10.1142/S0219749920400079.
  * [58] Morelli, S., Usui, A., Agudelo, E. & Friis, N.  Bayesian parameter estimation using gaussian states and measurements.  _Quantum Sci. Technol._ 6, 025018 (2021).  URL https://doi.org/10.1088/2058-9565/abd83d.
  * [59] Martínez-García, F., Vodola, D. & Müller, M.  Adaptive bayesian phase estimation for quantum error correcting codes.  _New J. Phys._ 21, 123027 (2019).  URL https://doi.org/10.1088/1367-2630/ab5c51.
  * [60] Berni, A. A. _et al._ Ab initio quantum-enhanced optical phase estimation using real-time feedback control.  _Nat. Photonics_ 9, 577–581 (2015).  URL https://doi.org/10.1038/nphoton.2015.139.
  * [61] Genoni, M. G. _et al._ Optical interferometry in the presence of large phase diffusion.  _Phys. Rev. A_ 85, 043817 (2012).  URL https://link.aps.org/doi/10.1103/PhysRevA.85.043817.
  * [62] Oh, C. & Son, W.  Sub shot-noise frequency estimation with bounded a priori knowledge.  _J. Phys. A Math. Theor._ 48, 045304 (2014).  URL https://doi.org/10.1088/1751-8113/48/4/045304.
  * [63] Macieszczak, K., Fraas, M. & Demkowicz-Dobrzański, R.  Bayesian quantum frequency estimation in presence of collective dephasing.  _New J. Phys._ 16, 113002 (2014).  URL https://doi.org/10.1088/1367-2630/16/11/113002.
  * [64] Glatthard, J. _et al._ Optimal cold atom thermometry using adaptive bayesian strategies (2022).  URL https://arxiv.org/abs/2204.11816.
  * [65] McMichael, R. D., Dushenko, S. & Blakley, S. M.  Sequential bayesian experiment design for adaptive ramsey sequence measurements.  _J. Appl. Phys._ 130, 144401 (2021).  URL https://doi.org/10.1063/5.0055630.  eprint https://doi.org/10.1063/5.0055630.
  * [66] Kleinegesse, S. & Gutmann, M. U.  Efficient bayesian experimental design for implicit models.  In _AISTATS_ (2019).
  * [67] Paninski, L.  Asymptotic theory of information-theoretic experimental design.  _Neural Comput._ 17, 1480–1507 (2005).  URL https://doi.org/10.1162/0899766053723032.
  * [68] Verdinelli, I.  A note on bayesian design for the normal linear model with unknown error variance.  _Biometrika_ 87, 222–227 (2000).  URL http://www.jstor.org/stable/2673577.
  * [69] Ryan, E., Drovandi, C., Thompson, H. & Pettitt, T.  Towards bayesian experimental design for nonlinear models that require a large number of sampling times 70, 45–60 (2014).  URL https://eprints.qut.edu.au/56522/.
  * [70] Nielsen, M. A. & Chuang, I. L.  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * [71] Braunstein, S. L. & Caves, C. M.  Statistical distance and the geometry of quantum states.  _Phys. Rev. Lett._ 72, 3439–3443 (1994).  URL https://link.aps.org/doi/10.1103/PhysRevLett.72.3439.
  * [72] Paris, M. G.  Quantum estimation for quantum technology.  _Int. J. Quantum Inf._ 7, 125–137 (2009).
  * [73] Holevo, A.  Statistical decision theory for quantum systems.  _J. Multivar. Anal._ 3, 337 – 394 (1973).  URL http://www.sciencedirect.com/science/article/pii/0047259X73900286.
  * [74] Pellonpää, J.-P. & Schultz, J.  Measuring the canonical phase with phase-space measurements.  _Phys. Rev. A_ 88, 012121 (2013).  URL https://link.aps.org/doi/10.1103/PhysRevA.88.012121.
  * [75] Escher, B. M., de Matos Filho, R. L. & Davidovich, L.  General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology.  _Nat. Phys._ 7, 406–411 (2011).  URL https://doi.org/10.1038/nphys1958.
  * [76] Wang, P. _et al._ Single ion qubit with estimated coherence time exceeding one hour.  _Nat. Commun._ 12, 233 (2021).  URL https://doi.org/10.1038/s41467-020-20330-w.
  * [77] Hall, M. J. W. & Wiseman, H. M.  Does nonlinear metrology offer improved resolution? answers from quantum information theory.  _Phys. Rev. X_ 2, 041006 (2012).  URL https://link.aps.org/doi/10.1103/PhysRevX.2.041006.
  * [78] Young, D. S.  _Handbook of regression methods_.  A Chapman and Hall Book (2017).
  * [79] DiMario, M. T. & Becerra, F. E.  Channel-noise tracking for sub-shot-noise-limited receivers with neural networks.  _Phys. Rev. Res._ 3, 013200 (2021).  URL https://link.aps.org/doi/10.1103/PhysRevResearch.3.013200.
  * [80] Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P.  Quantum metrology with nonclassical states of atomic ensembles.  _Rev. Mod. Phys._ 90, 035005 (2018).  URL https://link.aps.org/doi/10.1103/RevModPhys.90.035005.
  * [81] Qi, B., Lougovski, P., Pooser, R., Grice, W. & Bobrek, M.  Generating the local oscillator “locally” in continuous-variable quantum key distribution based on coherent detection.  _Phys. Rev. X_ 5, 041009 (2015).  URL https://link.aps.org/doi/10.1103/PhysRevX.5.041009.
  * [82] Soh, D. B. S. _et al._ Self-referenced continuous-variable quantum key distribution protocol.  _Phys. Rev. X_ 5, 041010 (2015).  URL https://link.aps.org/doi/10.1103/PhysRevX.5.041010.
  * [83] DiMario, M. T. & Becerra, F. E.  Robust measurement for the discrimination of binary coherent states.  _Phys. Rev. Lett._ 121, 023603 (2018).
  * [84] Allevi, A. _et al._ State reconstruction by on/off measurements.  _Phys. Rev. A_ 80, 022114 (2009).  URL https://link.aps.org/doi/10.1103/PhysRevA.80.022114.
  * [85] Nehra, R. _et al._ State-independent quantum state tomography by photon-number-resolving measurements.  _Optica_ 6, 1356–1360 (2019).  URL http://opg.optica.org/optica/abstract.cfm?URI=optica-6-10-1356.
  * [86] Huszár, F. & Houlsby, N. M. T.  Adaptive bayesian quantum tomography.  _Phys. Rev. A_ 85, 052120 (2012).  URL https://link.aps.org/doi/10.1103/PhysRevA.85.052120.
  * [87] Granade, C., Ferrie, C. & Flammia, S. T.  Practical adaptive quantum tomography.  _New J. Phys._ 19, 113017 (2017).  URL https://doi.org/10.1088/1367-2630/aa8fe6.
  * [88] D’Ariano, G. M. & Paris, M. G. A.  Adaptive quantum homodyne tomography.  _Phys. Rev. A_ 60, 518–528 (1999).  URL https://link.aps.org/doi/10.1103/PhysRevA.60.518.
  * [89] Rodríguez-García, M. A.  adaptive_photon_counting_for_coherent-state (2022).  URL https://github.com/Gateishion/adaptive_photon_counting_for_coherent-state.