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# Optimal estimation of pure states with displaced-null measurements

Federico Girotti  School of Mathematical Sciences, University of Nottingham,
United Kingdom  Centre for the Mathematics and Theoretical Physics of Quantum
Non-Equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, UK
Department of Mathematics, Polytechnic University of Milan, Milan, Piazza L.
da Vinci 32, 20133, Italy  Alfred Godley  School of Mathematical Sciences,
University of Nottingham, United Kingdom  Centre for the Mathematics and
Theoretical Physics of Quantum Non-Equilibrium Systems, University of
Nottingham, Nottingham, NG7 2RD, UK  Mădălin Guţă  School of Mathematical
Sciences, University of Nottingham, United Kingdom  Centre for the Mathematics
and Theoretical Physics of Quantum Non-Equilibrium Systems, University of
Nottingham, Nottingham, NG7 2RD, UK

###### Abstract

We revisit the problem of estimating an unknown parameter of a pure quantum
state, and investigate ‘null-measurement’ strategies in which the experimenter
aims to measure in a basis that contains a vector close to the true system
state. Such strategies are known to approach the quantum Fisher information
for models where the quantum Cramér-Rao bound is achievable but a detailed
adaptive strategy for achieving the bound in the multi-copy setting has been
lacking.

We first show that the following naive null-measurement implementation fails
to attain even the standard estimation scaling: estimate the parameter on a
small sub-sample, and apply the null-measurement corresponding to the
estimated value on the rest of the systems. This is due to non-identifiability
issues specific to null-measurements, which arise when the true and reference
parameters are close to each other. To avoid this, we propose the alternative
displaced-null measurement strategy in which the reference parameter is
altered by a small amount which is sufficient to ensure parameter
identifiability.

We use this strategy to devise asymptotically optimal measurements for models
where the quantum Cramér-Rao bound is achievable. More generally, we extend
the method to arbitrary multi-parameter models and prove the asymptotic
achievability of the the Holevo bound. An important tool in our analysis is
the theory of quantum local asymptotic normality which provides a clear
intuition about the design of the proposed estimators, and shows that they
have asymptotically normal distributions.

##  I Introduction and main results

The estimation of unknown parameters from measurement data is the central task
of quantum statistical inference Hayashi (2005); Paris (2008); Tóth and
Apellaniz (2014); Demkowicz-Dobrzański  _et al._ (2020); Banaszek  _et al._
(2013); Albarelli  _et al._ (2020); Sidhu and Kok (2020). In recent decades,
the area has witnessed an explosive growth covering a wealth of topics such as
quantum state tomography Gross  _et al._ (2010); Cramer  _et al._ (2010);
O’Donnell and Wright (2016); Haah  _et al._ (2016); Lanyon  _et al._ (2017);
Guţă and Kiukas (2017); Yang  _et al._ (2019); Lahiry and Nussbaum (2023);
Yuen (2023), multi-parameter estimation Szczykulska  _et al._ (2016); Nichols
_et al._ (2018); Demkowicz-Dobrzański  _et al._ (2020); Albarelli  _et al._
(2019, 2020); Liu  _et al._ (2020), sufficiency Mosonyi and Petz (2003);
Jenčová and Petz (2006), local asymptotic normality Guţă and Kahn (2006); Guţă
and Jencova (2007); Kahn and Guţă (2009); Gill and Guţă (2013); Butucea  _et
al._ (2016); Yamagata  _et al._ (2013); Fujiwara and Yamagata (2020, 2023),
shadow tomography Aaronson (2018); Huang  _et al._ (2020), Bayesian methods
Personick (1971); Tsang (2020); Rubio and Dunningham (2020); Rubio  _et al._
(2021), quantum metrology Giovannetti  _et al._ (2005); Fujiwara and Imai
(2008); Giovannetti  _et al._ (2011); Demkowicz-Dobrzański  _et al._ (2012);
Girolami  _et al._ (2014); Smirne  _et al._ (2016); Seveso  _et al._ (2017);
Haase  _et al._ (2018); Pang and Brun (2014); Rossi  _et al._ (2020); Zhou and
Jiang (2021), error correction methods Gorecki  _et al._ (2020); Zhou  _et
al._ (2018a), hamiltonian learning Yuan and Fung (2015); Yuan (2016),
thermometry Correa  _et al._ (2015); Mehboudi  _et al._ (2019), gravitational
waves detection Christensen and Meyer (2022); Abadie  _et al._ (2011),
magnetometry Jones  _et al._ (2009); Amorós-Binefa and Kołodyński (2021);
Brask  _et al._ (2015); Albarelli  _et al._ (2017), quantum sensing Degen  _et
al._ (2017); Marciniak  _et al._ (2022); Zwick and Álvarez (2023), imaging
Tsang  _et al._ (2016); Tsang (2021); Lupo  _et al._ (2020); Fiderer  _et al._
(2021); Oh  _et al._ (2021), semi-parametric estimation Tsang (2019); Tsang
_et al._ (2020) estimation of open systems Gambetta and Wiseman (2001); Guţă
(2011); Gammelmark and Mølmer (2014); Guţă and Kiukas (2015); Catana  _et al._
(2015); Guţă and Kiukas (2017); Ilias  _et al._ (2022); Fallani  _et al._
(2022), waveform Tsang  _et al._ (2011); Berry  _et al._ (2015) and noise Ng
_et al._ (2016); Norris  _et al._ (2016); Shi and Zhuang (2023); Sung  _et
al._ (2019); Tsang (2023) estimation.

A common feature of many quantum estimation problems is that ‘optimal’
measurements depend on the unknown parameter, so they can only be implemented
approximately, and the optimality is at best achieved in the limit of large
‘sample size’. This raises the question of how to interpret theoretical
results such as the quantum Cramér-Rao bound (QCRB) Holevo (2011); Yuen and
Lax (1973); Belavkin (1976); Helstrom (1976); Braunstein and Caves (1994);
Nagaoka (2005) and how to design adaptive measurement strategies which attain
the optimal statistical errors in the asymptotic limit. When multiple copies
of the state are available, the standard strategy is to use a sub-sample to
compute a rough estimator and then apply the optimal measurement corresponding
to the estimated value. Indeed this works well for the case of the symmetric
logarithmic derivative Gill and Massar (2000), an operator which saturates the
quantum Cramér-Rao bound for one-dimensional parameters. However, the QCRB
fails to predict the correct _attainable_ error for quantum metrology models
which consist of correlated states and exhibit Heisenberg (quadratic) scaling
for the mean square error Górecki  _et al._ (2020). This is due to the fact
that in order to saturate the QCRB one needs to know the parameter to a
precision comparable to what one ultimately hopes to achieve.

In this paper we uncover a somewhat complementary phenomenon, where the usual
adaptive strategy fails _precisely_ because it is applied to a ‘good’ guess of
the true parameter value. This happens in the standard multi-copy setting when
estimating a pure state by means of ‘null measurements’, where the
experimenter aims to measure in a basis that contains the unknown state. While
this can only be implemented approximately, the technique is known to exhibit
certain Fisher-optimality properties Pezze and Smerzi (2014); Pezzè  _et al._
(2017); Liu  _et al._ (2019) and has the intuitive appeal of ‘locking’ onto
the correct value as outcomes corresponding to other measurement vectors
become more and more unlikely.

In Theorem 1, which is our first main result, we show that the standard
adaptive strategy in which the parameter is first estimated on a sub-sample
and then the null-measurement for this rough value is applied to the rest of
the ensemble, fails to saturate the QCRB, and indeed does not attain the
standard rate of precision. Our result shows the importance of accompanying
mathematical properties with clear operational procedures that allow us to
draw statistical conclusions; this provides another example of the limitations
of the ‘local’ estimation approach based on the quantum Cramér-Rao bound Tsang
. Indeed the reason behind the failure of the standard adaptive strategy is
the fact that null-measurements suffer from non-identifiability issues when
the true parameter and the rough preliminary estimator are too close to each
other, i.e. when the latter is a reasonable estimator of the former.

Fortunately, it turns out that the issue can be resolved by deliberately
shifting the measurement reference parameter away from the estimated value by
a vanishingly small but sufficiently large amount to resolve the non-
identifiabilty issue. Using this insight we devise a novel adaptive
measurement strategy which achieves the Holevo bound for arbitrary multi-
parameter models, asymptotically with the sample size. This second main result
is described in Theorem 2. In particular our method can be used to achieve the
quantum Cramér-Rao bound for models where this is achievable, which was the
original theme of Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et
al._ (2019). The validity of the displaced-null strategy goes beyond the
setting of the estimation with independent copies and has already been
employed for optimal estimation of dynamical parameters of open quantum
systems by counting measurements Yang  _et al._ (2023). The extension of our
present results to the setting of quantum Markov chains will be presented in a
forthcoming publication Godley  _et al._ . In the rest of this section we give
a brief review of the main results of the paper.

#### The quantum Cramér-Rao bound and the symmetric logarithmic derivative

The quantum estimation problem is formulated as follows: given a quantum
system prepared in a state $\rho_{\theta}$ which depends on an unknown (finite
dimensional) parameter $\theta\in\Theta$, one would like to estimate $\theta$
by performing a measurement $M$ and constructing an estimator
$\hat{\theta}=\hat{\theta}(X)$ based on the (stochastic) outcome $X$. The
Cramér-Rao bound Lehmann and Casella (1998); van der Vaart (1998) shows that
for a given measurement $M$, the covariance of any unbiased estimator is lower
bounded as ${\rm Cov}(\hat{\theta})\geq I^{-1}_{M}(\theta)$ where
$I_{M}(\theta)$ is the classical Fisher information (CFI) of the measurement
outcome.

Since the right side depends on the measurement, this prompts a fundamental
and distinctive question in quantum statistics: what are the ultimate bounds
on estimation accuracy and what measurement designs achieve these limits? The
cornerstone result in this area is that, irrespective of the measurement $M$,
the CFI $I_{M}(\theta)$ is upper bounded by the quantum Fisher information
$F(\theta)$, the latter being an intrinsic property of the quantum statistical
model $\\{\rho_{\theta}\\}_{\theta\in\Theta}$. By combining the two bounds we
obtain the celebrated quantum Cramér-Rao bound (QCRB) Holevo (2011); Yuen and
Lax (1973); Belavkin (1976); Helstrom (1976); Braunstein and Caves (1994);
Nagaoka (2005) ${\rm Cov}(\hat{\theta})\geq F^{-1}(\theta).$ For one
dimensional parameters the QFI can be (formally) achieved by measuring an
observable $\mathcal{L}_{\theta}$ called the symmetric logarithmic derivative
(SLD), defined as the solution of the Lyapunov equation
$\frac{d\rho_{\theta}}{d\theta}=\frac{1}{2}(\rho_{\theta}\mathcal{L}_{\theta}+\mathcal{L}_{\theta}\rho_{\theta})$.
However, since the SLD depends on the unknown parameter $\theta$, this
measurement cannot be performed without its prior knowledge, and the formal
achievability is unclear without further operational specifications.

Fortunately, this apparent circularity issue can be solved in the context of
asymptotic estimation Hayashi (2005). In most practical applications one does
not measure a single system but deals with (large) ensembles of identically
prepared systems, or multi-partite correlated states as in quantum enhanced
metrology Pang and Brun (2014); Zhou  _et al._ (2018b) and continuous time
estimation of Markov dynamics Yang  _et al._ (2023); Guţă and Kiukas (2015,
2017); Gammelmark  _et al._ (2014). Here one considers issues such as the
scaling of errors with sample size, collective versus separable measurements,
and whether one needs fixed or adaptive measurements. In particular, in the
case of _one-dimensional_ models, the QCRB can be achieved _asymptotically_
with respect to the size $n$ of an ensemble of independent identically
prepared systems, by using a two steps _adaptive_ measurement strategy Gill
and Massar (2000). In the first step, a preliminary ‘rough’ estimator
$\tilde{\theta}_{n}$ is computed by measuring a sub-ensemble of
$\tilde{n}=o(n)$ systems, after which the SLD for parameter value
$\tilde{\theta}_{n}$ (our best guess at the optimal observable
$\mathcal{L}_{\theta}$) is measured on each of the remaining systems. In the
limit of large sample size $n$, the preliminary estimator $\tilde{\theta}_{n}$
approaches $\theta$ and the two step procedure achieves the QCRB in the sense
that the mean square error (MSE) of the final estimator scales as
$(nF(\theta))^{-1}$.

By implicitly invoking the above adaptive measurement argument, the quantum
estimation literature has largely focused on computing or estimating the QFI
of specific models, or designing input states which maximise the QFI in
quantum metrology settings. However, as shown in Górecki  _et al._ (2020), the
adaptive argument breaks down for models exhibiting quadratic (or Heisenberg)
scaling of the QFI where the _achievable_ MSE is larger by a constant factor
compared to the QCRB prediction, even asymptotically. In this work we show
that similar care needs to be taken even when considering standard estimation
problems involving ensembles of independent quantum systems and standard error
scaling.

#### Null measurements and their standard adaptive implementation

Figure 1: The figure illustrates the non-identifiability problem occurring
with null measurement (first row) and how it is fixed by displaced-null
measurement (second row). In the first column the red arc on the xz Bloch
sphere circle (in blue) represents the set of parameters after localisation
(confidence interval), the green disk represents the true parameter value
$\theta=\theta_{+}$ and the blue disk (panel a) is the parameter $\theta_{-}$
which is indistinguishable from the true one, in the null basis. The black
arrow represents the chosen measurement basis. The second column displays a
plot of the single count probability as a function of the parameter: in the
null measurement case such a function is not injective on the set of
parameters determined after the localisation (panel b). The third column shows
the phase space of a Gaussian model consisting of coherent states with unknown
displacement along the Q axis: the red interval is the parameter space, the
black dot corresponds to the number operator measured, the green disk to the
true coherent state and the blue disk (panel c) is the coherent state which is
indistinguishable from the true one in the null measurement case. The last
column plots the intensity of the number operator as a function of the
coherent state amplitude.

Specifically, we revisit the problem of estimating a parameter of a _pure
state model_ $\\{|\psi_{\theta}\rangle\\}_{\theta\in\Theta}$ and analyse a
measurement strategy Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et
al._ (2019), which we broadly refer to as _null measurement_. The premise of
the null measurement is the observation that if one measures
$|\psi_{\theta}\rangle$ in an orthonormal basis
$\mathcal{B}(\theta):=\\{|v_{1}\rangle,\dots,|v_{d}\rangle\\}$ such that
$|v_{1}\rangle=|\psi_{\theta}\rangle$ then the only possible outcome is $X=1$
and all other outcomes have probability zero. Since $\theta$ is unknown, in
practice one would measure in a basis $\mathcal{B}(\tilde{\theta})$
corresponding to an approximate value $\tilde{\theta}$ of the true parameter
$\theta$, and exploit the occurrence of low probability outcomes $X\neq 1$ in
order to estimate the deviation of $\theta$ from $\tilde{\theta}$. This
intuition is supported by the following property which is a specialisation to
one-dimensional parameters of a more general result derived in Pezze and
Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et al._ (2019): as
$\tilde{\theta}$ approaches $\theta$, the classical Fisher information
$I_{\tilde{\theta}}(\theta)$ associated with $\mathcal{B}(\tilde{\theta})$
converges to the QFI $F(\theta)$. This implies that null measurements can
achieve MSE rates scaling as $n^{-1}$ with constants that are _arbitrarily
close_ to $F^{-1}(\theta)$, by simply measuring all $n$ systems of an ensemble
in a basis $\mathcal{B}(\tilde{\theta})$ with a _fixed_ $\tilde{\theta}$ that
is close to $\theta$:

$n\mathbb{E}_{\theta}[(\hat{\theta}_{n}-\theta)^{2}]\to
I^{-1}_{\tilde{\theta}}(\theta)\approx F^{-1}(\theta).$

Do null measurements actually achieve the QCRB (asymptotically) or just ’come
close’ to it? In absence of a detailed multi-copy operational interpretation
in Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et al._ (2019), the
most natural strategy is to apply the same two step adaptive procedure which
worked well in the case of the SLD measurement. A preliminary estimator
$\tilde{\theta}_{n}$ is first computed by measuring $\tilde{n}$ systems and
the rest of the ensemble is subsequently measured in the basis
$\mathcal{B}(\tilde{\theta}_{n})$. Since $I_{\tilde{\theta}_{n}}(\theta)$
converges to $F(\theta)$ as $\tilde{\theta}_{n}$ approaches $\theta$, it would
appear that the QCRB is achieved asymptotically. One of our main results is to
show that this adaptive procedure actually _fails_ to achieve the QCRB even in
the simple qubit model

$|\psi_{\theta}\rangle=\cos\theta|0\rangle+\sin\theta|1\rangle,$ (1)

thus providing another example where caution is needed when using arguments
based on Fisher information, see Tsang  for other examples.

More precisely, we show that if the preliminary estimator $\tilde{\theta}_{n}$
is reasonably good (cf. section III for precise formulation), any final
estimator $\hat{\theta}_{n}$ computed from the outcomes of the null
measurement $\mathcal{B}(\tilde{\theta}_{n})$ is not only suboptimal but does
not even achieve the standard $n^{-1}$ estimation MSE rate. The reason for the
radically different behaviors of the SLD and null meaurement settings is that
the latter suffers from a _non-identifiability_ problem when the parameter
$\tilde{\theta}$ (which determines the null basis) is close to $\theta$.
Indeed, since at $\tilde{\theta}=\theta$ the null measurement has a
deterministic outcome, for $\tilde{\theta}\approx\theta$ the outcome
probabilities are quadratic in $\epsilon=\theta-\tilde{\theta}$ and therefore,
the parameters $\theta_{\pm}=\tilde{\theta}\pm\epsilon$ cannot be
distinguished (at least in second order). If $\tilde{\theta}_{n}$ is a
reasonably good estimator, then $\epsilon_{n}=|\theta-\tilde{\theta}_{n}|$ is
of the order $\tilde{n}^{-1/2}$, so the error in estimating $\theta$ is at
least of the order of the distance $|\theta_{+}-\theta_{-}|$ between the two
undistinguishable candidate parameters
$\theta_{\pm}=\tilde{\theta}_{n}\pm\epsilon_{n}$, which scales as
$\tilde{n}^{-1/2}$ instead of $n^{-1/2}$. Since $\tilde{n}=o(n)$ the mean
square error decreases slower that the standard rate $n^{-1}$. This argument
is illustrated in Figure 1a. for the simple case of the qubit rotation model
(1) which is discussed in detail in section III.

#### Asymptotic optimality of displaced-null measurements

Fortunately, the above explanation offers an intuitive solution to the non-
identifiability problem. Assuming that the preliminary estimator
$\tilde{\theta}_{n}$ satisfies standard concentration properties (e.g.
asymptotic normality), one finds that $\theta$ belongs (with high probability)
to a confidence interval $I_{n}$ centered at $\tilde{\theta}_{n}$, whose
length is slightly larger than the estimation uncertainty $\tilde{n}^{-1/2}$.
Therefore by displacing $\tilde{\theta}_{n}$ by a (vanishingly small) amount
$\delta_{n}>0$ that is larger than this uncertainty, we can make sure that
$I_{n}$ lies at the left side of
$\theta^{\prime}_{n}:=\tilde{\theta}_{n}+\delta_{n}$ and therefore measuring
in the basis $\mathcal{B}(\theta^{\prime}_{n})$ circumvents the non-
identifiability issue. This is illustrated in panels e. and f. of Figure 1.

The main aim of the paper is to investigate this method which we call a
_displaced-null_ measurement strategy and derive asymptotic optimality results
for the resulting estimators. In section IV.1 we show that the displaced-null
measurement achieves the QCRB in the one-parameter qubit model for which the
standard adaptive procedure failed; the corresponding second stage estimator
is a simple average of measurement outcomes and satisfies asymptotic
normality, thus allowing practitioners to define asymptotic confidence
intervals.

In section VI we extend the null-measurement strategy to _multi-parameter_
models of pure qudit states. In this case, the QCRB is typically not
attainable even asymptotically due to the incompatibility of optimal
measurements corresponding to different parameter components. However, we show
that the Holevo bound Holevo (2011) _can_ be achieved asymptotically. We first
consider the task of estimating a completely unknown pure state with respect
to the the Bures (fidelity) distance. In this case we show that the Holevo
bound can be achieved by using two separate displaced-null measurements, for
the real and imaginary parts of the state coefficients with respect to a basis
containing $|\psi_{\theta^{\prime}_{n}}\rangle$ as a vector. The second task
is to estimate a general $m$-dimensional model with respect to an arbitrary
locally quadratic distance on the parameter space. Here we show that the
Holevo bound is achievable by applying displaced-null measurements on copies
of the systems coupled with an ancilla in a fixed state. The proof relies on
the intuition gained from quantum local asymptotic normality theory and its
use in establishing the achievability of the Holevo bound Gill and Guţă
(2013); Demkowicz-Dobrzański  _et al._ (2020) by mapping the ensemble onto a
continuous variables system. However, unlike the latter, the displaced-null
technique only involves separate projective measurements on system-ancilla
pairs.

Finally, in section VI.6 we show that for multiparameter models where the QCRB
is achievable, this can be done using displaced-null measurements. This puts
related results of Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et
al._ (2019) on a firm operational basis.

#### Local asymptotic normality perspective

The theory of quantum local asymptotic normality (QLAN) Guţă and Kahn (2006);
Guţă and Jencova (2007); Kahn and Guţă (2009); Gill and Guţă (2013) offers an
alternative perspective on the displaced-null measurements strategy outlined
above. In broad terms, QLAN is a statistical tool that allows us to
approximate the i.i.d. model describing the joint state of an ensemble of
systems, by a single continuous variables Gaussian state whose mean encodes
information about the unknown parameter (cf. sections V.1 and VI.2 for more
details). By applying this approximation, the null measurement problem
discussed earlier can be cast into a Gaussian version formulated as follows.
Suppose we are given a one-mode continuous variables system prepared in a
coherent state $|u\rangle$ with unknown displacement $u\in\mathbb{R}$ along
the $Q$ axis, and assume that $|u|\leq a_{n}$ for some bound $a_{n}$ which
diverges with $n$. At $u=0$, the system state is the vacuum, and the
measurement of the number operator $N$ is a null measurement (see Figure 1c.).
However, for a given $u\neq 0$ the number operator has Poisson distribution
with intensity $|u|^{2}$, and therefore cannot distinguish between parameters
$u_{\pm}:=\pm u$, cf. Figure 1d. This means that any estimator will have large
MSEs of order $a_{n}^{2}$ for large values of $u$. In contrast, measuring the
quadrature $Q$ produces (optimal) estimators with fixed MSE given by the
vacuum fluctuations. However, the non-identifiability problem of the counting
measurement can be lifted by displacing the coherent state along the $Q$ axis
by an amount $\Delta_{n}>a_{n}$ and then measuring $N$. Equivalently, one can
measure the corresponding displaced number operator on the original coherent
state as illustrated in panels g. and h. of Figure 1. In this case the
intensity $(u-\Delta_{n})^{2}$ is in one-to-one correspondence with $u$ so the
parameter is identifiable. Moreover, for large $n$, the counting measurement
can be linearised and becomes equivalent to measuring the quadrature $Q$, a
well known fact from homodyne detection Leonhardt (1997).

QLAN shows that the Gaussian problem discussed above is the asymptotic version
of the one-parameter qubit rotation model (1) which we used earlier to
illustrate the concept of approximate and displaced null measurements. The
coherent state $|u\rangle$ corresponds to all qubits in the state
$|\psi_{u/\sqrt{n}}\rangle$ (assuming for simplicity that
$\tilde{\theta}_{n}=0$ and writing $\theta=u/\sqrt{n}$). The number operator
corresponds to measuring in the standard basis, which is an exact null
measurement at $u=0$. On the other hand, the displaced number operator
corresponds to measuring in the rotated basis with angle
$\delta_{n}=n^{-1/2}\Delta_{n}$.

The same Gaussian correspondence is used in section VI for more general
problems involving multiparameter estimation for pure qudit state models and
establishing the achievability of the Holevo bound, cf. Theorem 2. The general
strategy is to translate the i.i.d. problem into a Gaussian one, solve the
latter by using displaced number operators in a specific mode decomposition
and then translate this into qudit measurement with respect to specific
rotated bases.

This paper is organised as follows. Section II reviews the QCRB and the
conditions for its achievability. In section III we show that null
measurements based at reasonable preliminary estimators fail to achieve the
standard error scaling. In section IV.1 we introduce the idea of displaced-
null measurement and prove its optimality in the paradigmatic case of a one-
parameter qubit model. In section VI we treat the general case of $d$
dimensional systems and show how the Holevo bound is achieved on general
models, and deal with the case where the multi-parameter QCRB is achievable.

##  II Achievability of the quantum Cramér-Rao bound for pure states

In this section we review the quantum Cramér-Rao bound (QCRB) and the
conditions for its achievability in the case of models with _one-dimensional_
parameters, which will be relevant for the first part of the paper.

The estimation of multidimensional models and the corresponding Holevo bound
is discussed in section VI.

Consider a quantum statistical model given by a family of d-dimensional
density matrices $\rho_{\theta}$ which depend smoothly on an unknown parameter
$\theta\in\mathbb{R}$. Let $\mathcal{M}$ be a measurement on $\mathbb{C}^{d}$
with positive operator valued measure (POVM) elements
$\\{M_{0},\dots,M_{p}\\}$. By measuring $\rho_{\theta}$ we obtain an outcome
$X\in\\{0,\dots,p\\}$ with probabilities

$p_{\theta}(X=i)=p_{\theta}(i)={\rm Tr}(M_{i}\rho_{\theta}),\qquad
i=0,\dots,p.$

The classical Cramér-Rao bound states that the variance of any _unbiased_
estimator $\hat{\theta}=\hat{\theta}(X)$ of $\theta$ is lower bounded as

${\rm Var}(\hat{\theta}):=\mathbb{E}_{\theta}[(\hat{\theta}-\theta)^{2}]\geq
I_{\mathcal{M}}(\theta)^{-1}$ (2)

where $I_{\mathcal{M}}(\theta)$ is the classical Fisher information (CFI)

$I_{\mathcal{M}}(\theta)=\mathbb{E}_{\theta}\left[\left(\frac{d\log
p_{\theta}}{d\theta}\right)^{2}\right]=\sum_{i:p_{\theta}(i)>0}p_{\theta}^{-1}(i)\left(\frac{dp_{\theta}(i)}{d\theta}\right)^{2}.$
(3)

The CFI associated to any measurement is upper bounded by the quantum Fisher
information (QFI) Braunstein and Caves (1994); Braunstein  _et al._ (1996)

$I_{\mathcal{M}}(\theta)\leq F(\theta)$ (4)

where $F(\theta)={\rm Tr}(\rho_{\theta}\mathcal{L}^{2}_{\theta})$ and
$\mathcal{L}_{\theta}$ is the symmetric logarithmic derivatives (SLD) defined
by the Lyapunov equation

$\frac{d{\rho}_{\theta}}{d\theta}=\frac{1}{2}(\mathcal{L}_{\theta}\rho_{\theta}+\rho_{\theta}\mathcal{L}_{\theta}).$

By putting together (2) and (4) we obtain the quantum Cramér-Rao bound (QCRB)
Holevo (2011); Helstrom (1976)

${\rm Var}(\hat{\theta}):=\mathbb{E}_{\theta}[(\hat{\theta}-\theta)^{2}]\geq
F(\theta)^{-1}.$ (5)

which sets a fundamental limit to the estimation precision. A similar bound on
the covariance matrix of an unbiased estimator holds for multidimensional
models, cf. section VI.

An important question is which measurements saturate the bound (4), and what
is the statistical interpretation of the corresponding QCRB (5). For
completeness, we state the exact conditions in the following Proposition whose
formulation is adapted from Zhou  _et al._ (2018b). The proof is included in
appendix A.

######  Proposition 1.

Let $\rho_{\theta}$ be a one-dimensional quantum statistical model and let
$\mathcal{M}:=\\{M_{0},\dots,M_{p}\\}$ be a measurement with probabilities
$p_{\theta}(i):={\rm Tr}(\rho_{\theta}M_{i})$. Then $\mathcal{M}$ achieves the
bound (4) if and only if the following conditions hold:

1) if $p_{\theta}(i)>0$ there exists $\lambda_{i}\in\mathbb{R}$ such that

$M^{1/2}_{i}\rho^{1/2}_{\theta}=\lambda_{i}M_{i}^{1/2}\mathcal{L}_{\theta}\rho^{1/2}_{\theta}$
(6)

2) if $p_{\theta}(i)=0$ for some $i$ then ${\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta}\mathcal{L}_{\theta})=0$.

One can check that the conditions in Proposition 1 are satisfied, and hence
the bound (4) is saturated, if $\mathcal{M}$ is the measurement of the
observable $\mathcal{L}_{\theta}$. However, in general this observable depends
on the unknown parameter, so achieving the QFI does not have an immediate
statistical interpretation. Nevertheless, one can provide a meaningful
operational interpretation in the scenario in which a large number $n$ of
copies of $\rho_{\theta}$ is available. In this case one can apply the
adaptive scheme presented in the introduction: using a (small) sub-sample to
obtain a ‘rough’ preliminary estimator $\tilde{\theta}$ of $\theta$ and then
measuring $\mathcal{L}_{\tilde{\theta}}$ on the remaining copies. This
adaptive procedure provides estimators $\hat{\theta}_{n}$ which achieve the
Cramér-Rao bound asymptotically in the sense that (see e.g. Gill and Massar
(2000); Gill and Guţă (2013))

$n\mathbb{E}_{\theta}[(\hat{\theta}_{n}-\theta)^{2}]\to F^{-1}(\theta).$

_Pure state models._ While for full rank states $(\rho_{\theta}>0)$ the second
condition in Proposition 1 is irrelevant, this is not the case for rank
deficient states, and in particular for pure state models.

Indeed let us assume that the model consists of pure states
$\rho_{\theta}=|\psi_{\theta}\rangle\langle\psi_{\theta}|$ and let us choose
the phase dependence of the vector state such that
$\langle\dot{\psi}_{\theta}|\psi_{\theta}\rangle=0$ (alternatively, one can
use
$|\psi^{\perp}_{\theta}\rangle:=|\dot{\psi}_{\theta}\rangle-\langle\psi_{\theta}|\dot{\psi}_{\theta}\rangle|\psi_{\theta}\rangle$
instead of $|\dot{\psi}_{\theta}\rangle$ in the equations below). Then

$\mathcal{L}_{\theta}=2(|\dot{\psi}_{\theta}\rangle\langle\psi_{\theta}|+|\psi_{\theta}\rangle\langle\dot{\psi}_{\theta}|),\quad{\rm
and}\quad F(\theta)=4\|\dot{\psi}_{\theta}\|^{2}.$

Let $\mathcal{M}$ to be a projective measurement with
$M_{i}=|v_{i}\rangle\langle v_{i}|$ where
$\mathcal{B}:=\\{|v_{0}\rangle,\dots,|v_{d-1}\rangle\\}$ is an orthonormal
basis (ONB). Without loss of generality we can choose the phase factors such
that $\langle v_{i}|\psi_{\theta}\rangle\in\mathbb{R}$ at the particular value
of interest $\theta$. Equation (6) in Proposition 1 becomes $\langle
v_{i}|\dot{\psi}_{\theta}\rangle\in\mathbb{R},$ i.e. in the first order, the
statistical model is in the real span of the basis vectors. Condition 2
requires that if $\langle v_{i}|\psi_{\theta}\rangle=0$ then $\langle
v_{i}|\dot{\psi}_{\theta}\rangle=0$. Intuitively, this implies that, in the
first order, the model is restricted to the real subspace spanned by the basis
vectors with positive probabilities. For example if

$|\psi_{\theta}\rangle:=\cos\theta|0\rangle+\sin\theta|1\rangle\in\mathbb{C}^{2},$
(7)

then any measurement with respect to an ONB consisting of superpositions of
$|0\rangle$ and $|1\rangle$ with _nonzero_ _real_ coefficients achieves the
QCRB at $\theta=0$, and no other measurement does so. This model will be
discussed in detail in sections III and IV.

_Null measurements._ We now formally introduce the concept of a _null
measurement_ which will be the focus of our investigation. The general idea is
to choose a measurement basis such that one of its vectors is equal or close
to the unknown state. In this case, the corresponding outcome has probability
close to one while the occurrence of other outcomes can serve as a ‘signal’
about the deviation from the true state. Let us consider first an _exact null
measurement_ , i.e. one in which the measurement basis
$\mathcal{B}=\mathcal{B}(\theta)$ is chosen such that
$|v_{0}\rangle=|\psi_{\theta}\rangle$, e.g in the example in equation (7) the
null measurement at $\theta=0$ is determined by the standard basis. Such a
measurement does not satisfy the conditions for achieving the QCRB. Indeed, we
have $p_{\theta}(i)=\delta_{0,i}$ and condition 2 implies $\langle
v_{i}|\dot{\psi}_{\theta}\rangle=0$ for all $i=1,\dots,d-1$. However this is
impossible given that $|v_{0}\rangle=|\psi_{\theta}\rangle$ and
$\langle\dot{\psi}_{\theta}|\psi_{\theta}\rangle=0$. In fact, the exact null
measurement has zero CFI, which implies that there exists no (locally)
unbiased estimator. Indeed, since probabilities belong to $[0,1]$, and
$p_{\theta}(i)$ is either $0$ or $1$ for a null measurement, all first
derivatives at $\theta$ are zero so the CFI (3) is equal to zero, i.e.
$I_{\mathcal{B}(\theta)}(\theta)=0$.

One can rightly argue that the exact null measurement as defined above is not
an operationally useful concept and cannot be implemented experimentally as it
requires the exact knowledge of the unknown parameter. However, in a multi-
copy setting the measurement _can_ incorporate information about the
parameter, as this can be obtained by measuring a sub-ensemble of systems in a
preliminary estimation step, similarly to the SLD case. It is therefore
meaningful to consider _approximate null_ measurements, which satisfy the null
property at $\tilde{\theta}\approx\theta$, i.e. we measure in a basis
$\mathcal{B}(\tilde{\theta})=\\{|v_{0}^{\tilde{\theta}}\rangle,\dots,|v_{d-1}^{\tilde{\theta}}\rangle\\}$
with $|v_{0}^{\tilde{\theta}}\rangle=|\psi_{\tilde{\theta}}\rangle.$
Interestingly, while the exact null measurement has zero CFI, an approximate
null measurement $\mathcal{B}(\tilde{\theta})$ ‘almost achieve’ the QCRB in
the sense that the corresponding classical Fisher information
$I_{\mathcal{B}(\tilde{\theta})}(\theta)$ converges to $F(\theta)$ as
$\tilde{\theta}$ approaches $\theta$ Pezze and Smerzi (2014); Pezzè  _et al._
(2017); Liu  _et al._ (2019). This means that by using an approximate null
measurement we can achieve asymptotic error rates arbitrarily close (but not
equal) to the QCRB, by measuring in a basis $\mathcal{B}(\tilde{\theta})$ with
a fixed $\tilde{\theta}$ close to $\theta$.

The question is then, is it possible to achieve the QCRB asymptotically with
respect to the sample size by employing null measurements determined by an
_estimated_ parameter value, as in the case of the SLD measurement? References
Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu  _et al._ (2019) do not
address this question, aside from the above Fisher information convergence
argument.

To answer the question we allow for measurements which have the null property
at parameter values determined by _reasonable_ preliminary estimators based on
measuring a sub-sample of a large ensemble of identically prepared systems
(cf. section III for precise definition). We investigate such measurement
strategies and show that the natural two step implementation – use the rough
estimator as a vector in the second step measurement basis – _fails_ to
achieve the standard rate $n^{-1/2}$ on simple qubit models. We will see that
this is closely related to the fact that the CFI of the exact null measurement
is zero, unlike the SLD case.

Nevertheless, in section IV.1 we show that a modified strategy which we call a
_displaced-null measurement_ does achieve asymptotic optimality in the simple
qubit model discussed above. This scheme is then extended to general
multidimensional qudit models in section VI and shown to achieve the Holevo
bound for general multiparameter models.

##  III Why the naive implementation of a null measurement does not work

In this section we analyse the _null measurement_ scheme described in section
II, for the case of a simple one-parameter qubit rotation model. The main
result is Theorem 1 which shows that the naive/natural implementation of the
null-fails to achieve the QCRB.

Let

$|\psi_{\theta}\rangle=e^{-i\theta\sigma_{y}}|0\rangle=\cos(\theta)|0\rangle+\sin(\theta)|1\rangle$
(8)

be a one-parameter family of pure states which describes a circle in the $xz$
plane of the Bloch sphere. To simplify some of the arguments below we will
assume that $\theta$ is known to be in the open interval
$\Theta=(-\pi/8,\pi/8)$, but the analysis can be extended to completely
unknown $\theta$. The quantum Fisher information is

$F(\theta)=4{\rm
Var}(\sigma_{y})=4\langle\psi_{\theta}|\sigma_{y}^{2}|\psi_{\theta}\rangle-4\langle\psi_{\theta}|\sigma_{y}|\psi_{\theta}\rangle^{2}=4.$

We now consider the specific value $\theta=0$, so $|\psi_{0}\rangle=|0\rangle$
and $|\dot{\psi}_{0}\rangle=|1\rangle$. According to Proposition 1 any
measurement with respect to a basis consisting of real combinations of
$|0\rangle$ and $|1\rangle$ achieves the QCRB, with the exception of the basis
$\\{|0\rangle,|1\rangle\\}$ itself. Indeed, let

$|v_{0}^{\tau}\rangle=\exp(-i\tau\sigma_{y})|0\rangle,\quad|v_{1}^{\tau}\rangle=\exp(-i\tau\sigma_{y})|1\rangle$
(9)

be such a basis ($\tau\neq 0$) , then the probability distribution is

$p_{\theta}(0)=\cos^{2}(\theta-\tau),\qquad
p_{\theta}(1)=\sin^{2}(\theta-\tau)$

and the classical Fisher information is

$I_{\tau}(\theta=0)=\mathbb{E}_{\theta=0}\left[\left(\frac{d\log
p_{\theta}}{d\theta}\right)^{2}\right]=4.$

However, at $\tau=0$ we have $I_{0}(\theta=0)=0$ in agreement with the general
fact that exact null measurements have zero CFI. This reveals a curious
singularity in the space of optimal measurements, and our goal is to
understand to what extent this is mathematical artefact or it has a deeper
statistical significance.

To start, we note that the failure of the standard basis measurement can also
be understood as a consequence of parameter _non-identifiability_ around the
parameter value $0$. Indeed, for $\tau=0$ we have
$p_{\theta}(i)=p_{-\theta}(i)$ so this measurement cannot distinguish $\theta$
from $-\theta$. A similar issue exists for $\tau\neq 0$, if $\theta$ is
assumed to be completely unknown, or in an interval containing $\tau$, cf.
Figure 1. On the other hand, if $\theta$ is _known_ to belong to an interval
$I$ and $\tau$ is outside this interval, then the parameter _is_ identifiable
and the standard asymptotic theory applies. For instance, measuring
$\sigma_{x}$ leads to an identifiable statistical model for our quantum qubit
model.

Consider now the following two step procedure, which arguably is the most
natural way of implementing approximate-null measurements. A sub-ensemble of
$\tilde{n}$ systems is used to compute a preliminary estimator
$\tilde{\theta}_{n}$, and subsequently the remaining samples are measured in
the null-basis at angle $\tau=\tilde{\theta}_{n}$. For concreteness we assume
that $\tilde{n}=n^{1-\epsilon}$ for some small constant $\epsilon>0$, but our
results hold more generally for $\tilde{n}=o(n)$ and $\tilde{n}\to\infty$ with
$n$.

To formulate our theoretical result, we use the language of Bayesian
statistics which we temporarily adopt for this purpose. We consider that the
unknown parameter $\theta$ is random and is drawn from the uniform _prior
distribution_ $\pi(d\theta)=\frac{4}{\pi}d\theta$ over the parameter space
$\Theta$. Adopting a Bayesian notation we let
$p(d\tilde{\theta}_{n}|\theta):=p_{\theta}(d\tilde{\theta}_{n})$ be the
distribution of $\tilde{\theta}_{n}$ given $\theta$. The joint distribution of
$(\theta,\tilde{\theta}_{n})$ is then

$p(d\theta,d\tilde{\theta}_{n})=\pi(d\theta)p(d\tilde{\theta}_{n}|\theta)=p(d\tilde{\theta})\pi(d\theta|\tilde{\theta}_{n})$

where $\pi(d\theta|\tilde{\theta}_{n})$ is the _posterior distribution_ of
$\theta$ given $\tilde{\theta}_{n}$.

Reasonable estimator hypothesis: we assume that $\tilde{\theta}_{n}$ is a
_reasonable estimator_ in the sense that the following conditions are
satisfied for every $n\geq 1$:

  1. 1.

$\pi(d\theta|\tilde{\theta}_{n})$ has a density
$\pi(\theta|\tilde{\theta}_{n})$ with respect to the Lebesgue measure;

  2. 2.

For each $n$ there exist a set $A_{n}\subseteq\Theta$ such that
$\mathbb{P}(\tilde{\theta}_{n}\in A_{n})>c$ for some constant $c>0$, and the
following condition holds: for each $\tilde{\theta}_{n}\in A_{n}$, the
positive symmetric function

$g_{n,\tilde{\theta}_{n}}(r):={\rm
min}\\{\pi(\tilde{\theta}_{n}+r|\tilde{\theta}_{n}),\pi(\tilde{\theta}_{n}-r|\tilde{\theta}_{n})\\}$

satisfies

$\int_{r\geq\tau_{n}}\,g_{n,\tilde{\theta}_{n}}(r)dr\geq C$ (10)

where $\tau_{n}:=n^{-1/2+\epsilon/4}$ and $C>0$ is a constant independent on
$n$ and $\tilde{\theta}_{n}$.

Condition 2. means that the posterior distribution has significant mass on
_both_ sides of the preliminary estimator $\tilde{\theta}_{n}$, at a distance
which is larger than $n^{-1/2+\epsilon/4}$, as illustrated in Figure 2. Since
standard estimators such as maximum likelihood have asymptotically normal
posterior distribution with standard deviation
$\tilde{n}^{-1/2}=n^{-1/2+\epsilon/2}\gg n^{-(1-\epsilon/2)/2}$, condition 2.
is expected to hold quite generally, hence the name reasonable estimator. The
following lemma shows that the natural estimator in our model is indeed
reasonable.

######  Lemma 1.

Consider the measurement of $\sigma_{x}$ on a sub-ensemble of
$\tilde{n}=n^{1-\epsilon}$ systems, and let $\tilde{\theta}_{n}$ be the
maximum likelihood estimator. Then $\tilde{\theta}_{n}$ is a reasonable
estimator.

The proof of Lemma 1 can be found in Appendix C. The method can be extended to
a wide class of estimators, since it essentially relies on assumptions which
are quite standard in usual statistical problems.

The next Theorem is the main result of this section and shows that if a
reasonable (preliminary) estimator is used as reference for a null measurement
on the remaining samples, the MSE of the final estimator cannot achieve the
QCRB, indeed it cannot even achieve standard scaling.

Figure 2: For a reasonable estimator $\tilde{\theta}_{n}$, the posterior
distribution of $\theta$ is centred around $\tilde{\theta}_{n}$, and has width
of order $n^{-(1-\epsilon)/2}$. The assumption amounts to the fact that the
posterior has non-vanishing mass on either side of $\tilde{\theta}_{n}$ at
distance larger than $n^{-(1-\epsilon/2)/2}$ which is much smaller that the
typical standard deviation.

######  Theorem 1.

Assume that $\tilde{\theta}_{n}$ is a reasonable estimator as defined above,
obtained by measuring a sub-ensemble of size $\tilde{n}:=n^{1-\epsilon}$. Let
$\hat{\theta}_{n}$ be an estimator of $\theta$ based on measuring the
remaining $n-n^{1-\epsilon}$ sub-ensemble in the basis corresponding to angle
$\tilde{\theta}_{n}$. Then

$\lim_{n\to\infty}nR_{\pi}(\hat{\theta}_{n})=\infty$

where

$R_{\pi}(\hat{\theta}_{n})=\int_{\Theta}\pi(d\theta)\mathbb{E}_{\theta}[(\hat{\theta}_{n}-\theta)^{2}]$

is the average mean square error risk.

The proof of Theorem 1 can be found in Appendix D.

The fact that a reasonable estimator has a ‘balanced’ posterior was key in
obtaining the negative result in Theorem 1. This encodes the fact that the
null measurement cannot distinguish between possible parameter values
$\theta=\tilde{\theta}_{n}+\tau_{n}$ and $\theta=\tilde{\theta}_{n}-\tau_{n}$
leading to errors that are larger than $n^{-1/2}$. In section IV we show how
we can go around this problem by deliberately choosing the reference parameter
of the null measurement to be displaced away from a reasonable estimator
$\tilde{\theta}_{n}$ by an amount $\delta_{n}$ that is large enough to insure
identifiability, but small enough to still be in a shrinking neighbourhood of
$\theta$.

In the proof of Theorem 1 we made use of the fact that, for the statistical
model defined in equation (8), the law of the measurement in the basis
containing ${|{\psi_{\tilde{\theta}_{n}}}\rangle}$ could not distinguish
between $\tilde{\theta}_{n}\pm r$. Although for general pure state models this
might not be the case, in the appendix E we show that under some mild
additional assumptions, the result of Theorem 1 extends to weaker notions of
non-identifiability.

##  IV Displaced-null estimation scheme for optimal estimation of pure qubit
states

In section III we showed that a null measurement that uses a reasonable
preliminary estimator as reference parameter is sub-optimal. We will now show
that one can achieve the asymptotic version of the QCRB (5) by employing a
null measurement at a reference parameter that is _deliberately shifted_ away
from the reasonable estimator by a certain amount. We will call these
_displaced-null_ measurements.

###  IV.1 The displaced-null measurement for one parameter qubit models

We consider the one parameter model $|\psi_{\theta}\rangle$ defined in
equation (8) and assume that we are given $n$ identical copies of
$|\psi_{\theta}\rangle$. We apply the usual two step adaptive procedure: in
the first step we use a vanishingly small proportion of the samples containing
$\tilde{n}=n^{1-\epsilon}$ copies (where $\epsilon>0$ is a small parameter) to
perform a preliminary (non-optimal) estimation producing a reasonable
estimator $\tilde{\theta}_{n}$. For concreteness we assume that
$\tilde{\theta}_{n}$ is the estimator described in Lemma 1. Using Hoeffding’s
bound we find that $\tilde{\theta}_{n}$ satisfies the concentration bound

$\mathbb{P}_{\theta}(|\tilde{\theta}_{n}-\theta|>n^{-1/2+\epsilon})\leq
Ce^{-n^{\epsilon}r}$ (11)

for some constants $C,r>0$. This means that with high probability, $\theta$
belongs to the confidence interval
$I_{n}=(\tilde{\theta}_{n}-n^{-1/2+\epsilon},~{}\tilde{\theta}_{n}+n^{-1/2+\epsilon})$
whose size shrinks at a slightly slower rate than $n^{-1/2}$.

In the second step we would like to measure all remaining qubits in a basis
which contains a vector that is close to $|\psi_{\theta}\rangle$. However, as
argued in section III, the null measurement basis
$\\{|v^{\tilde{\theta}_{n}}_{0}\rangle,|v^{\tilde{\theta}_{n}}_{1}\rangle\\}$
satisfying
$|v^{\tilde{\theta}_{n}}_{0}\rangle=|\psi_{\tilde{\theta}_{n}}\rangle$ is
suboptimal. More generally, for any angle $\tau\in I_{n}$, the basis defined
by equation (9) suffers an identifiability problem as illustrated in panels a.
and b. of Figure 1. For this reason, in the second step we choose the
reference value

$\theta^{\prime}_{n}:=\tilde{\theta}_{n}+\delta_{n},\quad\delta_{n}:=n^{-1/2+3\epsilon},$

such that $\theta^{\prime}_{n}$ is well outside $I_{n}$ but nevertheless,
$\theta^{\prime}_{n}\to\theta$ for large $n$ (assuming $\epsilon<1/6$). The
$3\epsilon$ factor in the exponent is chosen such that the result of
Proposition 2 below holds, but any factor larger than $2\epsilon$ suffices. We
measure all remaining samples in the basis
$\\{|v_{0}^{\theta_{n}^{\prime}}\rangle,|v_{1}^{\theta_{n}^{\prime}}\rangle\\}$
(cf eq. (9)) to obtain outcomes $X_{1},\dots,X_{n}\in\\{0,1\\}$ with
probability distribution

$P^{(n)}_{\theta}=(1-p^{(n)}_{\theta},p^{(n)}_{\theta}),\quad
p^{(n)}_{\theta}=\sin^{2}(\theta-\theta^{\prime}_{n}).$

######  Proposition 2.

Assume that $\Theta$ is bounded and $\epsilon<1/10$ is fixed, and let
$\tilde{\theta}_{n}$ be the preliminary estimator based on
$\tilde{n}=n^{1-\epsilon}$ samples.

Let $\hat{\theta}_{n}$ be the estimator

$\hat{\theta}_{n}:=\tilde{\theta}_{n}+\frac{n^{-1/2+3\epsilon}}{2}-\frac{n^{1/2-3\epsilon}}{2}\hat{p}_{n}$

where $\hat{p}_{n}$ is the empirical estimator of $p^{(n)}_{\theta}$, i.e.

$\hat{p}_{n}=\frac{|\\{i:X_{i}=1,~{}i=1,\dots,n\\}|}{n}.$ (12)

Then $\hat{\theta}_{n}$ is asymptotically optimal in the sense that

$\lim_{n\to\infty}n\mathbb{E}_{\theta}[(\hat{\theta}_{n}-\theta)^{2}]=F^{-1}(\theta)=\frac{1}{4}.$

Moreover, $\hat{\theta}_{n}$ is asymptotically normal, i.e.

$\sqrt{n}(\hat{\theta}_{n}-\theta)\to N\left(0,\frac{1}{4}\right)$

where the convergence holds in distribution.

The proof of Proposition 2 can be found in Appendix F. Note that we chose to
identify $n$ and $n^{\prime}=n-n^{1-\epsilon}$ in order to simplify the
notation and the proofs, but it is immediate to adapt the reasoning in order
to deal with this technicality. We also remark that the assumption
$\epsilon<1/10$ is not essential and could be removed at the price of using
more involved analysis of the concentration properties of $\tilde{\theta}_{n}$
and the definition of the displacement parameter $\delta_{n}$.

##  V Displaced-null measurements in the asymptotic Gaussian picture

In this section we cast the null-measurement problem into a companion Gaussian
estimation problem which arises in the limit of large sample sizes. The
Gaussian approximation is described by the theory of quantum local asymptotic
normality (QLAN) developed in Guţă and Kahn (2006); Kahn and Guţă (2009); Guţă
and Jencova (2007); Gill and Guţă (2013). For reader’s convenience we review
the special case of pure qubit states in section V.1.

###  V.1 Brief review of local asymptotic normality for pure qubit states

The QLAN theory is closely related to the quantum Central Limit Theorem (QCLT)
and shows that for large $n$ the statistical model describing ensembles of $n$
identically prepared qubits can be approximated (locally in the parameter
space) by a single coherent state of a one-mode continuous variables (cv)
system, whose mean encodes the unknown qubit rotation angle. We refer to Guţă
and Kahn (2006); Guţă  _et al._ (2008) for mathematical details and focus here
on the intuitive correspondence between qubit ensembles and the cv mode.

We start with a completely unknown pure qubit state described by a one-
dimensional projection $P=|\psi\rangle\langle\psi|$. In the first step we
measure a sub-sample of $\tilde{n}=n^{1-\epsilon}$ systems and obtain a
preliminary estimator
$\tilde{P}_{n}=|\tilde{\psi}_{n}\rangle\langle\tilde{\psi}_{n}|$. We assume
that $\tilde{P}_{n}$ satisfies a concentration bound similar to the one in
equation (11) so that $P$ lies within a ball of size $n^{-1/2+\epsilon}$
around $\tilde{P}_{n}$ with high probability. For more about the localisation
procedure we refer to Appendix B.

We now choose the ONB $\\{|0\rangle,|1\rangle\\}$ such that
$|0\rangle:=|\tilde{\psi}_{n}\rangle$.

Thanks to parameter localisation we can focus our attention on ‘small’
rotations around $|0\rangle$ whose magnitude is of the order
$n^{-1/2+\epsilon}$ where $n$ is the sample size and $\epsilon>0$ is small. We
parametrise such states as

$|\psi_{{\bm{u}}/\sqrt{n}}\rangle:=U\left(\frac{\bm{u}}{\sqrt{n}}\right)|0\rangle=e^{-i(u_{1}\sigma_{y}-u_{2}\sigma_{x})/\sqrt{n}}|0\rangle,$

where ${\bm{u}}=(u_{1},u_{2})$ is a two-dimensional local parameter of
magnitude $|{\bm{u}}|<n^{\epsilon}$. The joint state of the ensemble of $n$
identically prepared qubits is then

$|\Psi_{\bm{u}}^{n}\rangle=|\psi_{{\bm{u}}/\sqrt{n}}\rangle^{\otimes n}.$

We now describe the _Gaussian shift model_ which approximates the i.i.d. qubit
model in the large sample size limit. A one mode cv system is specified by
canonical coordinates $Q,P$ satisfying $[Q,P]=i\mathbb{1}$. These act on a
Hilbert space $\mathcal{H}$ with a orthonormal Fock basis $\\{|k\rangle:k\geq
0\\}$, such that $a|k\rangle=\sqrt{k}|k-1\rangle$, where $a$ is the
annihilation operator $a=(Q+iP)/\sqrt{2}$. The coherent states are defined as

$|z\rangle:=e^{-|z|^{2}/2}\sum_{k=0}^{\infty}\frac{z^{k}}{\sqrt{k!}}|k\rangle,\quad
z\in\mathbb{C}$

and satisfy $\langle z|a|z\rangle=z$. In the coherent state ${|{z}\rangle}$,
the canonical coordinates $Q,P$ have normal distributions $N\left(\sqrt{2}{\rm
Re}z,\,\frac{1}{2}\right)$ and $N\left(\sqrt{2}{\rm
Im}z,\,\frac{1}{2}\right)$, respectively. In addition, the number operator
$N:=a^{*}a$ has Poisson distribution with intensity $|z|^{2}$.

We now outline two approaches to QLAN embodying different ways to express the
closeness of the multiqubits model
$\\{|\Psi_{\bm{u}}^{n}\rangle:|{\bm{u}}|\leq n^{\epsilon}\\}$ to the quantum
Gaussian shift model $\\{|u_{1}+iu_{2}\rangle:|{\bm{u}}|\leq n^{\epsilon}\\}$.
By applying the QCLT Petz (2008), one shows that the collective spin in the
‘transverse’ directions x and y have asymptotically normal distributions

$\displaystyle\frac{1}{\sqrt{2n}}S_{x}(n):=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}\sigma_{x}^{(i)}$
$\displaystyle\to$ $\displaystyle N\left(\sqrt{2}u_{1},\frac{1}{2}\right)$
$\displaystyle\frac{1}{\sqrt{2n}}S_{y}(n):=\frac{1}{\sqrt{2n}}\sum_{i=1}^{n}\sigma_{y}^{(i)}$
$\displaystyle\to$ $\displaystyle N\left(\sqrt{2}u_{2},\frac{1}{2}\right)$

where the arrows represent convergence in distribution with respect to
$|\Psi^{n}_{u}\rangle$. In fact the convergence holds for the whole ‘joint
distribution’ which we write symbolically as

$\left(\frac{1}{\sqrt{2n}}S_{x}(n),\frac{1}{\sqrt{2n}}S_{y}(n)\,:\,|\Psi_{\bm{u}}^{n}\rangle\right)\to\left(Q,P\,:\,|u_{1}+iu_{2}\rangle\right).$

So, in what concerns the collective spin observables, the joint qubit state
converges to a coherent state whose displacement is linear with respect to the
local rotation parameters.

An alternative way to formulate the convergence to the Gaussian model is to
show that the two models can be mapped into each other by means of physical
operations (quantum channels) with asymptotically vanishing error, uniformly
over all local parameters $|{\bm{u}}|\leq n^{\epsilon}$. Consider the
isometric embedding of the symmetric subspace
$\mathcal{S}_{n}=(\mathbb{C}^{2})^{\otimes_{s}^{n}}$ of the tensor product
$(\mathbb{C}^{2})^{\otimes n}$ into the Fock space

$\displaystyle V_{n}:=\mathcal{S}_{n}$ $\displaystyle\to$
$\displaystyle\mathcal{H}$ $\displaystyle|k,n\rangle$ $\displaystyle\mapsto$
$\displaystyle|k\rangle$

where $|k,n\rangle$ is the normalised projection of the vector
$|1\rangle^{\otimes k}\otimes|0\rangle^{\otimes n-k}$ onto $\mathcal{S}_{n}$.
The following limits hold Guţă and Kahn (2006)

$\displaystyle\lim_{n\to\infty}\sup_{|{\bm{u}}|\leq
n^{1/2-\eta}}\left\|V_{n}|\Psi_{\bm{u}}^{n}\rangle-|u_{1}+iu_{2}\rangle\right\|=0,$
$\displaystyle\lim_{n\to\infty}\sup_{|{\bm{u}}|\leq
n^{1/2-\eta}}\left\||\Psi_{\bm{u}}^{n}\rangle-
V_{n}^{*}|u_{1}+iu_{2}\rangle\right\|=0.$

where $\eta>0$ is an arbitrary fixed parameter. In particular, for
$\eta<1/2-\epsilon$ the supremum is taken over regions that contain all
$|{\bm{u}}|<n^{\epsilon}$, which means that the Gaussian approximation holds
uniformly over all values of the local parameter arising from the preliminary
estimation step.

We now move to describe the relationship between qubit rotations and Gaussian
displacements in the QLAN approximation. Let
$U^{n}(\bm{\Delta}):=U(n^{-1/2}\bm{\Delta})^{\otimes n}$ be a qubit rotation
by small angles ${\bm{\delta}}:=n^{-1/2}\bm{\Delta}$ and let
$D(\bm{\Delta})=\exp(-i\sqrt{2}(\Delta_{1}P-\Delta_{2}Q))$ be the
corresponding displacement operator. Then the following commutative diagram
shows how QLAN translates (small) rotations into displacements (asymptotically
with $n$ and uniformly over local parameters)

$\begin{CD}|\Psi_{\bm{u}}^{n}\rangle @>{V_{n}}>{}>|u_{1}+iu_{2}\rangle\\\
@V{}V{U^{n}(-\bm{\Delta})}V@V{}V{D(-\bm{\Delta})}V\\\
|\Psi_{\bm{u}-\bm{\Delta}}^{n}\rangle
@>{V_{n}}>{}>|u_{1}-\Delta_{1}+i(u_{2}-\Delta_{2})\rangle\end{CD}$

Notice that also the vertical arrow on the left of the diagram is true
asymptotically with $n$ and has to be intended as
$\lim_{n\rightarrow+\infty}\|U^{n}(-\bm{\Delta})|\Psi_{\bm{u}}^{n}\rangle-|\Psi_{\bm{u}-\bm{\Delta}}^{n}\rangle\|=0$.

Finally, we note that while the transverse spin components $S_{x},S_{y}$
converge to the canonical coordinates of the cv mode, the collective operator
related to the total spin in direction $z$ becomes the number operator $N$.
Indeed if $E_{n}:=(n\mathbb{1}-S_{z})/2$ then $E_{n}|k,n\rangle=k|k,n\rangle$
so $E_{n}=V^{*}_{n}NV_{n}$. This correspondence can be extended to small
rotations of such operators. Consider the collective operator

$N^{n}_{\bm{\Delta}}:=U^{n}(\bm{\Delta})(n\mathbb{1}-S_{z})U^{n}(-\bm{\Delta})$

which corresponds to measuring individual qubits in the basis

$|v^{\bm{\delta}}_{0}\rangle=U(\bm{\delta})|0\rangle,\quad|v^{\bm{\delta}}_{1}\rangle=U(\bm{\delta})|1\rangle$

and adding the resulting $\\{0,1\\}$ outcomes. In the limit Gaussian model,
this corresponds to measuring the displaced number operator
$N_{\bm{\Delta}}=D(\bm{\Delta})ND(-\bm{\Delta})$. More precisely, the binomial
distribution $p^{(n)}_{{\bm{u}},\bm{\Delta}}$ of $N^{n}_{\bm{\Delta}}$
computed in the state $|\Psi^{n}_{\bm{\Delta}}\rangle$ converges to the
Poisson distribution of $N_{\bm{\Delta}}$ with respect to the state
$|u_{1}+iu_{2}\rangle$

$\lim_{n\to\infty}p^{(n)}_{\bm{u},\bm{\Delta}}(k)=e^{-\|\bm{u}-\bm{\Delta}\|^{2}}\frac{\|\bm{u}-\bm{\Delta}\|^{2k}}{k!},\qquad
k\geq 0.$

###  V.2 Asymptotic perspective on displaced-null measurements via local
asymptotic normality

We now offer a complementary picture of the displaced-null measurement schemes
outlined in section IV.1, using the QLAN theory of section V.1. In the
Gaussian limit, the qubits ensemble is replaced by a single coherent state
while the qubit null measurement becomes the number operator measurement. The
Gaussian picture will illustrate why the null measurement does not work and
how this problem can be overcome by using the displaced null strategy.

Consider first the one dimensional model given by equation (8), and let us
assume for simplicity that the preliminary estimator takes the value
$\tilde{\theta}_{n}=0$. The general case can be reduced to this by a rotation
of the block sphere.

We write $\theta$ in terms of the local parameter $u$ as
$\theta=\tilde{\theta}_{n}+u/\sqrt{n}=u/\sqrt{n}$ with $|u|\leq n^{\epsilon}$.
By employing QLAN we map the i.i.d. model $|\Psi_{u}^{n}\rangle$
(approximately) into the limit coherent state model $|u\rangle$. At
$\tilde{\theta}_{n}=0$ the null measurement for an individual qubit is that of
$\sigma_{z}$ (standard basis). On the ensemble level this translates into
measuring the collective spin observable $S_{z}$, which converges to the
number operator $N$ in the limit model, cf. section V.1. Indeed, at $u=0$ the
coherent state is the vacuum which is an eigenstate of $N$.

As in the qubit case, the number measurement suffers from the non-
identifiabilty issue since both $|\pm u\rangle$ states produce the same
Poisson distribution (see panels c. and d. in Figure 1).

We now interpret the displaced-null measurement in the QLAN picture. Recall
that if we measure each qubit in the rotated basis

$|v^{\delta_{n}}_{0}\rangle=U((\delta_{n},0))|0\rangle,\quad|v^{\delta_{n}}_{1}\rangle=U((\delta_{n},0))|1\rangle,$

then the non-identifiability is lifted and the parameter can be estimated
optimally. The collective spin in this rotated basis is

$N^{n}_{(\Delta_{n},0)}:=U^{n}((\Delta_{n},0))(\mathbb{1}-S_{z})U^{n}((-\Delta_{n},0)).$

where $\Delta_{n}=n^{1/2}\delta_{n}=n^{3\epsilon}$ and by the QLAN
correspondence it maps to the displaced number operator

$N_{(\Delta_{n},0)}=D((\Delta_{n},0))ND((-\Delta_{n},0)).$

In this case the distribution with respect to the state $|u\rangle$ is ${\rm
Poisson}(|\Delta_{n}-u|^{2})$, and since $\Delta_{n}=n^{3\epsilon}\gg|u|$, the
model is identifiable, i.e. the correspondence the intensity
$|\Delta_{n}-u|^{2}$ and $u$ is one-to-one (see panels g. and h. in Figure 1).
Moreover, for large $n$ the measurement provides an optimal estimator of $u$.
Indeed by writing

$N_{(\Delta_{n},0)}=(a-\Delta_{n}\mathbf{1})^{*}(a-\Delta_{n}\mathbf{1})=a^{*}a-\Delta_{n}(a+a^{*})+\Delta_{n}^{2}\mathbf{1}$
(13)

and noting that the term $a^{*}a$ is $O(n^{2\epsilon})$ (for $|u|\leq
n^{\epsilon}$) we get

$\frac{1}{2}\Delta_{n}-\frac{1}{2\Delta_{n}}N_{(\Delta_{n},0)}=\frac{Q}{\sqrt{2}}+o(1)$
(14)

where we recover the well known fact that quadrature (homodyne) measurement
can be implemented by displacement and counting. By measuring the operator on
the lefthand side of (14) we obtain an asymptotically optimal estimator of
$u$, which corresponds to the qubit estimator constructed in section IV.1.

##  VI Multiparameter estimation for pure qudit states

In this section we discuss the general case of a multidimensional statistical
model for a $d$-dimensional quantum system (qudit).

The first two subsections review the theory of multiparameter estimation and
how QLAN is used to establish the asymptotic achievability of the Holevo
bound. This circle of ideas will be helpful in understanding the results in
the following sections which deal with displaced-null estimation of qudit
models. In particular we show that displaced-null measurements achieve the
following:

  * 1.

The Holevo bound for completely unknown pure state models where the figure of
merit is given by the Bures distance (Proposition 3);

  * 2.

The quantum Cramér-Rao bound in statistical models where the parameters can be
estimated simultaneously (Proposition 4), providing an operational
implementation for the results in Pezze and Smerzi (2014); Pezzè  _et al._
(2017); Liu  _et al._ (2019);

  * 3.

The Holevo bound for completely general pure state models and figures of merit
(Theorem 2).

Since the two stage strategy is discussed in detail in Appendix B, we do not
give a detailed account of the preliminary stage and assume that the parameter
has been localised in a neighbourhood of size $n^{-1/2+\epsilon}$ around a
preliminary estimator with probability that converges to $1$ exponentially
fast in $n$.

###  VI.1 Multiparameter estimation

Let us consider the problem of estimating the parameter $\bm{\theta}$
belonging to an open set $\Theta\subseteq\mathbb{R}^{m}$ given the
corresponding family of states $\rho_{\bm{\theta}}$ of a $d$-dimensional
quantum system. Given a measurement with POVM ${\cal
M}:=\\{M_{0},\dots,M_{p}\\}$, the CFI matrix is given by

$I_{\cal M}(\bm{\theta})_{ij}=\mathbb{E}_{\theta}\left[\frac{\partial\log
p_{\bm{\theta}}}{\partial\theta_{i}}\frac{\partial\log
p_{\bm{\theta}}}{\partial\theta_{j}}\right].$

The QFI matrix is $F(\theta)_{ij}={\rm
Tr}(\rho_{\bm{\theta}}\mathcal{L}^{i}_{\bm{\theta}}\circ\mathcal{L}^{j}_{\bm{\theta}})$
where $\mathcal{L}^{j}_{\bm{\theta}}$ are the SLDs satisfying
$\partial_{i}\rho_{\bm{\theta}}=\mathcal{L}^{j}_{\bm{\theta}}\circ\rho_{\bm{\theta}}$
and $\circ$ denotes the symmetric product $A\circ B=(AB+BA)/2$. If
$\hat{\bm{\theta}}$ is an unbiased estimator then the multidimensional QCRB
states that its covariance matrix is lower bounded as

${\rm
Cov}_{{\bm{\theta}}}(\hat{\bm{\theta}}):=\mathbb{E}_{\bm{\theta}}[(\hat{\bm{\theta}}-{\bm{\theta}})(\hat{\bm{\theta}}-{\bm{\theta}})^{T}]\geq
I_{\cal M}(\bm{\theta})^{-1}\geq F(\bm{\theta})^{-1}.$ (15)

In general, the second lower bound is not achievable even asymptotically.
Roughly, this is due to the fact that the optimal measurements for estimating
the different components of ${\bm{\theta}}$ are incompatible with each other.
The precise condition for the achievability of the QCRB is Ragy  _et al._
(2016); Demkowicz-Dobrzański  _et al._ (2020)

$\mbox{Tr}(\rho_{\bm{\theta}}[{\cal L}^{i}_{\bm{\theta}},{\cal
L}^{j}_{\bm{\theta}}])=0,\quad i,j=1,\dots,m.$ (16)

which in the case of a pure statistical model ${|{\psi_{\bm{\theta}}}\rangle}$
becomes

${\rm
Im}(\langle\partial_{\theta_{i}}\psi|\partial_{\theta_{j}}\psi\rangle)=0,\quad
i,j=1,\dots,m.$ (17)

When the QCRB is not achievable, one may look for measurements that optimise a
specific figure of merit. The simplest example is that of a quadratic form
with positive weight matrix $W$

$R_{W}(\hat{\bm{\theta}},{\bm{\theta}})=\mathbb{E}_{\theta}[(\hat{\bm{\theta}}-\bm{\theta})^{T}W(\hat{\bm{\theta}}-\bm{\theta})]$

This choice is not as restrictive as it may seem since many interesting loss
functions have a local quadratic approximation which determines the leading
term of the asymptotic risk. A straightforward lower bound on $R_{W}$ can be
obtained by taking the trace with $W$ in (15) but this bound is not achievable
either. A better one is the Holevo bound Holevo (2011)

$\displaystyle\mbox{Tr}(W{\rm
Cov}_{{\bm{\theta}}}(\hat{\bm{\theta}}))\geq{\cal H}^{W}(\bm{\theta})$
$\displaystyle:=$ $\displaystyle\min_{{\bf X}}{\rm Tr}({\rm Re}(Z({\bf
X}))W)+{\rm Tr}\left|W^{1/2}{\rm Im}(Z({\bf X}))W^{1/2}\right|$

where the minimum runs over m-tuples of selfadjoint operators
$\bm{X}=(X_{1},\dots,X_{m})^{T}$ acting on the system, which satisfy the
constraints
$\mbox{Tr}(\nabla_{\bm{\theta}}\rho_{\bm{\theta}}\bm{X}^{T})=\bm{1}$, and
${Z}({\bf X})$ is the $m\times m$ complex matrix with entries ${Z}({\bf
X})_{ij}={\rm Tr}(\rho_{\bm{\theta}}X_{i}X_{j})$. Unlike the multidimensional
QCRB, the Holevo bound is achievable asymptotically in the i.i.d. scenario
Gill and Guţă (2013); Demkowicz-Dobrzański  _et al._ (2020). In the next two
section we will give an intuitive explanation based on the QLAN theory.

###  VI.2 Gaussian shift models and QLAN

Quantum Gaussian shift models play a fundamental role in quantum estimation
theory Holevo (2011). Such models are fairly tractable in that the Holevo
bound is achievable with simple linear measurements. More importantly,
Gaussian shift models arise as limits of i.i.d. models in the QLAN theory,
which offers a recipe for constructing estimators which achieve the Holevo
bound asymptotically in the i.i.d. setting.

For the purposes of this work, the asymptotic Gaussian limit offers a clean
intuition about the working of the proposed estimators, but is not explicitly
used in deriving the mathematical results. We therefore keep the presentation
on a intuitive level and refer to the papers Gill and Guţă (2013); Demkowicz-
Dobrzański  _et al._ (2020); Butucea  _et al._ (2016) for more details.

In this subsection we recall the essentials of multiparameter estimation in a
pure quantum Gaussian shift model and of QLAN theory for pure states of finite
dimensional quantum systems, extending what we already presented in the case
of qubits in Section V.

####  VI.2.1 Achieving the Holevo bound in a pure Gaussian shift model

Consider a cv system consisting of $(d-1)$ modes. The corresponding Hilbert
space ${\cal H}$ is the multimode Fock space which will be identified with the
tensor product of $d-1$ copies of the single mode spaces, with ONB given by
the Fock vectors $|{\bf
k}\rangle:=|k_{1}\rangle\otimes\dots\otimes|k_{d-1}\rangle$, with ${\bf
k}=(k_{1},\dots,k_{d-1})\in\mathbb{N}^{d-1}$. The creation/annihilation
operators, canonical coordinates and number operator of the individual modes
are denoted $a_{i}^{*}$, $a_{i}$, $Q_{i}=(a_{i}+a_{i}^{*})/\sqrt{2}$,
$P_{i}=(a_{i}-a_{i}^{*})/(\sqrt{2}i)$ and $N_{i}=a^{*}_{i}a_{i}$ for
$i=1,\dots,{d-1}$.

We denote by
${|{\bm{z}}\rangle}={|{z_{1}}\rangle}\otimes\dots\otimes{|{z_{d-1}}\rangle}$
the multimode coherent states with
$\bm{z}=(z_{1},\dots,z_{d-1})\in\mathbb{C}^{d-1}$, so that $Q_{i}$ and $P_{i}$
have normal distribution with variance $1/2$ and mean $\sqrt{2}{\rm
Re}(z_{i})$ and $\sqrt{2}{\rm Im}(z_{i})$, respectively, while $N_{i}$ have
Poisson distributions with intensities $|z_{i}|^{2}$. We denote by ${\bf
R}:=(Q_{1},\dots,Q_{d-1},P_{1},\dots,P_{d-1})^{T}$ the vector of canonical
coordinates which satisfy commutation relations $[R_{i},R_{j}]=i\Omega_{ij}$
where $\Omega$ is the $2(d-1)\times 2(d-1)$ symplectic matrix

$\Omega=\begin{pmatrix}\bf{0}&\bf{1}\\\ -\bf{1}&\bf{0}\end{pmatrix}.$

Let ${\bm{u}}\in\mathbb{R}^{m}$ be an unknown parameter and let $\mathcal{G}$
be the _quantum Gaussian shift model_

$\mathcal{G}:=\\{{|{C\bm{u}}\rangle}:{\bm{u}}\in\mathbb{R}^{m}\\}$

where $C:\mathbb{R}^{m}\rightarrow\mathbb{C}^{d-1}$ is a linear map. The goal
is to estimate ${\bm{u}}$ optimally for a given figure of merit.

Denoting the entries of $C$ as $C_{k,j}=c^{q}_{kj}+ic^{p}_{kj}$ for
$k=1,\dots,d-1$ and $j=1,\dots,m$, we call $D$ the real $2(d-1)\times m$
matrix with elements
$D_{k,j}=\sqrt{2}c^{q}_{kj},D_{k+(d-1),j}=\sqrt{2}c^{p}_{kj}$ with $k=1,\dots
d-1$; notice that $\mathbb{E}_{\bm{u}}[{\bf R}]=D{\bm{u}}$. We remark that
$\bm{u}$ is identifiable if and only if $D$ has rank equal to $m$. The quantum
Fisher information matrix is independent of ${\bm{u}}$ and is given by
$F=2D^{T}D>0$.

Let us first consider the case when the QCRB is achievable (in which case it
leads to the Holevo bound by tracing with $W$). Condition (17) amounts to
$C^{*}C$ being a _real_ matrix which is equivalent to $D^{T}\Omega D=0$ and
the fact that the generators of the Gaussian shift model $\mathcal{G}$

$S_{j}=\sum_{k=1}^{d-1}c_{kj}^{q}P_{k}-c_{kj}^{p}Q_{k}=(D^{T}\Omega{\bf
R})_{j},\quad j=1,\dots m$

commute with each other.

An optimal unbiased measurement consists of simultaneously measuring the
commuting operators $\bm{Z}=\Sigma^{-1}D^{T}{\bf R}$, where
$\Sigma:=D^{T}D=F/2$. Indeed

  * •

$[{\bf Z},{\bf Z}^{T}]=\Sigma^{-1}D^{T}\Omega D\Sigma^{-1}=0$ (commutativity),

  * •

$\mathbb{E}_{\bm{u}}[{\bf Z}]=\Sigma^{-1}D^{T}D{\bm{u}}={\bm{u}}$
(unbiasedeness),

  * •

${\rm Cov}_{\bm{u}}({\bf Z})=\Sigma^{-1}/2$ (achieves the QCRB).

Consider now the case when the QCRB is not achievable. For a given positive
weight matrix $W$, the corresponding Holevo bound is given by

$\displaystyle{\rm Tr}({\rm Cov}_{\bm{u}}(\hat{\bm{u}})W)\geq{\cal
H}^{W}(\mathcal{G})$ (19)
$\displaystyle:=\min_{B}\frac{1}{2}\left(\mbox{Tr}(WBB^{T})+\mbox{Tr}(|\sqrt{W}B\Omega
B^{T}\sqrt{W}|)\right),$

where ${\hat{\bm{u}}}$ is an unbiased estimator and the minimum is taken over
real $m\times 2(d-1)$ matrices $B$ such that $BD=\bf{1}$. The Holevo bound can
be saturated by coupling the system with another ancillary $(d-1)$-dimensional
cv system in the vacuum state and with position and momentum vector that we
denote by ${\bf
R}^{\prime}=(Q^{\prime}_{1},\dots,Q^{\prime}_{d-1},P^{\prime}_{1},\dots,P_{d-1}^{\prime})^{T}$.
In order to estimate ${\bm{u}}$, we consider a vector of quadratures of the
form $\bm{Z}=B{\bf R}+B^{\prime}{\bf R}^{\prime}$ for $B,B^{\prime}$ real
$m\times(d-1)$ matrices and we require that $\bm{Z}$ is unbiased and belongs
to a commutative family:

  * •

$B^{\prime}\Omega B^{\prime T}=-B\Omega B^{T}$ (commutativity of the
$Z_{i}$’s),

  * •

$\langle C{\bm{u}}\otimes{\bf 0}|{\bf Z}|C{\bm{u}}\otimes{\bf
0}\rangle={\bm{u}}\Leftrightarrow BD=\mathbf{1}$ (unbiasedeness).

The corresponding risk is

$R_{\bf Z}=\frac{1}{2}\left(\mbox{Tr}(WBB^{T})+\mbox{Tr}(WB^{\prime}B^{\prime
T})\right)$

and by minimizing over $B$ and $B^{\prime}$ one obtains the expression of the
Holevo bound in Equation (19). Therefore, given a minimiser
$(B^{\star},B^{\prime\star})$, the corresponding vector of quadratures
$\bm{Z}^{\star}$ is an optimal estimator for any $\bm{u}$.

To summarise, in the pure Gaussian shift model there always exists a set of
commuting quadratures ${\bf Z}^{\star}$ of a doubled up system that achieves
the Holevo bound; in the case when the QCRB is achievable, one does not need
an ancilla.

For the discussion in section VI.4 it is useful to consider the following
implementation of the optimal measurement. Let
$(\tilde{Q}_{1},\dots,\tilde{Q}_{2(d-1)},\tilde{P}_{1},\dots,\tilde{P}_{2(d-1)})$
be a choice of vacuum modes of the doubled-up cv system such that ${\bf
Z}^{\star}=T\tilde{\bf Q}$ where $\tilde{\bf
Q}=(\tilde{Q}_{1},\dots,\tilde{Q}_{m})^{T}$ for some $m\times m$ invertible
matrix $T$ with real entries. Up to classical post-processing, measuring
$Z_{1},\dots,Z_{m}$ is equivalent to measuring
$\tilde{Q}_{1},\dots,\tilde{Q}_{m}$. If we denote the outcomes of the latter
by $\tilde{\bf q}:=(\tilde{q}_{1},\dots\tilde{q}_{m})$ then an optimal
unbiased estimator of ${\bf u}$ is given by $\hat{\bf u}=T\tilde{\bf q}$.

####  VI.2.2 QLAN for i.i.d. pure qudit models

The idea of QLAN is that the states in a shrinking neigbourhood of a fixed
state can be approximated by a Gaussian shift model. In the next section we
will show how this can be used as an estimation tool, but here we describe the
general structure of QLAN for _pure_ qudit states.

We choose the centre of the neighbourhood to be the first vector of an ONB
$\\{|0\rangle,\dots,|d-1\rangle\\}$, and parametrise the local neighborhood of
states around ${|{0}\rangle}$ as

${|{\psi_{{\bm{u}}/\sqrt{n}}}\rangle}=\exp\left(-i\sum_{k=1}^{d-1}(u_{1}^{k}\sigma_{y}^{k}-u_{2}^{k}\sigma_{x}^{k})/\sqrt{n}\right){|{0}\rangle}$
(20)

for $\bm{u}=({\bm{u}}_{1},\bm{u}_{2})\in\mathbb{R}^{2(d-1)}$,
$\|{\bm{u}}\|\leq n^{\epsilon}$,
$\sigma_{y}^{k}=i{|{k}\rangle}{\langle{0}|}-i{|{0}\rangle}{\langle{k}|}$ and
$\sigma_{x}^{k}={|{k}\rangle}{\langle{0}|}+{|{0}\rangle}{\langle{k}|}$. As in
the qubit case, the appropriately rescaled collective variables converge to
position, momentum and number operators in ‘joint distribution’ with respect
to
${|{\Psi^{n}_{\bm{u}}}\rangle}:={|{\psi_{{\bm{u}}/\sqrt{n}}}\rangle}^{\otimes
n}$

$\displaystyle\left(\frac{1}{\sqrt{2n}}S^{k}_{x}(n),\frac{1}{\sqrt{2n}}S^{k}_{y}(n),n\mathbb{1}-S^{k}_{z}(n)\,:\,|\Psi_{\bm{u}}^{n}\rangle\right)$
$\displaystyle\to\left(Q_{k},P_{k},N_{k}\,:\,{|{{\bf
z}=\bm{u}_{1}+i\bm{u}_{2}}\rangle}\right),$

where $S^{k}_{\alpha}(n)=\sum_{l=1}^{n}(\sigma_{\alpha}^{k})^{(l)}$ for
$\alpha\in\\{x,y,z\\}$. More generally, we have a (real) linear map between
the orthogonal complement of $|0\rangle$ and Gaussian quadratures: for every
vector ${|{v}\rangle}=\sum_{k=1}^{d-1}(v_{x}^{k}+iv_{y}^{k}){|{k}\rangle}$ we
construct the corresponding Pauli operator
$\sigma(v)={|{v}\rangle}{\langle{0}|}+{|{0}\rangle}{\langle{v}|}$ and the
following CLT holds

$\left(\frac{1}{\sqrt{2n}}S_{v}(n):{|{\Psi^{n}_{u}}\rangle}\right)\to\left(X(v):{|{{\bf
z}=\bm{u}_{1}+i\bm{u}_{2}}\rangle}\right)$ (21)

where $S_{v}(n):=\sum_{l=1}^{n}\sigma(v)^{(l)}$ and
$X(v):=\sum_{k=1}^{n}v_{x}^{k}Q_{k}+v_{y}^{k}P_{k}$.

In addition to the QCLT, the following strong QLAN statement holds: the
statistical model $\\{{|{\Psi_{\bm{u}}^{n}}\rangle}\\}$ can be approximated by
a pure Gaussian shift model in the sense that

$\displaystyle\lim_{n\to\infty}\sup_{|{\bm{u}}|\leq
n^{1/2-\eta}}\left\|V_{n}|\Psi_{\bm{u}}^{n}\rangle-|{\bm{u}}_{1}+i{\bm{u}}_{2}\rangle\right\|=0,$
(22) $\displaystyle\lim_{n\to\infty}\sup_{|{\bm{u}}|\leq
n^{1/2-\eta}}\left\||\Psi_{\bm{u}}^{n}\rangle-
V_{n}^{*}|{\bm{u}}_{1}+i{\bm{u}}_{2}\rangle\right\|=0$ (23)

for any fixed $0<\eta<1/2$. $V_{n}$ is the isometric embedding of the
symmetric subspace
$\mathcal{S}^{(n)}_{d}:=\left(\mathbb{C}^{d}\right)^{\otimes_{s}n}$ into a
$(d-1)$-mode Fock space ${\cal H}$ (cf. previous section) characterised by

$\displaystyle V_{n}:\mathcal{S}^{(n)}_{d}$ $\displaystyle\to$
$\displaystyle\mathcal{H}$ $\displaystyle|{\bf k};n\rangle$
$\displaystyle\mapsto$ $\displaystyle|{\bf k}\rangle$ (24)

where ${|{{\bf k};n}\rangle}$ denotes the normalised vector obtained by
symmetrising

${|{1}\rangle}^{\otimes k_{1}}\otimes\dots\otimes{|{{d-1}}\rangle}^{\otimes
k_{d-1}}\otimes{|{0}\rangle}^{\otimes(n-(k_{1}+\dots+k_{d-1}))}.$

As in the qubits case, the Gaussian approximation maps small rotations into
displacements of the coherent states. Consider collective qubit rotations by
small angles ${\bm{\delta}}:=n^{-1/2}\bm{\Delta}$

$U^{n}(\bm{\Delta}):=\left(\exp\left(-i\sum_{k=1}^{d-1}(n^{-1/2}\Delta_{1}^{k}\sigma^{k}_{y}-n^{-1/2}\Delta_{2}^{k}\sigma^{k}_{x})\right)\right)^{\otimes
n}$

and the corresponding displacement operators

$D(\bm{\Delta})=\exp\left(-i\sum_{k=1}^{d-1}(\Delta_{1}^{k}P_{k}-\Delta_{2}^{k}Q_{k})\right).$

The diagram below conveys the asymptotic covariance between rotations and
displacements, where the arrows should be interpreted in the same way as the
strong convergence equations (22) and (23)

$\begin{CD}|\Psi_{\bm{u}}^{n}\rangle
@>{V_{n}}>{}>|{\bm{u}}_{1}+i{\bm{u}}_{2}\rangle\\\
@V{}V{U^{n}(-{\bm{\Delta}})}V@V{}V{D(-{\bm{\Delta}})}V\\\
|\Psi_{{\bm{u}}-{\bm{\Delta}}}^{n}\rangle
@>{V_{n}}>{}>|{\bm{u}}_{1}-{\bm{\Delta}}_{1}+i({\bm{u}}_{2}-{\bm{\Delta}}_{2})\rangle\end{CD}$

A similar correspondence holds for measurements with respect to rotated bases
and displaced number operators

$\begin{CD}|\Psi_{\bm{u}}^{n}\rangle
@>{V_{n}}>{}>|{\bm{u}}_{1}+i{\bm{u}}_{2}\rangle\\\
@V{}V{N^{i}_{\bm{\Delta}}(n)}V@V{}V{N^{i}_{\bm{\Delta}}}V\\\
p^{n}({\bm{u}},{\bm{\Delta}})@>{V_{n}}>{}>{\rm
Poisson}(\|{\bm{u}}_{1}-{\bm{\Delta}}_{1}+i({\bm{u}}_{2}-{\bm{\Delta}}_{2})\|^{2})\end{CD}$

More precisely, suppose we measure the commuting family of operators
$\\{N^{i}_{\bm{\Delta}}(n),i=1,\dots,d-1\\}$ given by

$N^{i}_{\bm{\Delta}}(n):=U^{n}(-\bm{\Delta})(n\mathbb{1}-S^{i}_{z}(n))U^{n}(\bm{\Delta})\quad
i=1,\dots,d-1,$

which amounts to measuring individual qudits in the basis

$|v^{\bm{\delta}}_{i}\rangle=U(\bm{\delta})|i\rangle\quad i=0,\dots,d-1$

and collecting the total counts for individual outcomes in
$\\{0,\dots,d-1\\}$. In the Gaussian model this corresponds to measuring the
displaced number operators
$N^{i}_{\bm{\Delta}}=D(-\bm{\Delta})N^{i}D(\bm{\Delta})$, and by QLAN, the
multinomial distribution $p^{n}(u,\bm{\Delta})$ of $N^{i}_{\bm{\Delta}}(n)$
converges to the law of the vector of Poisson random variables obtained by
measuring $N^{i}_{\bm{\Delta}}$ with respect to the state
$|\bm{u}_{1}+i\bm{u}_{2}\rangle$.

###  VI.3 Achieving the Holevo bound for pure qudit states via QLAN

We will now treat a general pure states statistical model and show how one can
use QLAN to achieve the Holevo bound (VI.1) asymptotically with the sample
size. Let ${|{\psi_{\bm{\theta}}}\rangle}$ be a statistical model where
$\bm{\theta}=(\theta^{j})_{j=1}^{m}$ belongs to some open set
$\Theta\subset\mathbb{R}^{m}$ with $m\leq 2(d-1)$ and the parameter is assumed
to be identifiable. Given an ensemble of $n$ copies of the unknown state, we
would like to devise a measurement strategy and estimation procedure which
attains the smallest average error (risk), asymptotically with $n$. For mixed
states, a general solution has been discussed in Demkowicz-Dobrzański  _et
al._ (2020) where it is shown how the Holevo bound can be achieved
asymptotically using the QLAN machinery. Here we adapt this method to the case
of pure state models.

In brief, the procedure involves three steps. We first use
$\tilde{n}=n^{1-\epsilon}$ samples to produce a preliminary estimator
$\tilde{\bm{\theta}}_{n}$ and write
${\bm{\theta}}=\tilde{\bm{\theta}}_{n}+{\bm{u}}/\sqrt{n}$ where ${\bm{u}}$ is
the local parameter satisfying $\|{\bm{u}}\|\leq n^{\epsilon}$ (with high
probability). We chooose an ONB $\\{|0\rangle,\dots|d-1\rangle\\}$ such that
$|\psi_{\tilde{\bm{\theta}}_{n}}\rangle=|0\rangle$ and use the QLAN isometry
$V_{n}$ (cf. equation 24) to map the remaining qubits
$|\Psi^{n}_{\bm{u}}\rangle:=|\psi_{\tilde{\bm{\theta}}_{n}+{\bm{u}}/\sqrt{n}}\rangle^{\otimes
n}$ approximately into the Gaussian state $|C{\bm{u}}\rangle$. We then use the
method described in section VI.2.1 to estimate the unknown parameter and
achieve the Holevo bound.

We start by expressing the local states as small rotations around $|0\rangle$

$\displaystyle|\psi_{\bm{\tilde{\theta}}_{n}+{\bm{u}}/\sqrt{n}}\rangle$ (25)
$\displaystyle=\exp\left(-i\sum_{k=1}^{d-1}\left(f^{q}_{k}\left(\frac{{\bm{u}}}{\sqrt{n}}\right)\sigma^{k}_{y}-f^{p}_{k}\left(\frac{{\bm{u}}}{\sqrt{n}}\right)\sigma^{k}_{x}\right)\right)|0\rangle$

where $f^{q}_{k}$ and $f^{p}_{k}$ are real functions and $\sigma_{y}^{k}$ and
$\sigma_{x}^{k}$ are the Pauli matrices of equation (20). We now ‘linearise’
the generators of the rotations and define

$|\tilde{\psi}_{{\bm{u}}/\sqrt{n}}\rangle:=\exp\left(-i\sum_{j=1}^{m}u_{j}S_{j}/\sqrt{n}\right)|0\rangle$
(26)

where

$S_{j}=\sum_{k=1}^{d-1}(c^{q}_{kj}\sigma^{k}_{y}-c^{p}_{kj}\sigma^{k}_{x}),\quad
c^{q,p}_{kj}=\left.\partial_{j}f^{q,p}_{k}({\bm{u}})\right|_{{\bm{u}}={\bf
0}}.$

We denote the ensemble state of the linearised model
$|\widetilde{\Psi}^{n}_{\bm{u}}\rangle:=|\tilde{\psi}_{{\bm{u}}/\sqrt{n}}\rangle^{\otimes
n}$. The following lemma shows that the original and the ‘linearised’ models
are locally undistinguishable in the asymptotic limit.

######  Lemma 2.

With the above notations if $\epsilon<1/6$ one has

$\lim_{n\to\infty}\sup_{\|{\bm{u}}\|\leq
n^{\epsilon}}\||\Psi^{n}_{\bm{u}}\rangle\langle\Psi^{n}_{\bm{u}}|-|\widetilde{\Psi}^{n}_{\bm{u}}\rangle\langle\widetilde{\Psi}^{n}_{\bm{u}}|\|_{1}=0$

where $\|\cdot\|_{1}$ denotes the trace distance.

The proof of Lemma 2 can be found in Appendix G. Thanks to such uniform
approximation results, one can replace the original model with the linearised
one without affecting the asymptotic estimation analysis. We denote the latter
by

$\mathcal{Q}_{n}:=\\{|\widetilde{\Psi}^{n}_{\bm{u}}\rangle:\,{\bm{u}}\
\in\mathbb{R}^{m},\|{\bm{u}}\|\leq n^{\epsilon}\\}.$

Let us now consider the second ingredient of the estimation problem, the risk
(figure of merit). We fix a loss function
$L:\Theta\times\Theta\to\mathbb{R}_{+}$, so that the risk of an estimator
$\hat{\bm{\theta}}_{n}$ at ${\bm{\theta}}$ is
$R(\hat{\bm{\theta}}_{n},{\bm{\theta}})=\mathbb{E}_{\bm{\theta}}[L(\hat{\bm{\theta}}_{n},{\bm{\theta}})]$.
We assume that the loss function is locally quadratic around any point and in
particular

$L(\tilde{\bm{\theta}}_{n}+{\bm{u}},\tilde{\bm{\theta}}_{n}+{\bm{v}})\approx\sum_{i,j=1}^{m}w_{ij}(\tilde{\bm{\theta}}_{n})(u_{i}-v_{i})(u_{j}-v_{j})$

for a strictly positive weight matrix function ${\bm{\theta}}^{\prime}\mapsto
W({\bm{\theta}}^{\prime})=(w_{ij}({\bm{\theta}}^{\prime}))$ (which we assume
to be continuous in ${\bm{\theta}}^{\prime}$). In asymptotics,
$\tilde{\bm{\theta}}_{n}\to{\bm{\theta}}$ and the loss function can be
replaced by its quadratic approximation at the true parameter ${\bm{\theta}}$
without affecting the leading contribution to the estimation risk. We denote
$W:=W({\bm{\theta}})$.

Returning to the original estimation problem, we now show how QLAN can be used
to construct an estimator which achieves the Holevo bound asymptotically.

We couple each system with a $d$-dimensional ancillary system in state
$|0^{\prime}\rangle$ and fix an ONB for the ancilla ${\cal
B}^{\prime}=\\{{|{0^{\prime}}\rangle},\dots,{|{d-1^{\prime}}\rangle}\\}$. The
extended i.i.d. statistical model is
${|{\Psi^{n}_{\bm{u}}}\rangle}\otimes{|{0^{\prime}}\rangle}^{\otimes n}$. By
quantum LAN, the joint ensemble can be approximated by a pure Gaussian shift
model coupled with an ancillary $(d-1)$-modes cv system prepared in the
vacuum: ${|{C\bm{u}}\rangle}\otimes{|{\bf 0}\rangle}$ where $C$ is the
$(d-1)\times m$ complex matrix with entries $C_{kj}=c_{kj}^{q}+ic_{kj}^{p}$;
more precisely we map the two qudit ensembles into their Fock spaces by means
of a tensor of isometries as in equation (24) and we consider the $2(d-1)$
modes which correspond to the linear space $\mathcal{L}:={\rm
Lin}\\{|0\rangle\otimes|i^{\prime}\rangle,|i\rangle\otimes|0^{\prime}\rangle:i=1,\dots
d-1\\}$ (which contains
$\\{{|{\psi_{\bm{\theta}}}\rangle}\otimes{|{0^{\prime}}\rangle}\\}_{{\bm{\theta}}\in\Theta}$).
Alternatively, one can map the original ensemble to the the cv space and
_then_ add a second cv system in the vacuum state. The reason we chose to add
an ancillary ensemble at the beginning is because this same setup will be used
in the next section in the context of displaced-null measurements.

We now apply the optimal measurement for the Gaussian shift model
$|C{\bm{u}}\rangle$ with weight matrix $W$, as described in section VI.2.1.
This involves measuring commuting quadratures of the doubled up cv system,
such that the resulting estimator $\hat{\bm{u}}_{n}$ achieves the Gaussian
Holevo bound (19) in the limit of large $n$.

Thanks to the parameter localisation and LAN, the asymptotic (rescaled) risk
of the corresponding ’global’ estimator
$\hat{\bm{\theta}}_{n}=\tilde{\bm{\theta}}_{n}+\hat{\bm{u}}_{n}/\sqrt{n}$
satisfies

$\lim_{n\to\infty}nR(\hat{\bm{\theta}}_{n},{\bm{\theta}})=\mathcal{H}^{W}(\mathcal{G}).$

Finally we note that the expressions of the Holevo bound (VI.1) in i.i.d.
model $|\psi_{\bm{\theta}}\rangle$ with loss function $L$, and the
corresponding Gaussian shift model $|C{\bm{u}}\rangle$ with weight matrix $W$
coincide: $\mathcal{H}^{W}({\bm{\theta}})=\mathcal{H}^{W}(\mathcal{G})$.
Indeed, since
$\rho_{\bm{\theta}}=|\psi_{\bm{\theta}}\rangle\langle\psi_{\bm{\theta}}|$ is a
pure state, the minimisation in (VI.1) can be restricted to operators
${\bm{X}}=(X_{1},\dots,X_{m})$ such that $PX_{i}P=P^{\perp}X_{i}P^{\perp}=0$
where $P=\rho_{\bm{\theta}},P^{\perp}=\mathbf{1}-P$. In this case the two
Holevo bounds coincide after making the identification $B_{j,k}=\sqrt{2}{\rm
Re}\langle k|X_{j}|0\rangle,B_{j+d-1,k}=\sqrt{2}{\rm Im}\langle
k|X_{j}|0\rangle$.

###  VI.4 Achieving the Holevo bound with displaced-null measurements

In this section we show how displaced-null measurements offer an alternative
strategy to the one presented in the previous section, for optimal estimation
in a general finite dimensional pure statistical model
${|{\psi_{\bm{\theta}}}\rangle}$ with
${\bm{\theta}}\in\Theta\subset\mathbb{R}^{m}$. As before, we assume that the
risk function $L:\Theta\times\Theta\rightarrow\mathbb{R}_{+}$ has a continuous
quadratic local approximation given by the matrix valued function
$W({\bm{\theta}})$.

The first steps are the same as in the estimation procedure in section VI.3:
we use $\tilde{n}=n^{1-\epsilon}$ samples to produce a preliminary estimator
$\tilde{\bm{\theta}}_{n}$ and we write
${\bm{\theta}}=\tilde{\bm{\theta}}_{n}+{\bm{u}}/\sqrt{n}$ where ${\bm{u}}$ is
the local parameter such that $\|{\bm{u}}\|\leq n^{\epsilon}$ with high
probability. We choose an ONB ${\cal
B}=\\{{|{0}\rangle},\dots,{|{d-1}\rangle}\\}$ such that
${|{0}\rangle}:={|{\psi_{\bm{\tilde{\theta}}_{n}}}\rangle}$ and apply Lemma 2
to approximate the local model as in equation (26). We couple each system with
an ancillary qudit in state $|0^{\prime}\rangle$. By QLAN, the joint model is
approximated by the Gaussian shift model consisting of coherent states
${|{C{\bm{u}}}\rangle}\otimes{|{0}\rangle}$ of a $2(d-1)$-modes cv system.

As detailed in section VI.2.1, the Holevo bound for the Gaussian shift can be
attained by measuring a certain set of canonical coordinates
$\tilde{Q}:=(\widetilde{Q}_{1},\dots,\widetilde{Q}_{m})$ of the doubled-up
systems. In turn, this provides an asymptotically optimal measurement for the
i.i.d. qudit model as explained in section VI.3. Instead of measuring these
quadratures, here we adopt the displaced-null measurements philosophy used in
section IV, which achieves the same asymptotic risk. This means that one
measures the commuting set of displaced number operators
$\tilde{N}^{j}_{{\bm{\Delta}}_{n}}=D(-{{\bm{\Delta}}_{n}})\widetilde{N}^{j}D({\bm{\Delta}}_{n})$
where $\widetilde{N}^{j}=\widetilde{a}^{*}_{j}\widetilde{a}_{j}$ is the number
operator corresponding to the mode $(\widetilde{Q}_{j},\widetilde{P}_{j})$ and

$D({\bm{\Delta}}_{n})=\exp\left(-i\Delta_{n}\sum_{k=1}^{m}\tilde{P}_{k}\right),\quad\Delta_{n}=\sqrt{n}\delta_{n}=n^{3\epsilon}.$

We note that

$\tilde{N}^{j}_{{\bm{\Delta}}_{n}}=(\widetilde{a}_{j}-n^{3\epsilon}{\bf
1})^{*}(\widetilde{a}_{j}-n^{3\epsilon}{\bf 1})=n^{6\epsilon}{\bf
1}-\sqrt{2}\widetilde{Q}_{j}n^{3\epsilon}+\widetilde{N}^{j},$

so for large $n$, measuring $\tilde{N}^{j}_{{\bm{\Delta}}_{n}}$ is equivalent
to measuring $\widetilde{Q}_{j}$. We recall that by measuring ${\bf
Z}^{\star}:=T\widetilde{{\bf Q}}$ we obtain an optimal unbiased estimator of
${\bf u}$, where $T$ is the invertible matrix defined at the end of section
VI.2.1. Therefore, using the above equation we can construct an
(asymptotically) optimal estimator given by the outcomes of the following set
of commuting operators

$\sum_{k=1}^{m}T_{jk}\left(\frac{n^{3\epsilon}}{\sqrt{2}}{\bf
1}-\frac{n^{-3\epsilon}}{\sqrt{2}}\tilde{N}^{k}_{{\bm{\Delta}}_{n}}\right)\approx
Z_{i}$

We are now ready to translate the above cv measurement into its corresponding
projective qudit measurement using the correspondence between displaced number
operators measurements and rotated bases, described in section VI.2.2.

Using the general CLT map (21), we identify vectors
$\\{{|{\tilde{1}}\rangle},\dots,{|{\widetilde{m}}\rangle}\\}$ in the
orthogonal complement of
$|\tilde{0}\rangle={|{0}\rangle}\otimes{|{0^{\prime}}\rangle}$ such that their
corresponding limit quadratures are $X(|\tilde{k}\rangle)=\tilde{Q}_{k}$, for
$k=1,\dots,m$. By virtue of the CLT the vectors
$|\tilde{0}\rangle,{|{\tilde{1}}\rangle},\dots,{|{\widetilde{m}}\rangle}$ are
normalised and orthogonal to each other, so we can complete the set to an ONB
$\tilde{\mathcal{B}}:=\\{|\tilde{0}\rangle,\dots|\widetilde{d^{2}-1}\rangle\\}$
of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ where the remaining vectors are
chosen arbitrarily. Now let $\tilde{\mathcal{B}}_{n}$ be the rotated basis

$|v^{\delta_{n}}_{j}\rangle=U(\delta_{n})|\tilde{j}\rangle=\exp\left(-i\delta_{n}\sum_{k=1}^{m}\sigma(i\tilde{k})\right)|\tilde{j}\rangle$

for $\delta_{n}=n^{-1/2+3\epsilon}$ and
$\sigma(i\tilde{k}):=-i|\tilde{0}\langle\tilde{k}|+i|\tilde{k}\rangle\langle\tilde{0}|.$
Note that $\tilde{\mathcal{B}}_{n}$ is a small rotation of the basis
$\mathcal{B}$ which contains the reference state
$|\tilde{0}\rangle=|0\rangle\otimes{|{0^{\prime}}\rangle}$, so the
corresponding measurement is a of the displaced-null type.

We measure each of the qudits in the basis $\tilde{\mathcal{B}}_{n}$ and
obtain i.i.d. outcomes $X_{1},\dots,X_{n}$ taking values in
$\\{0,\dots,d^{2}-1\\}$, and let $p_{{\bm{u}}}^{(n)}$ be their distribution:

$p^{(n)}_{{\bm{u}}}(j)=|\langle\psi_{{\bm{u}}/\sqrt{n}}\otimes
0^{\prime}|v^{\delta_{n}}_{j}\rangle|^{2},\quad j=0,\dots,d^{2}-1.$

The following Theorem is one of the main results of the paper and shows that
the Holevo bound can be attained by using displaced-null measurements.

######  Theorem 2.

Assume we are given $n$ samples of the qudit state
$|\psi_{\bm{\theta}}\rangle$ where $\bm{\theta}\in\Theta\subset\mathbb{R}^{m}$
is unknown. We further assume that assume that $\Theta$ is bounded and
$\epsilon<1/10$. Using $\tilde{n}=n^{1-\epsilon}$ samples, we compute a
preliminary estimator $\tilde{\bm{\theta}}_{n}$, and we measure the rest of
the systems in the ONB $\tilde{{\cal B}}_{n}$, as defined above. Let

$\hat{\bm{\theta}}_{n}:=\tilde{\bm{\theta}}_{n}+\hat{\bm{u}}_{n}/\sqrt{n}$

be the estimator with

$\hat{u}_{n}^{j}=\sum_{k=1}^{m}T_{jk}\left(\frac{n^{3\epsilon}}{\sqrt{2}}-\frac{n^{1-3\epsilon}}{\sqrt{2}}\hat{p}_{n}(k)\right),\quad
j=1,\dots,m$

where $\hat{p}_{n}(j)$ is the empirical estimator of $p^{(n)}_{{\bm{u}}}(j)$,
i.e.

$\hat{p}_{n}(j)=\frac{|\\{i:X_{i}=j,~{}i=1,\dots,n\\}|}{n},$

for $j=1,\dots,m$.

Then $\hat{\bm{\theta}}_{n}$ is asymptotically optimal in the sense that for
every ${\bm{\theta}}\in\Theta$

$\lim_{n\to\infty}nR_{n}(\hat{\bm{\theta}}_{n},{\bm{\theta}})={\cal
H}^{W({\bm{\theta}})}({\bm{\theta}})$

Moreover, $\sqrt{n}(\hat{\bm{\theta}}_{n}-{\bm{\theta}})$ converges in law to
a centered normal random variable with covariance given by $TT^{T}/2$.

The proof of Theorem 2 can be found in Appendix H.

Our measurement has been obtained by modifying the optimal linear measurement
for the limiting Gaussian shift to displaced counting one, and translating
this to a qudit and ancilla measurement with repect to a displaced-null basis.
Interestingly, this resulting measurement is closely connected to the optimal
measurement described in Matsumoto (2002). The connection is discussed in
Appendix I.

###  VI.5 Estimating a completely unknown pure state with respect to the Bures
distance

In this section we consider the problem of estimating a completely unknown
pure qudit state, when the loss function (figure of merit) is defined as the
squared Bures distance

$d^{2}_{b}(|\psi\rangle\langle\psi|,|\phi\rangle\langle\phi|)=2(1-|\langle\psi|\phi\rangle|).$

In this particular case, we will show that one can asymptotically achieve the
Holevo bound using diplaced-null measurement without the need of using any
ancillary system.

We parametrise a neighbourhood of the preliminary estimator
${|{0}\rangle}:=|\tilde{\psi}_{n}\rangle$ as

$|\psi_{{\bm{u}}/\sqrt{n}}\rangle=\exp\left(-i\sum_{k=1}^{d-1}(u_{1}^{k}\sigma^{k}_{y}-u_{2}^{k}\sigma^{k}_{x})/\sqrt{n}\right)|0\rangle$

where
${\bm{u}}=(u_{1}^{1},u_{2}^{1},\dots,u_{1}^{d-1},u_{2}^{d-1})\in\mathbb{R}^{2(d-1)}$
satisfies $\|{\bm{u}}\|\leq n^{\epsilon}$ with high probability.

For small deviations from $|0\rangle$ the Bures distance has the quadratic
approximation

$d^{2}_{b}\left(|\psi_{\frac{\bm{u}}{\sqrt{n}}}\rangle\langle\psi_{\frac{\bm{u}}{\sqrt{n}}}|,|\psi_{\frac{{\bm{u}}^{\prime}}{\sqrt{n}}}\rangle\langle\psi_{\frac{{\bm{u}}^{\prime}}{\sqrt{n}}}|\right)=\frac{1}{n}\|{\bm{u}}-{\bm{u}}^{\prime}\|^{2}+o(n^{-1+2\epsilon})$

which determines the optimal measurement and error rate in the asymptotic
regime.

The Gaussian approximation consists in the model
${|{{\bm{u}}_{1}+i{\bm{u}}_{2}}\rangle}$ and the optimal measurement with
respect to the identity cost matrix would be to measure the $Q_{k}$’s and
$P_{k}$’s. In order to estimate ${\bm{u}}$, instead of usign an acilla, we
split the ensemble of $n$ qudits in two equal sub-ensembles and perform
separate ‘displaced-null’ measurements on each of them in the following bases
which are obtained by rotating $\\{|0\rangle,\dots|d-1\rangle\\}$ by (small)
angles of size $\delta_{n}=n^{-1/2+3\epsilon}$

$\displaystyle|v^{\delta_{n}}_{j}\rangle$ $\displaystyle=$ $\displaystyle
U_{1}(\delta_{n})|j\rangle=\exp\left(-i\delta_{n}\sum_{k=1}^{d-1}\sigma^{k}_{y}\right)|j\rangle$
(27) $\displaystyle|w^{\delta_{n}}_{j}\rangle$ $\displaystyle=$ $\displaystyle
U_{2}(\delta_{n})|j\rangle=\exp\left(i\delta_{n}\sum_{k=1}^{d-1}\sigma^{k}_{x}\right)|j\rangle.$
(28)

Therefore in the asymptotic picture, the proposed measurements are effectively
joint measurements of $\\{Q_{i},i=1,\dots d-1\\}$ and respectively
$\\{P_{i},i=1,\dots d-1\\}$ which are known to be optimal measurements for the
local parameter ${\bm{u}}$ in the Gaussian shift model when performed on two
separate copies of $|({\bm{u}}_{1}+i{\bm{u}}_{2})/\sqrt{2}\rangle$ obtained
from the original state by using a beamsplitter.

Let $X_{1},\dots,X_{n/2}$ and $Y_{1},\dots,Y_{n/2}$ be the independent
outcomes of the two types of measurements, taking values in
$\\{0,\dots,d-1\\}$, and let $p_{{\bm{u}}}^{(n)}$ and $q_{{\bm{u}}}^{(n)}$ be
their respective distributions

$p^{(n)}_{{\bm{u}}}(j)=|\langle\psi_{{\bm{u}}/\sqrt{n}}|v^{\delta_{n}}_{j}\rangle|^{2},\quad
q^{(n)}_{{\bm{u}}}(j)=|\langle\psi_{{\bm{u}}/\sqrt{n}}|w^{\delta_{n}}_{j}\rangle|^{2}.$
(29)

######  Proposition 3.

Assume $\epsilon<1/10$ and let

$|\hat{\psi}\rangle:=|\psi_{\hat{\bm{u}}/\sqrt{n}}\rangle$

be the state estimator with local parameter $\hat{\bm{u}}_{n}$ defined as

$\displaystyle\hat{u}^{j}_{1}$ $\displaystyle=$
$\displaystyle\frac{n^{3\epsilon}}{2}-\frac{n^{1-3\epsilon}}{2}\hat{p}_{n}(j),$
$\displaystyle\hat{u}^{j}_{2}$ $\displaystyle=$
$\displaystyle\frac{n^{3\epsilon}}{2}-\frac{n^{1-3\epsilon}}{2}\hat{q}_{n}(j),\quad
j=1,\dots,{d-1},$

where $\hat{p}_{n}$, $\hat{q}_{n}$ are the empirical estimator of
$p^{(n)}_{{\bm{u}}}$ and $q^{(n)}_{{\bm{u}}}$, respectively, i.e.

$\displaystyle\hat{p}_{n}(j)$ $\displaystyle=$
$\displaystyle\frac{|\\{i:X_{i}=j,~{}i=1,\dots,n/2\\}|}{n/2},$
$\displaystyle\hat{q}_{n}(j)$ $\displaystyle=$
$\displaystyle\frac{|\\{i:Y_{i}=j,~{}i=1,\dots,n/2\\}|}{n/2},$

for $j=1,\dots,d-1$.

Then under $\mathbb{P}_{\bm{u}}$, $\sqrt{n}(\hat{\bm{u}}_{n}-{\bm{u}})$ is
asymptotically distributed as a centered Gaussian random vector with
covariance $\bm{1}/2$ and $|\hat{\psi}_{n}\rangle$ is asymptotically optimal
in the sense that it achieves the Holevo bound:

$\lim_{n\to\infty}n\mathbb{E}_{{|{\psi}\rangle}}[d_{b}^{2}(|\psi\rangle\langle\psi|,|\hat{\psi}_{n}\rangle\langle\hat{\psi}_{n}|)]=d-1.$

The proof of Proposition 3 can be found in see Appendix J.

###  VI.6 Achieving the QCRB with displaced-null measurements

We now consider quantum statistical models for which the QCRB is
(asymptotically) achievable. In contrast to models discussed in sections VI.4
and VI.5, in this case all parameter components can be estimated
simultaneously at maximum precision. We will provide a class a displaced-null
measurements which achieve the QCRB asymptotically.

Let us consider the statistical model $\\{{|{\psi_{\bm{\theta}}}\rangle}\\}$,
$\bm{\theta}\in\Theta\subset\mathbb{R}^{m}$ with $m\leq 2(d-1)$ and assume
that the parameter is identifiable and that the QCRB is achievable for all
$\theta\in\Theta$. This is equivalent to condition (17) for all
${\bm{\theta}}\in\Theta$. The QFI is given by

$F({\bm{\theta}})_{ij}=4\langle\partial_{i}\psi_{\bm{\theta}}|\partial_{j}\psi_{\bm{\theta}}\rangle-4\langle\psi_{\bm{\theta}}|\partial_{j}\psi_{\bm{\theta}}\rangle\langle\partial_{i}\psi_{\bm{\theta}}|\psi_{\bm{\theta}}\rangle,$

for $i,j=1,\dots,m$. Let
${|{0}\rangle}:={|{\psi_{\tilde{\bm{\theta}}_{n}}}\rangle}$ be the preliminary
estimator. We write ${\bm{\theta}}=\tilde{\bm{\theta}}_{n}+{\bm{u}}/\sqrt{n}$
with ${\bm{u}}$ the local parameter satisfying $\|{\bm{u}}\|\leq n^{\epsilon}$
with high probability. We assume that the phase of $|\psi_{\theta}\rangle$ has
been chosen such that $\langle\dot{\psi}_{i}|0\rangle=0$ for all $i$, and
denote $\dot{\psi}_{i}:=\partial_{i}\psi_{\tilde{\bm{\theta}}_{n}}$.

We now describe a class of measurements that will be shown to achieve the QCRB
asymptotically. We choose an orthonormal basis
$\mathcal{B}:=\\{|0\rangle,|1\rangle,\dots,|d-1\rangle\\}$ whose first vector
is $|0\rangle$ and the other vectors satisfy

$c_{ki}:=\langle k|\dot{\psi}_{i}\rangle\in\mathbb{R},\qquad i=1,\dots,m,\quad
k=1,\dots,d-1.$ (30)

This condition is similar to equation (7) in Pezzè  _et al._ (2017), but
unlike this reference we do not impose additional conditions for the case when
$\langle k|\dot{\psi}_{i}\rangle=0$ for all $i=1,\dots,m$. If we assume that
the parameter $\bm{\theta}$ is identifiable, then the matrix $C=(c_{ki})$
needs to have rank $m$.

We will further rotate $\mathcal{B}$ with a unitary $U=\exp(-i\delta_{n}G)$
where $\delta_{n}=n^{-1/2+3\epsilon}$ and

$G=\sum_{k=1}^{d-1}g_{k}\sigma^{k}_{y},\quad\sigma_{y}^{k}=-i{|{0}\rangle}{\langle{k}|}+i{|{k}\rangle}{\langle{0}|}$

where $g_{k}\neq 0$ are arbitrary real coefficients. We obtain the ONB
$\\{|v_{0}^{\delta_{n}}\rangle,\dots|v_{d-1}^{\delta_{n}}\rangle\\}$ with

$|v_{k}^{\delta_{n}}\rangle=U|k\rangle,\qquad k=0,\dots d-1.$

We measure all the systems in the basis $\tilde{\mathcal{B}}$ and obtain
i.i.d. outcomes $X_{1},\dots,X_{n}\in\\{0,\dots,d-1\\}$ and denote by
$\hat{p}_{n}$ the corresponding empirical frequency. We denote by $T=(T_{ij})$
the $m\times(d-1)$ matrix defined as

$T=(C^{T}C)^{-1}C^{T}.$

######  Proposition 4.

Assume that $\Theta$ be bounded and $\epsilon<1/10$. Let
$\hat{\bm{\theta}}_{n}=\tilde{\bm{\theta}}_{n}+\hat{\bm{u}}_{n}/\sqrt{n}$ be
the estimator determined by

$\hat{\bm{u}}^{j}_{n}=\sum_{k=1}^{d-1}T_{jk}\left(\frac{g_{k}n^{3\epsilon}}{2}-\frac{n^{1-3\epsilon}}{2g_{k}}\hat{p}_{n}(k)\right).$

Then $\hat{\bm{\theta}}_{n}$ achieves the QCRB, i.e.

$\lim_{n\to\infty}n\mathbb{E}_{\bm{\theta}}[(\hat{\bm{\theta}}_{n}-{\bm{\theta}})(\hat{\bm{\theta}}_{n}-{\bm{\theta}})^{T}]=F({\bm{\theta}})^{-1}.$

The proof of Proposition 4 can be found in Appendix K.

We now give a QLAN interpretation of the above construction. The fact that
$c_{ki}$ are real implies that the linearisation of the model around the
preliminary estimation is given by

$|\tilde{\psi}_{{\bm{u}}/\sqrt{n}}\rangle=\exp\left(-i\sum_{j=1}^{m}u_{j}S_{j}/\sqrt{n}\right)|0\rangle$

with

$S_{j}=\sum_{k=1}^{d-1}c_{kj}\sigma^{k}_{y},\qquad c_{kj}=\langle
k|\dot{\psi}_{j}\rangle.$

By QLAN, the corresponding Gaussian model consists of coherent states
$|C{\bm{u}}\rangle$ of a $(d-1)$-modes cv system where
$C:\mathbb{R}^{m}\to\mathbb{C}^{d-1}$ is given by the _real_ coefficients
$c_{kj}=\langle k|\dot{\psi}_{j}\rangle$. This means that each of the $(d-1)$
modes is in a coherent state whose displacement is along the $Q$ axis, so
$\langle C{\bm{u}}|P_{k}|C{\bm{u}}\rangle=0$ for all $k$, while

$q_{k}:=\langle
C{\bm{u}}|Q_{k}|C{\bm{u}}\rangle=\sqrt{2}\sum_{j=1}^{m}c_{kj}u_{j}.$

As we mentioned in Section VI.2.1, the QCRB is achievable for the limit model
too and the simultaneous measurement of all $Q_{k}$ is optimal. This is
asymptotically obtained by the counting in the rotated basis.

##  VII Conclusions and outlook

In this paper we showed that the framework of displaced-null measurements
provides a general scheme for optimal estimation of unknown parameters
${\bm{\theta}}\in\mathbb{R}^{m}$ of pure states models
$|\psi_{\bm{\theta}}\rangle\in\mathbb{C}^{d}$. In particular, displaced-null
measurements achieve the quantum Cramér-Rao bound (QCRB) for models in which
the bound is achievable, and the Holevo bound for general qudit models.

Our method is related to previous works Pezze and Smerzi (2014); Pezzè  _et
al._ (2017); Liu  _et al._ (2019) that deal with the achivebility of the QCRB
for pure state models $|\psi_{\bm{\theta}}\rangle$. These works exhibit a
class of parameter-dependent orthonormal bases
$\mathcal{B}(\tilde{\bm{\theta}})$ whose associated classical Fisher
information $I_{\tilde{\bm{\theta}}}({\bm{\theta}})$ converges to the quantum
Fisher information $F({\bm{\theta}})$ of $|\psi_{\bm{\theta}}\rangle$ as
$\tilde{\bm{\theta}}$ approaches the true unknown state parameter
${\bm{\theta}}$. The measurement basis $\mathcal{B}(\tilde{\bm{\theta}})$ has
the special feature that it contains the state
$|\psi_{\tilde{\bm{\theta}}}\rangle$ as one of its elements, so that at
$\tilde{\bm{\theta}}={\bm{\theta}}$ the measurement has only one outcome,
while for $\tilde{\bm{\theta}}\approx{\bm{\theta}}$ the occurrence of other
outcomes can be interpreted as signaling the deviation from the reference
value $\tilde{\bm{\theta}}$. With this in mind we called such measurements,
null measurements.

However, the references Pezze and Smerzi (2014); Pezzè  _et al._ (2017); Liu
_et al._ (2019) do not provide an explicit operational implementation of a
strategy that achieves the QCRB. The naive solution would be to choose the
reference parameter as a preliminary estimator $\tilde{\bm{\theta}}_{n}$
obtained by measuring a sub-sample of $\tilde{n}\ll n$ systems, and to apply
the approximate null measurement $\mathcal{B}_{\tilde{\bm{\theta}}_{n}}$ to
the rest of the systems. Surprisingly, it turned out that this adaptive
strategy fails to achieve the QCRB, and indeed does not even reach the
standard $n^{-1}$ scaling of precision, when the preliminary estimator
satisfies certain natural assumptions. This is due to the fact that
$\tilde{\bm{\theta}}_{n}$ lies in the interior of a confidence interval of
${\bm{\theta}}$ and the measurement cannot distinguish positive and negative
deviations from the reference since probabilities depend on the square of the
deviations. This is an important finding which shows the pitfalls of drawing
statistical conclusions based solely on Fisher information arguments.

To avoid this issue, we proposed to displace the preliminary estimator by a
small amount ${\bm{\delta}}_{n}$ which is however sufficiently large to ensure
that the new reference parameter $\tilde{\bm{\theta}}_{n}+{\bm{\delta}}_{n}$
is outside the confidence interval of ${\bm{\theta}}$. Building on this idea
we showed the achievability of the QCRB in the setting of Pezze and Smerzi
(2014); Pezzè  _et al._ (2017); Liu  _et al._ (2019). Furthermore, for general
pure state models and locally quadratic loss functions, we devised displaced-
null measurements which achieve the Holevo bound asymptotically for arbitrary
qudit models.

The theory of quantum local asymptotic normality (QLAN) has played an
important role in our investigations. The QLAN machinery translates the multi-
copy estimation problem into one about estimating the mean of a multi-mode
coherent state. In the latter case, counting measurements are paradigmatic
example of null-measurements, while appropriately displacing the number
operators provides the basis for displaced-null measurements. Using the QLAN
correspondence, this translates into a simple prescription for rotating a
basis containing the preliminary estimator
$|\psi_{\tilde{\bm{\theta}}_{n}}\rangle$ into that of the displaced-null
measurement. Interestingly, the obtained measurement turned out to be closely
related to the parameter-dependent measurements proposed by Matsumoto in
Matsumoto (2002), and our approach offers an alternative asymptotic
perspective on this work.

An exciting area of applications for displaced-null measurements is that of
optimal estimation of dynamical parameters of open systems Gambetta and
Wiseman (2001); Guţă (2011); Gammelmark and Mølmer (2014); Guţă and Kiukas
(2015); Catana  _et al._ (2015); Guţă and Kiukas (2017); Ilias  _et al._
(2022); Fallani  _et al._ (2022). Recent works Godley and Guţă (2023); Yang
_et al._ (2023) have shown out that quantum post-processing by means of
coherent absorbers allows for optimal estimation of such parameters. In
particular Yang  _et al._ (2023) pointed out that a basic measurement such as
photon counting constitutes a null-measurement, thus opening the route for
devising optimal measurements for multidimensional estimation of Markov
dynamics. An asymptotic analysis of displaced null measurements in this
context will be the subject of a forthcoming publication Godley  _et al._ .

Another area of future interest is to extend the method to models consisting
of mixed states. While this will probably not work in general, the ideas
presented here may be useful for models consisting of states with a high
degree of purity which is the relevant setup in many quantum technology
applications. Another important extension is towards refining the methodology
for optimal estimation in the finite sample rather than asymptotic regime.
Finally, we would like to better understand how displaced-null measurements
can be used in the context of quantum metrology and interferometry Schnabel
(2017); Hans-A. Bachor (2019).

Acknowledgements: This work was supported by the EPSRC grant EP/T022140/1. We
acknoweledge fruitful discussions with Dayou Yang, Rafał Demkowicz-Dobrzański,
Janek Kołodyński and Richard Gill.

## References

  * Hayashi (2005) M. Hayashi,  _Asymptotic theory of quantum statistical inference: Selected papers_ (World Scientific Publishing Co., 2005).
  * Paris (2008) M. G. Paris, International Journal of Quantum Information 7, 125 (2008).
  * Tóth and Apellaniz (2014) G. Tóth and I. Apellaniz, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
  * Demkowicz-Dobrzański  _et al._ (2020) R. Demkowicz-Dobrzański, W. Gorecki, and M. Guţă, Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020).
  * Banaszek  _et al._ (2013) K. Banaszek, M. Cramer, and D. Gross, New Journal of Physics 15, 125020 (2013).
  * Albarelli  _et al._ (2020) F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, Physics Letters A 384, 126311 (2020).
  * Sidhu and Kok (2020) J. S. Sidhu and P. Kok, AVS Quantum Science 2, 014701 (2020).
  * Gross  _et al._ (2010) D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert, Phys. Rev. Lett. 105, 150401 (2010).
  * Cramer  _et al._ (2010) M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Nature Communications 1, 149 (2010).
  * O’Donnell and Wright (2016) R. O’Donnell and J. Wright, in  _Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing_, STOC ’16 (2016) p. 899.
  * Haah  _et al._ (2016) J. Haah, A. W. Harrow, Z. Ji, X. Wu, and N. Yu, in  _Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing_, STOC ’16 (2016) p. 913.
  * Lanyon  _et al._ (2017) B. Lanyon, C. Maier, M. Holzäpfel, T. Baumgratz, C. Hempel, P. Jurcevic, I. Dhand, A. Buyskikh, A. Daley, M. Cramer, M. B. Plenio, R. Blatt, and C. F. Roos, Nature Physics 13, 1158–1162 (2017).
  * Guţă and Kiukas (2017) M. Guţă and J. Kiukas, Journal of Mathematical Physics 58, 052201 (2017).
  * Yang  _et al._ (2019) Y. Yang, G. Chiribella, and M. Hayashi, Commun. Math. Phys. 368, 223–293 (2019).
  * Lahiry and Nussbaum (2023) S. Lahiry and M. Nussbaum, “Minimax estimation of low-rank quantum states and their linear functionals,” (2023), arXiv:2111.03279 [math.ST] .
  * Yuen (2023) H. Yuen, Quantum 7, 890 (2023).
  * Szczykulska  _et al._ (2016) M. Szczykulska, T. Baumgratz, and A. Datta, Advances in Physics: X 1, 621 (2016).
  * Nichols  _et al._ (2018) R. Nichols, P. Liuzzo-Scorpo, P. A. Knott, and G. Adesso, Physical Review A 98, 012114 (2018).
  * Albarelli  _et al._ (2019) F. Albarelli, J. F. Friel, and A. Datta, Phys. Rev. Lett. 123, 200503 (2019).
  * Liu  _et al._ (2020) J. Liu, H. Yuan, X.-M. Lu, and X. Wang, J. Phys. A: Math. Theor. 53, 023001 (2020).
  * Mosonyi and Petz (2003) M. Mosonyi and D. Petz, Letters in Mathematical Physics 68, 19 (2003).
  * Jenčová and Petz (2006) A. Jenčová and D. Petz, Communications in Mathematical Physics 263, 259 (2006).
  * Guţă and Kahn (2006) M. Guţă and J. Kahn, Physical Review A 73, 052108 (2006).
  * Guţă and Jencova (2007) M. Guţă and A. Jencova, Communications in Mathematical Physics 276, 341 (2007).
  * Kahn and Guţă (2009) J. Kahn and M. Guţă, Communications in Mathematical Physics 289, 597 (2009).
  * Gill and Guţă (2013) R. D. Gill and M. Guţă, IMS Collections 9, 105 (2013).
  * Butucea  _et al._ (2016) C. Butucea, M. Guţă, and M. Nussbaum, Annals Statist. 46, 3676 (2016).
  * Yamagata  _et al._ (2013) K. Yamagata, A. Fujiwara, and R. D. Gill, The Annals of Statistics 41, 2197 (2013).
  * Fujiwara and Yamagata (2020) A. Fujiwara and K. Yamagata, Bernoulli 26, 2105 (2020).
  * Fujiwara and Yamagata (2023) A. Fujiwara and K. Yamagata, Ann. Statist. 51, 1159 (2023).
  * Aaronson (2018) S. Aaronson, in  _Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing_, STOC 2018 (2018) p. 325.
  * Huang  _et al._ (2020) H. Y. Huang, R. Kueng, and J. Preskill, Nature Physics 16, 1050 (2020).
  * Personick (1971) S. D. Personick, IEEE Transactions on Information Theory 17, 240 (1971).
  * Tsang (2020) M. Tsang, Phys. Rev. A 102, 062217 (2020).
  * Rubio and Dunningham (2020) J. Rubio and J. Dunningham, Physical Review A 101, 032114 (2020).
  * Rubio  _et al._ (2021) J. Rubio, J. Anders, and L. A. Correa, Physical Review Letters 127, 190402 (2021).
  * Giovannetti  _et al._ (2005) V. Giovannetti, S. Lloyd, and L. Maccone, Physical Review Letters 96, 010401 (2005).
  * Fujiwara and Imai (2008) A. Fujiwara and H. Imai, Journal of Physics A: Mathematical and Theoretical 41, 255304 (2008).
  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).
  * Demkowicz-Dobrzański  _et al._ (2012) R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nature Communications 3, 1063 (2012).
  * Girolami  _et al._ (2014) D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Phys. Rev. Lett. 112, 210401 (2014).
  * Smirne  _et al._ (2016) A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 116, 120801 (2016).
  * Seveso  _et al._ (2017) L. Seveso, M. A. C. Rossi, and M. G. A. Paris, Phys. Rev. A 95, 012111 (2017).
  * Haase  _et al._ (2018) J. F. Haase, A. Smirne, J. Kołodyński, R. Demkowicz-Dobrzański, and S. F. Huelga, New Journal of Physics 20, 053009 (2018).
  * Pang and Brun (2014) S. Pang and T. A. Brun, Physical Review A - Atomic, Molecular, and Optical Physics 90, 022117 (2014).
  * Rossi  _et al._ (2020) M. A. C. Rossi, F. Albarelli, D. Tamascelli, and M. G. Genoni, Phys. Rev. Lett. 125, 200505 (2020).
  * Zhou and Jiang (2021) S. Zhou and L. Jiang, PRX Quantum 2, 010343 (2021).
  * Gorecki  _et al._ (2020) W. Gorecki, S. Zhou, L. Jiang, and R. Demkowicz-Dobrzański, Quantum 4, 288 (2020).
  * Zhou  _et al._ (2018a) S. Zhou, M. Zhang, J. Preskill, and L. Jiang, Nature Communications 9, 78 (2018a).
  * Yuan and Fung (2015) H. Yuan and C. H. F. Fung, Physical Review Letters 115, 110401 (2015).
  * Yuan (2016) H. Yuan, Phys. Rev. Lett. 117, 160801 (2016).
  * Correa  _et al._ (2015) L. A. Correa, M. Mehboudi, G. Adesso, and A. Sanpera, Physical Review Letters 114, 220405 (2015).
  * Mehboudi  _et al._ (2019) M. Mehboudi, A. Sanpera, and L. A. Correa, Journal of Physics A: Mathematical and Theoretical 52, 303001 (2019).
  * Christensen and Meyer (2022) N. Christensen and R. Meyer, Reviews of Modern Physics 94 (2022), 10.1103/REVMODPHYS.94.025001.
  * Abadie  _et al._ (2011) J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, _et al._ , Nature Physics 7, 962 (2011).
  * Jones  _et al._ (2009) J. A. Jones, S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Benjamin, G. A. D. Briggs, and J. J. Morton, Science 324, 1166 (2009).
  * Amorós-Binefa and Kołodyński (2021) J. Amorós-Binefa and J. Kołodyński,  23, 123030 (2021).
  * Brask  _et al._ (2015) J. B. Brask, R. Chaves, and J. Kolodynski, Physical Review X 5, 031010 (2015).
  * Albarelli  _et al._ (2017) F. Albarelli, M. A. Rossi, M. G. Paris, and M. G. Genoni, New J. Phys. 19, 123011 (2017).
  * Degen  _et al._ (2017) C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics 89, 035002 (2017).
  * Marciniak  _et al._ (2022) C. D. Marciniak, T. Feldker, I. Pogorelov, R. Kaubruegger, D. V. Vasilyev, R. van Bijnen, P. Schindler, P. Zoller, R. Blatt, and T. Monz, Nature 603, 604 (2022).
  * Zwick and Álvarez (2023) A. Zwick and G. A. Álvarez, Journal of Magnetic Resonance Open 16-17, 100113 (2023).
  * Tsang  _et al._ (2016) M. Tsang, R. Nair, and X.-M. Lu, Phys. Rev. X 6, 031033 (2016).
  * Tsang (2021) M. Tsang, Phys. Rev. A 104, 052411 (2021).
  * Lupo  _et al._ (2020) C. Lupo, Z. Huang, and P. Kok, Phys. Rev. Lett. 124, 080503 (2020).
  * Fiderer  _et al._ (2021) L. J. Fiderer, T. Tufarelli, S. Piano, and G. Adesso, PRX Quantum 2, 020308 (2021).
  * Oh  _et al._ (2021) C. Oh, S. Zhou, Y. Wong, and L. Jiang, Phys. Rev. Lett. 126, 120502 (2021).
  * Tsang (2019) M. Tsang, Phys. Rev. Res. 1, 033006 (2019).
  * Tsang  _et al._ (2020) M. Tsang, F. Albarelli, and A. Datta, Physical Review X 10, 031023 (2020).
  * Gambetta and Wiseman (2001) J. Gambetta and H. M. Wiseman, Phys. Rev. A 64, 042105 (2001).
  * Guţă (2011) M. Guţă, Physical Review A 83, 062324 (2011).
  * Gammelmark and Mølmer (2014) S. Gammelmark and K. Mølmer, Phys. Rev. Lett. 112, 170401 (2014).
  * Guţă and Kiukas (2015) M. Guţă and J. Kiukas, Communications in Mathematical Physics 335, 1397 (2015).
  * Catana  _et al._ (2015) C. Catana, L. Bouten, and M. Guţă, Journal of Physics A: Mathematical and Theoretical 48, 365301 (2015).
  * Ilias  _et al._ (2022) T. Ilias, D. Yang, S. F. Huelga, and M. B. Plenio, PRX Quantum 3, 010354 (2022).
  * Fallani  _et al._ (2022) A. Fallani, M. A. C. Rossi, D. Tamascelli, and M. G. Genoni, PRX Quantum 3, 020310 (2022).
  * Tsang  _et al._ (2011) M. Tsang, H. M. Wiseman, and C. Caves, Phys. Rev. Lett. 106, 090401 (2011).
  * Berry  _et al._ (2015) D. W. Berry, M. Tsang, M. J. Hall, and H. M. Wiseman, Physical Review X 5 (2015), 10.1103/PhysRevX.5.031018.
  * Ng  _et al._ (2016) S. Ng, S. Z. Ang, T. A. Wheatley, H. Yonezawa, A. Furusawa, E. H. Huntington, and M. Tsang, Phys. Rev. A 93, 042121 (2016).
  * Norris  _et al._ (2016) L. M. Norris, G. A. Paz-Silva, and L. Viola, Phys. Rev. Lett. 116, 150503 (2016).
  * Shi and Zhuang (2023) H. Shi and Q. Zhuang, npj Quantum Inf 9, 27 (2023).
  * Sung  _et al._ (2019) Y. Sung, F. Beaudoin, L. M. Norris, F. Yan, D. K. Kim, J. Y. Qiu, U. von Lüpke, J. L. Yoder, T. P. Orlando, S. Gustavsson, _et al._ , Nature communications 10, 3715 (2019).
  * Tsang (2023) M. Tsang, Phys. Rev. A 107, 012611 (2023).
  * Holevo (2011) A. Holevo,  _Probabilistic and Statistical Aspects of Quantum Theory_ (Edizioni della Normale, 2011).
  * Yuen and Lax (1973) H. Yuen and M. Lax, IEEE Trans. Inf. Theory 19, 740 (1973).
  * Belavkin (1976) V. Belavkin, Theoretical and Mathematical Physics 26, 213 (1976).
  * Helstrom (1976) C. W. Helstrom,  _Quantum detection and estimation theory_ (Academic Press, 1976).
  * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * Nagaoka (2005) H. Nagaoka, in  _Asymptotic Theory of Quantum Statistical Inference: Selected Papers_ (World Scientific, 2005) p. 100–112, originally published as IEICE Technical Report, 89, 228, IT 89–42, 9–14 (1989).
  * Gill and Massar (2000) R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).
  * Górecki  _et al._ (2020) W. Górecki, R. Demkowicz-Dobrzański, H. M. Wiseman, and D. W. Berry, Phys. Rev. Lett. 124, 030501 (2020).
  * Pezze and Smerzi (2014) L. Pezze and A. Smerzi, Atom Interferometry 188, 691 (2014).
  * Pezzè  _et al._ (2017) L. Pezzè, M. A. Ciampini, N. Spagnolo, P. C. Humphreys, A. Datta, I. A. Walmsley, M. Barbieri, F. Sciarrino, and A. Smerzi, Phys. Rev. Lett. 119, 130504 (2017).
  * Liu  _et al._ (2019) J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).
  * (95) M. Tsang, “Caveats of the Crámer-Rao bound,” https://blog.nus.edu.sg/mankei/caveats-of-the-cramer-rao-bound/.
  * Yang  _et al._ (2023) D. Yang, S. F. Huelga, and M. B. Plenio, Phys. Rev. X 13, 031012 (2023).
  * (97) A. Godley, F. Girotti, and M. Guţă, “Asymptotic estimation theory for quantum Markov chains,” .
  * Lehmann and Casella (1998) E. L. Lehmann and G. Casella, Design 41 (1998), 10.2307/1270597.
  * van der Vaart (1998) A. W. van der Vaart,  _Asymptotic Statistics_ (Cambridge University Press, 1998).
  * Zhou  _et al._ (2018b) S. Zhou, C.-L. Zou, and L. Jiang, Quantum Science and Technology 5, 025005 (2018b).
  * Gammelmark  _et al._ (2014) S. Gammelmark _et al._ , Physical Review A - Atomic, Molecular, and Optical Physics 89, 043839 (2014).
  * Leonhardt (1997) U. Leonhardt,  _Measuring the Quantum State of Light_ (Cambridge University Press, 1997).
  * Braunstein  _et al._ (1996) S. L. Braunstein, C. M. Caves, and G. J. Milburn, Annals of Physics 247, 135 (1996).
  * Guţă  _et al._ (2008) M. Guţă, B. Janssens, and J. Kahn, Commun. Math. Phys. 277, 127 (2008).
  * Petz (2008) D. Petz, Theoretical and Mathematical Physics (2008), 10.1007/978-3-540-74636-2.
  * Ragy  _et al._ (2016) S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrzański, Phys. Rev. A 94, 052108 (2016).
  * Matsumoto (2002) K. Matsumoto, Journal of Physics A: Mathematical and General 35, 3111 (2002).
  * Godley and Guţă (2023) A. Godley and M. Guţă, Quantum 7, 973 (2023).
  * Schnabel (2017) R. Schnabel, Physics Reports 684, 1 (2017).
  * Hans-A. Bachor (2019) T. C. R. Hans-A. Bachor,  _A Guide to Experiments in Quantum Optics_ (Wiley‐VCH Verlag GmbH & Co. KGaA, 2019).
  * Young and Smith (2005) G. A. Young and R. L. Smith,  _Essentials of Statistical Inference_ (Cambridge University Press, 2005).

##  Appendix A Proof of the QCRB-saturating POVM

In this appendix we present the necessary and sufficient conditions for a
quantum Cramér-Rao bound saturating POVM, based on the proof by Zhou et al.
Zhou  _et al._ (2018b), highlighting the features that allow this POVM to
saturate the bound. The classical Fisher information corresponding to the
measurement $\mathcal{M}=\\{M_{i}\\}$ is given by

$\displaystyle I_{\mathcal{M}}(\theta)$ $\displaystyle=\sum_{i:{\rm
Tr}(M_{i}\rho_{\theta})\neq 0}\frac{({\rm
Tr}(M_{i}\partial_{\theta}\rho_{\theta}))^{2}}{{\rm Tr}(M_{i}\rho_{\theta})}$
$\displaystyle=\sum_{i:{\rm Tr}(M_{i}\rho_{\theta})\neq 0}\frac{({\rm
Re}[Tr(M_{i}\mathcal{L}_{\theta}\rho_{\theta})])^{2}}{{\rm
Tr}(M_{i}\rho_{\theta})},$

where we have used the Lyapunov equation, see section II, and identified that
the resulting term corresponds to the above real component. Clearly

$\displaystyle\sum_{i:{\rm Tr}(M_{i}\rho_{\theta})\neq 0}$
$\displaystyle\frac{({\rm Re}[{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta})])^{2}}{{\rm
Tr}(M_{i}\rho_{\theta})}$ $\displaystyle\leq\sum_{i:{\rm
Tr}(M_{i}\rho_{\theta})\neq 0}\frac{|{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta})|^{2}}{{\rm
Tr}(M_{i}\rho_{\theta})},$

where we have equality when ${\rm Im}[{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta})]=0$. We now use the Cauchy-Schwarz
inequality to cancel the denominator, identifying the terms
$M_{i}^{\frac{1}{2}}\rho_{\theta}^{\frac{1}{2}}$ and
$M_{i}^{\frac{1}{2}}\mathcal{L}_{\theta}\rho_{\theta}^{\frac{1}{2}}$ within
the expression above, finding

$\sum_{i:{\rm Tr}(M_{i}\rho_{\theta})\neq 0}\frac{|{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta})|^{2}}{{\rm
Tr}(M_{i}\rho_{\theta})}\leq\sum_{i:{\rm Tr}(M_{i}\rho_{\theta})\neq 0}{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta}\mathcal{L}_{\theta}).$ (31)

The equality holds if

$M_{i}^{\frac{1}{2}}\rho_{\theta}^{\frac{1}{2}}=\lambda_{i}M_{i}^{\frac{1}{2}}\mathcal{L}_{\theta}\rho_{\theta}^{\frac{1}{2}}$
(32)

for all $i$ such that ${\rm Tr}(M_{i}\rho_{\theta})\neq 0$ for some
$\lambda_{i}\in\mathbb{C}$. Finally, since $M_{i}$ represents a POVM we have

$\sum_{i:{\rm Tr}(M_{i}\rho_{\theta})\neq 0}{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta}\mathcal{L}_{\theta})\leq{\rm
Tr}(\mathcal{L}_{\theta}^{2}\rho_{\theta})\equiv F(\theta).$ (33)

where $F(\theta)$ is the quantum Fisher information. Equality is achieved when
${\rm Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta}\mathcal{L}_{\theta})=0$ for
all $i$ such that ${\rm Tr}(M_{i}\rho_{\theta})=0$. Note: this ensures that
all the information on the parameter is contained in measurable outcomes with
non-zero probabilities.

In summary, to achieve the QFI the POVM $\\{M_{i}\\}$ needs to satisfy the 3
following conditions:

  1. 1.

If ${\rm Tr}(M_{i}\rho_{\theta})>0$ then ${\rm Im}[{\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta})]=0$,

  2. 2.

If ${\rm Tr}(M_{i}\rho_{\theta})>0$ then
$M_{i}^{\frac{1}{2}}\rho_{\theta}^{\frac{1}{2}}=\lambda_{i}M_{i}^{\frac{1}{2}}\mathcal{L}_{\theta}\rho_{\theta}^{\frac{1}{2}}$
,$\lambda_{i}\in\mathbb{C}$,

  3. 3.

If ${\rm Tr}(M_{i}\rho_{\theta})=0$ then ${\rm
Tr}(M_{i}\mathcal{L}_{\theta}\rho_{\theta}\mathcal{L}_{\theta})=0$.

We can combine conditions 1. and 2. into the following condition:

  1. 4.

If ${\rm Tr}(M_{i}\rho_{\theta})>0$ then
$M_{i}^{\frac{1}{2}}\rho_{\theta}^{\frac{1}{2}}=\lambda_{i}M_{i}^{\frac{1}{2}}\mathcal{L}_{\theta}\rho_{\theta}^{\frac{1}{2}}$,
$\lambda_{i}\in\mathbb{R}$ .

##  Appendix B Parameter localisation via a two step adaptive procedure

Here we discuss in more detail the general _parameter localisation_ principle
to which we refer repeatedly in the paper. The principle is formulated for
one-dimensional models, but can be extended staightforwardly to
multidimensional ones.

Suppose we are given a large number $n$ of independent, identically prepared
systems in the state $\rho_{\theta}$, depending smoothly on a parameter
$\theta$ which lies in an open set $\Theta\subset\mathbb{R}$. To avoid
pathological cases we assume that $F(\theta)>f>0$ for all $\theta\in\Theta.$
Even though the set $\Theta$ is a priori ‘large’, we can ‘localise’ the
parameter and subsequently perform measurements adapted to the parameter
value, by using the following two step procedure.

Consider a measurement $\mathcal{M}$ such that $\theta$ is _identifiable_ ,
i.e. no two different parameters produce outcomes with identical probability
distributions. In the first step we apply $\mathcal{M}$ to each system
belonging to a vanishingly small proportion $\tilde{n}=o(n)$ of the samples,
with $\tilde{n}$ growing with $n$. For concreteness we assume that
$\tilde{n}=n^{1-\epsilon}\ll n$, with $\epsilon>0$ a small number, but the
arguments hold for a generic choice. Using the data obtained from measuring
this sub-ensemble we construct a preliminary rough estimator
$\tilde{\theta}_{n}$ of $\theta$.

Naturally, one would like the estimator $\tilde{\theta}_{n}$ to be ‘pretty
good’ (given the used sample size), but not necessarily optimal. There are
several properties that could embody this requirement; for example one could
require that the mean square error (MSE) scales at the standard rate
$\tilde{n}^{-1}=n^{-1+\epsilon}$ and $\tilde{\theta}_{n}$ is asymptotically
normal, i.e. it concentrates around $\theta$ at rate $\tilde{n}^{-1/2}$ with

$\sqrt{\tilde{n}}(\tilde{\theta}_{n}-\theta)\longrightarrow N(0,V_{\theta})$

where the convergence to the normal distribution holds in law as $n\to\infty$
and the variance satisfies $V_{\theta}\geq F(\theta)^{-1}$. Standard
estimators such as maximum likelihood typically satisfy this property Lehmann
and Casella (1998). In particular, this implies that the (confidence) interval

$I_{n}=(\tilde{\theta}_{n}-n^{-1/2+\epsilon},~{}\tilde{\theta}_{n}+n^{-1/2+\epsilon})$

contains $\theta$ with probability converging to one exponentially fast. This
follows from the fact that the ratio between the size of $I_{n}$ and the
standard deviation of $\tilde{\theta}_{n}-\theta$ diverges as
$|I_{n}|/n^{(-1+\epsilon)/2}=n^{\epsilon/2}$.

On the other hand, if one adopts a Bayesian viewpoint and assumes the
existence of a prior distribution on $\Theta$ with density $\pi(\theta)$, then
it is natural to require the asymptotic normality of the posterior density

$p(\theta|\tilde{\theta}_{n}):=\frac{\pi(\theta)p(\tilde{\theta}_{n}|\theta)}{p(\tilde{\theta}_{n})}$

where $p(\tilde{\theta}_{n}|\theta)=p_{\theta}(\tilde{\theta}_{n})$ and
$p(\tilde{\theta}_{n})=\int p(\tilde{\theta}_{n}|\theta)\pi(d\theta)$.
Intuitively this follows from the asymptotic normality of
$\tilde{\theta}_{n}$. Indeed if the prior $\pi(\theta)$ and the variance
$V_{\theta}$ are sufficiently regular with respect to $\theta$ and
$V_{\theta}$ is bounded away from zero, then
$p(\theta|\tilde{\theta}_{n})\propto\exp(-\tilde{n}(\tilde{\theta}_{n}-\theta)^{2}/2V_{\theta})$
concentrates around $\tilde{\theta}_{n}$ with approximately normal
distribution. For more details on asymptotic normality theory we refer to van
der Vaart (1998); Young and Smith (2005). For our purposes, it will suffice to
assume that $\tilde{\theta}_{n}$ is a ‘reasonable’ estimator in the sense that
the posterior distribution is ‘balanced’ with respect to $\tilde{\theta}_{n}$
in a sense that is precisely defined in section III. In particular, this means
that we exclude ‘dishonest’ estimators for which the mass of the posterior
distribution lies largely on one side of the estimator. For instance, taking a
reasonable estimator $\tilde{\theta}_{n}$ and adding a constant $\delta_{n}$
such that $\delta_{n}/\tilde{n}^{-1/2}\to\infty$ for large $n$ would be an
example of a dishonest estimator. As we will see later this distinction
becomes important since the preliminary estimator enters the definition of the