# Quantum Learnability is Arbitrarily Distillable

Joe H. Jenne  Cavendish Laboratory, Department of Physics, University of
Cambridge, Cambridge CB3 0HE, United Kingdom  David R. M. Arvidsson-Shukur
Hitachi Cambridge Laboratory, J. J. Thomson Avenue, CB3 0HE, Cambridge, United
Kingdom  Cavendish Laboratory, Department of Physics, University of Cambridge,
Cambridge CB3 0HE, United Kingdom  Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

###### Abstract

Quantum learning (in metrology and machine learning) involves estimating
unknown parameters $\bm{\theta}=(\theta_{1},\dots,\theta_{M})$ from
measurements of quantum states $\hat{\rho}_{\bm{\theta}}$. The quantum Fisher
information matrix can bound the average amount of information learnt about
$\bm{\theta}$ per experimental trial. In several scenarios, it is advantageous
to concentrate information in as few states as possible. Here, we present two
“go-go” theorems proving that negativity, a narrower nonclassicality concept
than noncommutation, enables unbounded and lossless distillation of Fisher
information about multiple parameters in quantum learning.

††preprint: APS/123-QED

The use of experimental data to estimate unknown parameters
$\bm{\theta}=(\theta_{1},\theta_{2},\dots,\theta_{M})$ is a quintessential
task in metrology and many machine-learning algorithms. In quantum metrology
and quantum machine-learning, nonclassical phenomena are used to improve the
learning of $\bm{\theta}$ based on measurements of quantum states
$\hat{\rho}_{\bm{\theta}}$ Giovannetti  _et al._ (2011); Maccone (2013);
Szczykulska  _et al._ (2016); Kiani  _et al._ (2021). Here, we show that
negativity Arvidsson-Shukur  _et al._ (2020a), a narrower concept than
noncommutation, enables unbounded and lossless distillation of information
about multiple parameters in quantum learning.

A common measure of an experiment’s usefulness in learning (estimating)
multiple unknown parameters $\bm{\theta}$ is the Fisher information matrix
$I(\bm{\theta})$ Braunstein and Caves (1994); Liu  _et al._ (2019); Albarelli
_et al._ (2020). $I(\bm{\theta})$ quantifies the average information learned
about $\bm{\theta}$ from one experimental trial. The covariance matrix of a
locally unbiased estimator $\bm{\theta}^{\mathrm{e}}$ is lower-bounded by the
Cramér-Rao inequality:
$\Sigma(\bm{\theta}^{\mathrm{e}})\geq\left[NI(\bm{\theta})\right]^{-1}$, where
$N$ is the number of independent experimental trials Rao (1992); Cramér
(2016). Theoretically, the learning task is then to adjust the experimental
input state and final measurement to optimize the Fisher information matrix
and to minimize the estimator’s risk with respect to some risk function
Ballester (2004); Imai and Fujiwara (2007); Genoni  _et al._ (2013); Humphreys
_et al._ (2013); Pezzè  _et al._ (2017); Chen and Yuan (2017). However, such
results are not necessarily representative of optimal experimental
strategies—especially in quantum experiments.

Whilst a theorist aims to optimize the Fisher information, an experimentalist
must manage her cost Liuzzo-Scorpo  _et al._ (2018); Lipka-Bartosik and
Demkowicz-Dobrzański (2018). Recent works, theoretical and practical, have
focused on limiting experimental costs associated with the measurement and
post-processing of output states. Weak-value amplification Dressel  _et al._
(2014); Harris  _et al._ (2017); Xu  _et al._ (2020) and postselected
metrology Arvidsson-Shukur  _et al._ (2020b); Lupu-Gladstein et al. (prep)
allows the rate of output states per unit time to be lowered whilst a
significant fraction of the information about a single parameter $\theta_{1}$
is retained. This enables detectors to operate at lower intensities and can,
if the postselection is experimentally cheap, reduce temporal overheads
associated with measurements and postprocessing. The protocols cannot increase
the information content, but can reduce the experimental costs of accessing
it. A major shortcoming of most previous information-distillation protocols is
that they require perfect knowledge of all-but-one experimental parameter—an
often unrealistic setting.111Initial studies of weak-value amplification with
specific forms of multiparameter unitaries are given in Vella  _et al._
(2019); Xia  _et al._ (2020); Ho and Kondo (2021).

Given the important role of multiparameter learning in quantum metrology and
quantum machine learning, a generalization of these results is crucial for
both practical and foundational reasons. A generalization will help facilitate
postselected metrology in diverse experiments, where several parameters are
(partially) unknown, as well as in quantum machine-learning, where the
overhead associated with the postprocessing of output data can be monumental.
From a foundational perspective, a generalization could provide useful
knowledge about the nature of negativity and noncommutation as quantum
resources, as well as about the fundamental limits of encoding information in
quantum states.

In this Article, we provide this generalization. First, we review theoretical
results, establishing that scalar risk functions based on the quantum Fisher
information matrix are suitable objects to minimize, when optimizing quantum
learning. Second, we derive a formula for the distilled (postselected) quantum
Fisher information matrix. Third, we use a Kirkwood-Dirac quasiprobability
distribution Kirkwood (1933); Dirac (1945); Yunger Halpern  _et al._ (2018) (a
diversified cousin of the Wigner function) to find classical and nonclassical
bounds on the entries in the quantum Fisher information matrix (Thm. 1).222In
this work, if the experiment is described by an operationally defined
quasiprobability distribution (see below) that does not equal a classical
probability distribution, we call the experiment nonclassical. We prove that
the presence of negative quasiprobabilities allows the quantum Fisher
information matrix to take anomalous entries, outside the classical bounds.
Fourth, we design a quantum-learning protocol in which the useful information
in an arbitrarily large number of states $\hat{\rho}_{\bm{\theta}}$ is
distilled into an arbitrarily small number of states
$\hat{\rho}_{\bm{\theta}}^{\mathrm{ps}}$ (Thm. 2). Our protocol is lossless:
no information is wasted in the distillation (postselection) procedure. Fifth,
we discuss how our results can be applied to improve quantum learning in the
presence of imperfect detectors or postprocessing costs.

##  I Preliminaries

Consider an experiment with finite and discrete outcomes $k$ with
corresponding probabilities $p(k|\bm{\theta})$. The Fisher information matrix
is defined as

$I_{i,j}(\bm{\theta})=\sum_{k}p(k|\bm{\theta})\left\\{\partial_{i}\log[p(k|\bm{\theta})]\right\\}\left\\{\partial_{j}\log[p(k|\bm{\theta})]\right\\},$
(1)

where $\partial_{i}\equiv\frac{\partial}{\partial\theta_{i}}$ Cover and Thomas
(2006). The Fisher information matrix lower-bounds the covariance matrix
$\Sigma(\bm{\theta}^{\mathrm{e}})$ via the Cramér-Rao inequality:
$\Sigma(\bm{\theta}^{\mathrm{e}})\geq\left[NI(\bm{\theta})\right]^{-1}$.
Choosing a positive, real, $M\times M$ weight matrix $W$, introduces a scalar
Cramér-Rao bound:

$s(\Sigma(\bm{\theta}^{\mathrm{e}}),W)\equiv\operatorname{Tr}{\left[W\Sigma(\bm{\theta}^{\mathrm{e}})\right]}\geq\frac{1}{N}\operatorname{Tr}{\left[WI^{-1}(\bm{\theta})\right]}.$
(2)

If, e.g., $W=\mathbbm{1}$ and $\bm{\theta}^{\mathrm{e}}$ is an unbiased
estimator, the scalar risk function $s(\Sigma(\bm{\theta}^{\mathrm{e}}),W)$
equals the sum of the individual mean-square errors of the parameters in
$\bm{\theta}^{\mathrm{e}}$. See Albarelli  _et al._ (2020) for a review. For
unbiased, or “reasonable”, estimators $\bm{\theta}^{\mathrm{e}}$ and
$N\to\infty$, Ineq. (2) is saturated Lehmann and Casella (2006). In what
follows, we shall assume these conditions, such that
$s(\Sigma(\bm{\theta}^{\mathrm{e}}),W)\equiv
s(I(\bm{\theta}),W)=\operatorname{Tr}{\left[WI^{-1}(\bm{\theta})\right]}/N$.

From a learnability perspective, it is often useful to consider the most
informative experiment that extracts (Fisher) information from quantum states
$\hat{\rho}_{\bm{\theta}}$:

$\displaystyle s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)\equiv
s\left(\max_{\mathcal{M}}I(\bm{\theta}),W\right)=\frac{1}{N}\min_{\mathcal{M}}\operatorname{Tr}{\left[WI^{-1}(\bm{\theta})\right]}.$
(3)

Here, $\mathcal{M}$ is the set of all possible measurements.

The Fisher information matrix is upper-bounded by the quantum Fisher
information matrix Helstrom (1967); Liu  _et al._ (2019); Albarelli  _et al._
(2020): $I(\bm{\theta})\leq\mathcal{I}(\bm{\theta}|\hat{\rho}_{\bm{\theta}})$.
The quantum Fisher information matrix is defined by

$\mathcal{I}_{i,j}(\bm{\theta}|\hat{\rho}_{\bm{\theta}})=\operatorname{Tr}\left(\hat{L}_{j}\partial_{i}\hat{\rho}_{\bm{\theta}}\right).$
(4)

Here, $\hat{L}_{j}$ is the logarithmic derivative operator, which is not
uniquely defined Liu  _et al._ (2019). It can be defined using a symmetric
logarithmic derivative (SLD),
$2\partial_{i}\hat{\rho}_{\bm{\theta}}=\hat{L}^{\mathrm{(SLD)}}_{i}\hat{\rho}_{\bm{\theta}}+\hat{\rho}_{\bm{\theta}}\hat{L}^{\mathrm{(SLD)}}_{i}$,
or with a right logarithmic derivative (RLD),
$\partial_{i}\hat{\rho}_{\bm{\theta}}=\hat{\rho}_{\bm{\theta}}\hat{L}^{\mathrm{(RLD)}}_{i}$.
In the multiparameter scenario ($M>1$), noncommutation often forbids
measurements such that $I_{i,j}(\bm{\theta})=\mathcal{I}_{i,j}(\bm{\theta})$
for all $i,j$. Thus,
$I(\bm{\theta})\leq\mathcal{I}(\bm{\theta}|\hat{\rho}_{\bm{\theta}})$ cannot
commonly be saturated. Either the symmetric-logarithmic-derivative or the
right-logarithmic-derivative quantum Fisher information matrix can give a
bound that lies closer to the achievable bound. For pure states
$\hat{L}^{\mathrm{(SLD)}}_{i}=2\hat{L}^{\mathrm{(RLD)}}_{i}=2\partial_{i}\hat{\rho}_{\bm{\theta}}$,
and the symmetric-logarithmic-derivative quantum Fisher information matrix
[Eq. (4)] is

$\mathcal{I}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})=4\Re\left[\braket{\partial_{i}\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}-\braket{\partial_{i}\psi_{\bm{\theta}}}{\psi_{\bm{\theta}}}\braket{\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}\right],$
(5)

where
$\hat{\rho}_{\bm{\theta}}\equiv\ket{\psi_{\bm{\theta}}}\bra{\psi_{\bm{\theta}}}$
Liu  _et al._ (2019). In this theoretical proof-of-principle study, we proceed
with an investigation of pure states and the symmetric-logarithmic-derivative
quantum Fisher information matrix. An investigation of distilled quantum
learning in the presence of noise is left for an upcoming paper.

The quantum Fisher information matrix yields a scalar Cramér-Rao bound
Albarelli  _et al._ (2020):

$\displaystyle
s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)\geq\frac{1}{N}\operatorname{Tr}{\left[W\mathcal{I}^{-1}(\bm{\theta})\right]}.$
(6)

It is this bound that (directly or indirectly) leads quantum machine-learning
algorithms to optimize the quantum Fisher information matrix of their
subroutines Abbas  _et al._ (2020); Haug  _et al._ (2021); Meyer (2021).
However, Eq. (6) “only” provides a lower bound on
$s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)$. Consequently, it is
reasonable to ask: How good a measure of learnability is the quantum Fisher
information matrix? From an information theoretic perspective, the answer
Albarelli  _et al._ (2019); Carollo  _et al._ (2019) is given by

$\frac{1}{N}\operatorname{Tr}{\left[W\mathcal{I}^{-1}(\bm{\theta})\right]}\leq
h(\bm{\theta},W)\leq(1+\mathcal{Q})\frac{1}{N}\operatorname{Tr}{\left[W\mathcal{I}^{-1}(\bm{\theta})\right]},$
(7)

where $h(\bm{\theta},W)$ is Holevo’s lower bound of the Cramér-Rao inequality
Holevo (1977). The “geometric quantumness” measure $\mathcal{Q}$ (see Appendix
C) satisfies $0\leq\mathcal{Q}\leq 1$. Generally, it is hard to calculate
$h(\bm{\theta},W)$ (see Albarelli  _et al._ (2020) for the exact form).
Nevertheless, for pure states,
$s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)=h(\bm{\theta},W)$ Matsumoto
(2002).

For the purpose of the theoretical pure-state investigation in this work, the
formulae above can be summarized as

$\frac{1}{N}\operatorname{Tr}{\left[W\mathcal{I}^{-1}(\bm{\theta})\right]}\leq
s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)\leq
2\frac{1}{N}\operatorname{Tr}{\left[W\mathcal{I}^{-1}(\bm{\theta})\right]}.$
(8)

Within a factor of $2$, $\mathcal{I}(\bm{\theta})$ sets
$s^{(\mathrm{MI})}(\hat{\rho}_{\bm{\theta}},W)$. This constitutes our main
motivation for focusing on $\mathcal{I}(\bm{\theta})$ as a measure of quantum
learnability. Further, empirical motivation, can be found in Refs. Abbas  _et
al._ (2020); Haug  _et al._ (2021); Meyer (2021).

##  II Postselected Quantum Fisher Information Matrix

Here, we consider an experiment where an initial state, $\rho_{0}$, is evolved
sequentially by a series of $M$ unitary operators,
$\hat{U}(\bm{\theta})\equiv\prod_{m=M}^{1}\hat{U}_{m}(\theta_{m})$:
$\hat{\rho}_{0}\rightarrow\hat{\rho}_{\bm{\theta}}\equiv\hat{U}(\bm{\theta})\hat{\rho}_{0}\hat{U}^{\dagger}(\bm{\theta})$,
and then subject to a postselective measurement
$\\{\hat{F}_{1}=\hat{F},\hat{F}_{2}=\hat{1}-\hat{F}\\}$. $\hat{F}_{i}$ need
not be projective. The experiment is depicted in Fig. 1. We assume a discrete
Hilbert space of dimension $D$ and that $\hat{U}_{m}(\theta_{m})$ satisfies
Stone’s theorem on one-parameter unitary groups Stone (1932) such that
$\hat{U}_{m}(\theta_{m})=e^{i\theta_{m}\hat{A}_{m}}\;\forall m\in
1,\ldots,M$.333Experiments where $\hat{U}_{m}(\theta_{m})\neq
e^{i\theta_{m}\hat{A}_{m}}$ can often be transformed into the required form
via an artificial reparametrization. The Hermitian generators $\hat{A}_{m}$
are in general noncommuting.

Figure 1: Preparation and distillation of quantum states. First, $M$ unknown
parameters $\bm{\theta}$ are encoded in the initial state $\hat{\rho}_{0}$ by
the unitary $\hat{U}(\bm{\theta})$:
$\hat{\rho}_{0}\rightarrow\hat{\rho}_{\bm{\theta}}$. Second, the encoded state
$\hat{\rho}_{\bm{\theta}}$ is past through a postselective measurement
$\\{\hat{F}_{1}=\hat{F},\hat{F}_{2}=\hat{1}-\hat{F}\\}$. The postselection is
used to discard the quantum states unless outcome
$\hat{F}=\hat{K}^{\dagger}\hat{K}$ happens. Third, the experiment outputs the
distilled states
$\hat{\rho}_{\bm{\theta}}^{\textrm{ps}}=\hat{K}\hat{\rho}_{\bm{\theta}}\hat{K}^{\dagger}/p_{\bm{\theta}}^{\mathrm{ps}}$
with success probability
$p_{\bm{\theta}}^{\mathrm{ps}}=\mathrm{Tr}(\hat{F}\hat{\rho}_{\bm{\theta}})$.

We now present a formula for the quantum Fisher information matrix of the
distilled fraction of the output states in Fig. 1. These output states are
given by
$\ket{\psi_{\bm{\theta}}^{\mathrm{ps}}}\equiv\hat{K}\ket{\psi_{\bm{\theta}}}/\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}$,
where
$p_{\bm{\theta}}^{\mathrm{ps}}=\operatorname{Tr}[\hat{F}\hat{\rho}_{\bm{\theta}}]$
is the probability of successful postselection and $\hat{K}$ is the Kraus
operator that sets the postselection: $\hat{F}=\hat{K}^{\dagger}\hat{K}$. In
Appendix A, we evaluate Eq. (4) for $\ket{\psi^{\mathrm{ps}}_{\bm{\theta}}}$
and we find that

$\displaystyle\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=$
$\displaystyle
4\Re\bigg{\\{}\frac{1}{p_{\bm{\theta}}^{\mathrm{ps}}}\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]$
$\displaystyle-\frac{1}{(p_{\bm{\theta}}^{\mathrm{ps}})^{2}}\operatorname{Tr}\left[\hat{F}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\bigg{\\}}.$
(9)

Here,
$\hat{\tilde{A}}_{m}\equiv\left[\prod_{i=M}^{m+1}\hat{U}_{i}(\theta_{i})\right]\hat{A}_{m}\left[\prod_{j=m+1}^{M}\hat{U}^{\dagger}_{j}(\theta_{j})\right]$
for $m<M$. For $m=M$, $\hat{\tilde{A}}_{M}=\hat{A}_{M}$. The eigenspectra of
$\hat{\tilde{A}}_{m}$ and $\hat{A}_{m}$ are identical.

##  III Quasiprobabilistic analysis

We use quasiprobabilistic techniques to bound
$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ with
respective to classical and quantum statistics. Quasiprobability distributions
are mathematical objects that behave similar to probability distributions:
They sum to unity, and marginalizing over all but one of the arguments yields
a classical probability distribution. However, individual quasiprobabilities
can be nonclassical by having values outside $[0,1]$. This enables the
distributions to describe noncommuting quantum mechanics. The best-known
quasiprobability distribution is the complementary- and continuous-variable
Wigner function Wigner (1932). However, most of modern quantum information
research is framed in terms of discrete systems, e.g., systems of qubits; and
observables of interest are not necessarily complementary.

The complex-valued Kirkwood-Dirac (KD) quasiprobability distribution Kirkwood
(1933); Dirac (1945) is a relative of the Wigner function that describes,
straightforwardly, discrete systems—even qubits. The KD distribution has
recently illuminated quantum effects in weak-value amplification Steinberg
(1995); Dressel (2015); Yunger Halpern  _et al._ (2018), measurement
disturbance Hofmann (2011); Dressel and Jordan (2012); Dressel (2015); Monroe
_et al._ (2020), tomography Johansen (2007); Lundeen  _et al._ (2011); Lundeen
and Bamber (2012); Bamber and Lundeen (2014); Thekkadath  _et al._ (2016),
quantum chaos Swingle  _et al._ (2016); Yunger Halpern  _et al._ (2018);
Halpern  _et al._ (2019); González Alonso  _et al._ (2019); Landsman  _et al._
(2019); Mohseninia  _et al._ (2019), metrology Arvidsson-Shukur  _et al._
(2020b); Lupu-Gladstein et al. (prep), thermodynamics Levy and Lostaglio
(2019); Lostaglio (2020), and the foundations of quantum mechanics Griffiths
(1984); Goldstein and Page (1995); Hartle (2004); Hofmann (2011, 2012, 2014,
2015, 2016); Halliwell (2016); Stacey (2019). By optimizing a formula with
respect to a classical (real and non-negative) and a quantum (complex)
Kirkwood-Dirac distribution, classical and quantum bounds can be found,
respectively. Below we deploy this technique.

A KD distribution represents a quantum state $\hat{\rho}$ in terms of $k\geq
2$ sets of measurement operators. Equation (9) can be decomposed naturally in
terms of a KD distribution defined by a discrete $\hat{\rho}$ and $k=3$ sets
of measurement operators. Two sets are composed of the projectors onto the
subspaces of distinct eigenvalues of $\hat{\tilde{A}}_{i}$ and
$\hat{\tilde{A}}_{j}$, and one set contains the postselection measurement
operators:

$\displaystyle\left\\{\hat{\Pi}^{(i)}_{k}\,:\,\hat{\Pi}^{(i)}_{k}\hat{\tilde{A}}_{i}=a^{(i)}_{k}\hat{\Pi}^{(i)}_{k}\right\\},$
$\displaystyle\left\\{\hat{\Pi}^{(j)}_{l}\,:\,\hat{\Pi}^{(j)}_{l}\hat{\tilde{A}}_{j}=a^{(j)}_{l}\hat{\Pi}^{(j)}_{l}\right\\},$
$\displaystyle\left\\{\hat{F}_{1}=\hat{F},\hat{F}_{2}=\hat{1}-\hat{F}\right\\}.$

We order the eigenvalues of $\hat{\tilde{A}}_{i}$ and $\hat{\tilde{A}}_{j}$
ascendingly: $a_{1}^{(i)}\leq\cdots\leq a_{D}^{(i)}$, and define the spectral
eigengap $\Delta a^{(i)}\equiv a_{D}^{(i)}-a_{1}^{(i)}$ etc. We can now define
our operational KD distribution with respect to the operators above:

$\left\\{q^{\hat{\rho}}_{k,l,m}\right\\}\equiv\left\\{\operatorname{Tr}\left[\hat{\Pi}^{(i)}_{k}\hat{F}_{m}\hat{\Pi}^{(j)}_{l}\hat{\rho}\right]\right\\}.$
(10)

The KD distribution obeys an analogue of Bayes’ Theorem Johansen (2007);
Yunger Halpern  _et al._ (2018). Consequently, we can define a distribution
that corresponds to $\\{q^{\hat{\rho}}_{k,l,m}\\}$ conditioned on the
postselection yielding outcome $\hat{F}$:

$\displaystyle\left\\{Q^{\hat{\rho}}_{k,l}\right\\}$
$\displaystyle\equiv\left\\{\dfrac{q^{\hat{\rho}}_{k,l,m=1}}{\sum\limits_{k,l,m=1}q^{\hat{\rho}}_{k,l,m}}\right\\}=\left\\{\operatorname{Tr}\left[\hat{\Pi}^{(i)}_{k}\hat{F}\hat{\Pi}^{(j)}_{l}\hat{\rho}\right]/p_{\bm{\theta}}^{\mathrm{ps}}\right\\}.$
(11)

When $\\{Q^{\hat{\rho}}_{k,l}\\}$ is classical, all
$|Q^{\hat{\rho}}_{k,l}|\leq 1$. Negative quasiprobabilities allow the
denominators of Eq. (11) to approach $0$ even for finite numerators. Then,
$|Q^{\hat{\rho}}_{k,l}|$ can be arbitrarily large. Such negativity, an example
below shows, enables
$\left|\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\right|$
to be anomalously large, compared to experiments described by classical
distributions. This can increase distilled states’ multiparameter information
to nonclassically large values.

######  Theorem 1 (Necessary condition for anomalous postselected quantum
Fisher information matrix).

Suppose that a postselected quantum Fisher information matrix has some entry
$\left|\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\right|>\Delta
a^{(i)}\Delta a^{(j)}$. Then, an underlying KD distribution
$\left\\{Q^{\hat{\rho}}_{k,l}\right\\}$ necessarily contains at least one
negative value.

Proof of Thm. 1: We prove this theorem by contradiction. First, we use
Distribution (11) to recast Eq. (9):

$\displaystyle\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=$
$\displaystyle
4\Re\bigg{\\{}\sum\limits_{k,l}a^{(i)}_{k}a^{(j)}_{l}Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}$
$\displaystyle-\bigg{(}\sum\limits_{k^{\prime},l^{\prime}}a^{(i)}_{k^{\prime}}Q^{\hat{\rho}_{\bm{\theta}}}_{k^{\prime},l^{\prime}}\bigg{)}\bigg{(}\sum\limits_{k^{\prime\prime},l^{\prime\prime}}a^{(j)}_{l^{\prime\prime}}Q^{\hat{\rho}_{\bm{\theta}}}_{k^{\prime\prime},l^{\prime\prime}}\bigg{)}\bigg{\\}}.$
(12)

Equation (III) is a quantum extension of a covariance, where
$Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}$ replaces classical joint probabilities.
Second, we assume that $\\{Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\\}$ is
classical. Third, ignoring the specific form of
$\\{Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\\}$, we maximize and minimize Eq. (III)
over all classical distributions. When
$Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\in[0,1]$ and $i\neq j$, Eq. (III) has the
form of ($4$ times) a classical covariance with maximum and minimum values
$\Delta a^{(i)}\Delta a^{(j)}$ and $-\Delta a^{(i)}\Delta a^{(j)}$,
respectively.444Applying the Cauchy-Schwarz inequality to a covariance of
random variables $X$ and $Y$ yields
$|\mathrm{Cov}(X,Y)|\leq\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}$. When
$Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\in[0,1]$ and $i=j$, Eq. (III) is upper-
bounded by $(\Delta a^{(i)})^{2}$ and lower-bounded by $0$ Arvidsson-Shukur
_et al._ (2020b). Per definition, an anomalous QFIM entry breaks these bounds,
such that the assumption of a classical distribution
$\\{Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\\}$ cannot be satisfied. Consequently,
if
$\left|\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\right|>\Delta
a^{(i)}\Delta a^{(j)}$, then $\\{Q^{\hat{\rho}_{\bm{\theta}}}_{k,l}\\}$ is
nonclassical. The form of Eq. (III) implies that any nonreal values cancel.
Thus, the nonclassicality must be in the form of negativity.$\square$

An immediate corollary follows:

######  Corollary 1.

In a classically commuting theory, a theory in which operators commute, the
quantum Fisher information matrix satisfies
$\left|\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\right|\leq\Delta
a^{(i)}\Delta a^{(j)}$.

Proof: Reference Arvidsson-Shukur  _et al._ (2020a) proves that noncommutation
is necessary for nonclassical KD distributions.555In fact, noncommutation is
necessary, but not sufficient, for KD nonclassicality Arvidsson-Shukur  _et
al._ (2020a). The corollary thus follows from Thm. 1.$\square$

##  IV Distilling quantum learnability

If an underlying KD distribution possesses negative values, it is possible to
use postselection to distil quantum Fisher information such that
$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})>\Delta
a^{(i)}\Delta a^{(j)}$. However,
$\mathcal{I}^{-1}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ bounds
$s^{(\mathrm{MI})}(\psi^{\mathrm{ps}}_{\bm{\theta}},W)$ via matrix
inequalities [Ineqs. (8)], and it is generally hard to know which
$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ would be
beneficial to amplify. Furthermore, setting a postselection operator $\hat{F}$
to optimize one entry in
$\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ could have a
detrimental effect on another entry. Below, we show that it is possible to
chose $\hat{F}$ such that
$\det{\mathcal{I}}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\rightarrow\infty$,
and $s^{(\mathrm{MI})}(\psi^{\mathrm{ps}}_{\bm{\theta}},W)\rightarrow 0$. The
price to pay for larger portions of
$\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ is smaller success
chances $p_{\bm{\theta}}^{\mathrm{ps}}$. First, we provide a guiding example
of two-parameter estimation of a postselected qubit. Then, we present general
theory.

###  IV.I Example

Consider a qubit in an initial state $\ket{\psi_{0}}=\ket{0}$. The quantum
circuit of interest is parametrized by two parameters
$\bm{\theta}=(\theta_{1},\theta_{2})$ and represented by the unitary
$\hat{U}(\bm{\theta})=e^{i(\hat{\sigma}_{x}+\hat{\sigma}_{z})\theta_{2}/\sqrt{2}}e^{i\hat{\sigma}_{x}\theta_{1}}$,
where $\hat{\sigma}_{k}$ is the $k^{\mathrm{th}}$ Pauli operator. The quantum
Fisher information matrix of the output state
$\ket{\psi_{\bm{\theta}}}=\hat{U}(\bm{\theta})\ket{0}$ is

$\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}})=\begin{pmatrix}4\;\;\;&2\sqrt{2}\\\
2\sqrt{2}\;\;\;&3-\cos{(4\theta_{1})}\end{pmatrix}.$ (13)

We assume that our initial guess of $\bm{\theta}$ is off by $1/10$ for both
$\theta_{1}$ and $\theta_{2}$:
$\bm{\theta}^{0}=(\theta_{1}+\frac{1}{10},\theta_{2}+\frac{1}{10})$. We set
the Kraus operator to
$\hat{K}=(\frac{1}{\sqrt{10}}-1)\hat{\rho}_{\bm{\theta}^{0}}+\hat{1}$. The
probability of a successful postselection is given by
$p_{\bm{\theta}}^{\mathrm{ps}}=\operatorname{Tr}[\hat{K}^{\dagger}\hat{K}\hat{\rho}_{\bm{\theta}}]\approx
1/10$. Moreover, the postselected (distilled) quantum Fisher information
matrix is given by

$\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}}^{\mathrm{ps}})\approx
10\begin{pmatrix}4\;\;\;&2\sqrt{2}\\\
2\sqrt{2}\;\;\;&3-\cos{(4\theta_{1})}\end{pmatrix}.$ (14)

All entries of $\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}}^{\mathrm{ps}})$
break their classical maximum of $\Delta a^{(i)}\Delta a^{(j)}=4$. By reducing
the number of quantum states that will reach the final detector by a factor of
ten, we have also achieved a tenfold increase of the information content of
the remaining states.

###  IV.II General theory

Here, we outline how to achieve a diverging quantum Fisher information matrix
in the general scenario. We give the following theorem

######  Theorem 2 (Arbitrary distillation of quantum learnability).

For a sufficiently accurate initial estimate, the theoretically attainable
average information per trial about the unknown parameter vector $\bm{\theta}$
has no upper limit: It is possible to distill quantum states such that
$\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})\rightarrow\infty$
and $s^{(\mathrm{MI})}(\psi^{\mathrm{ps}}_{\bm{\theta}},W)\rightarrow 0$ in a
lossless fashion.

Proof of Thm. 2: Our proof is constructive. We present a specific protocol
that achieves the objective; other protocols might exist. Our results assume
that we possess an initial estimate of $\bm{\theta}$, $\bm{\theta}^{0}$, that
is sufficiently close to the true value:
$\bm{\theta}^{0}\approx\bm{\theta}$.666Also in weak-value amplification and
single-parameter metrology, conducting the optimal measurement generally
requires a good initial estimate of the unknown parameter of interest.
Moreover, many variational quantum algorithms, e.g. for quantum computational
chemistry, require good initial estimates $\bm{\theta}^{0}\approx\bm{\theta}$
Tang  _et al._ (2019); Grimsley  _et al._ (2019); Yordanov  _et al._ (2020);
McArdle  _et al._ (2020); Lavrijsen  _et al._ (2020); Bittel and Kliesch
(2021).  In the limit of many trials $N\rightarrow\infty$, we can always
“sacrifice” a vanishingly small fraction of the trials to achieve such an
initial estimate. $\bm{\theta}^{0}$ can also be improved iteratively, suitably
using a Kalman filter Zarchan  _et al._ (2000). Defining
$\delta_{m}\equiv\theta_{m}-\theta_{m}^{0}$ such that $M\delta^{2}\approx 0$,
$\hat{\rho}_{\bm{\theta}^{0}}$ is given by

$\displaystyle\hat{\rho}_{\bm{\theta}^{0}}\equiv\hat{U}(\bm{\theta}^{0})\hat{\rho}_{0}\hat{U}^{\dagger}(\bm{\theta}^{0})=\hat{\rho}_{\bm{\theta}}+\left[\hat{\rho}_{\bm{\theta}}\,,\,\hat{D}\right]+\mathcal{O}\left(\delta^{2}\right),$
(15)

where $\hat{D}\equiv-i\sum_{m=1}^{M}\delta_{m}\hat{\tilde{A}}_{m}$ and, as
before,
$\hat{\tilde{A}}_{m}\equiv\left[\prod_{i=M}^{m+1}\hat{U}_{i}(\theta_{i})\right]\hat{A}_{m}\left[\prod_{j=m+1}^{M}\hat{U}^{\dagger}_{j}(\theta_{j})\right]$.

We consider the setup depicted in Fig. 1, with postselected quantum Fisher
information given by Eqs. (9) and (III). We set the Kraus operator $\hat{K}$
with respect to the initial estimate of the quantum state before
postselection:777This choice of $\hat{K}$ generalizes the technique used by
Lupu Gladstein et al. in single-parameter metrology of optical qubits Lupu-
Gladstein et al. (prep).

$\hat{K}=(t-1)\hat{\rho}_{\bm{\theta}^{0}}+\hat{1},$ (16)

where $0\leq t\leq 1$. Physically, this choice of $\hat{K}$ generates a
postselection (distillation) procedure that transmits the expected state
$\hat{\rho}_{\bm{\theta}^{0}}$ with probability $t^{2}$ and transmits fully
any state orthogonal to $\hat{\rho}_{\bm{\theta}^{0}}$. Substituting $\hat{K}$
and $\hat{\rho}_{\bm{\theta}^{0}}$ into
$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ [Eq. (9)]
yields

$\displaystyle\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=\frac{1}{t^{2}}\mathcal{I}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})+\mathcal{O}\left(\delta^{2}\right).$
(17)

Equation (17) is derived in Appendix B and requires that $M\delta^{2}\ll
t^{2}$. $\hat{K}$ is independent of $i,j$, such that our distillation
technique amplifies all nonzero entries of
$\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ simultaneously:
$\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}})/t^{2}+\mathcal{O}\left(\delta^{2}\right)$.
Combining this result with Ineqs. (8),
$s^{(\mathrm{MI})}(\psi^{\mathrm{ps}}_{\bm{\theta}},W)\rightarrow 0$ when
$M\delta^{2}\ll t^{2}\rightarrow 0$.888We have assumed that the nonpostslected
quantum Fisher information $\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}})$ is
nonsingular. If it is singular, one has to remove the singularity-producing
parameters from the analysis. Finally, the probability of successful
postselection is given by
$p_{\bm{\theta}}^{\mathrm{ps}}=t^{-2}+\mathcal{O}(\delta^{2})$ (see App. B).
Consequently, the distillation of information is lossless:
$p_{\bm{\theta}}^{\mathrm{ps}}\times\mathcal{I}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}})+\mathcal{O}(\delta^{2})$.999Appendix
C shows that the geometric quantumness $\mathcal{Q}$ in Ineqs. (7) is constant
[to $\mathcal{O}(\delta^{2})$] with respect to the postselection. This
concludes our constructive proof.$\square$

##  V Applications

By distilling the multiparameter Fisher information, the intensity of output
states is reduced. This can lead to learnability improvements by allowing
metrologists and machine learners to use an intensity of input states that
normally would have caused the output detectors to saturate. The information
content available in the distilled, low-intensity output is identical to what
the nondistilled, high-intensity output would have been.

As an example, consider encoding an image in quantum states
$\hat{\rho}_{\bm{\theta}}$. $\bm{\theta}=(\theta_{1},...,\theta_{M})$ is a
vector of the image’s pixels’ intensities. Perhaps our task is to find
imperfections in the image-encoding procedure of a certain image
$\bm{\theta}^{\star}$. Then $\hat{\rho}_{\bm{\theta}^{\star}}$ is a good
initial guess to learn the imperfectly encoded, true image
$\bm{\theta}\approx\bm{\theta}^{\star}$. Or perhaps we want to learn an image
that deviates slightly from a blank image. Then $\bm{\theta}^{0}=\bm{0}$ and
$\hat{\rho}_{\bm{\theta}^{0}}=\hat{\rho}_{0}$ is a good initial guess. Our
distillation protocol allows us to both avoid detector saturation and increase
sensitivity, without losing information, when measuring
$\hat{\rho}_{\bm{\theta}}$ to learn the image.

A particle-number detector will suffer from a dead time, the time needed to
reset the detector after triggering it. In the jargon of experimental costs:
The dead time associates a temporal cost with the measurement Lupu-Gladstein
et al. (prep). Also, measurements call for postprocessing, which cost further
time and computation. Under the right conditions, our distillation protocol
enables an experimentalist to incur the final-measurement’s cost only when the
probe state carries a great deal of information. The “right conditions” are
when the postselection is experimentally cheaper than the final measurement.

Many quantum schemes can be sped up by using several quantum processors in
parallel Tang  _et al._ (2019); Grimsley  _et al._ (2019); Yordanov  _et al._
(2020). By using our protocol to distill the output from parallel processors,
it could be possible to reduce the number of final-measurement apparatuses in
setups, decreasing the monetary cost of parallel-processor schemes.

One can also envision scenarios where the encoding and final measurements are
spatially separated and connected by quantum channels. Our distillation
protocol allows the rate of quantum-state transmission to decrease, whilst
keeping the average information flow constant.

##  VI Conclusion

The quantum Fisher information matrix enables scalar quantification of quantum
learnability in multiparameter metrology and machine learning. We have shown
that there exist upper and lower classical bounds on the entries in the
quantum Fisher information matrix. Kirkwood-Dirac negativity, a narrower
nonclassicality concept than noncommutation, allows the entries to break these
bounds (Thm. 1). Motivated by this result, we designed a protocol that uses a
quantum analogue of Bayes’ theorem to amplify uniformly the nonzero entries in
the quantum Fisher information matrix. This translates into the ability to
probabilistically distill quantum learnability in a lossless fashion. We
proved (Thm. 2) that there is no upper bound on how much multiparameter
information can be distilled into a small number of states. From a theoretical
perspective, our results shed new light on the quantum Fisher information
matrix and generalizes, to the multiparameter-quantum-learnability regime,
previous results in single-parameter postselected metrology and weak-value
amplification. From a practical perspective, our results could mitigate the
impact of detector imperfections and enable simplified setups in parallelized
quantum schemes.

Acknowledgements.—The authors would like to thank Crispin Barnes, Rafal
Demkowicz-Dobrzanski, Bobak Kiani, Aleks Lasek, Zi-Wen Liu, Seth Lloyd, Noah
Lupu Gladstein, Milad Marvian, Yordan Yordanov, and Nicole Yunger Halpern for
useful discussions. This work was supported by the EPSRC, Lars Hierta’s
Memorial Foundation, and Girton College.

##  Appendix A Derivation of Eq. (9)

This appendix derives Eq. (9). We proceed by changing the quantum state in

$\mathcal{I}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})=4\Re\left[\braket{\partial_{i}\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}-\braket{\partial_{i}\psi_{\bm{\theta}}}{\psi_{\bm{\theta}}}\braket{\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}\right],$
(18)

to
$\ket{\psi_{\bm{\theta}}}\rightarrow\ket{\psi_{\bm{\theta}}^{\mathrm{ps}}}\equiv\hat{K}\ket{\psi_{\bm{\theta}}}/\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}$.
Remember that $\hat{F}=\hat{K}^{\dagger}\hat{K}$.

The first inner product is given by

$\displaystyle\left(\partial_{i}\frac{\bra{\psi_{\bm{\theta}}}\hat{K}^{\dagger}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)\cdot\left(\partial_{j}\frac{\hat{K}\ket{\psi_{\bm{\theta}}}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)=$
$\displaystyle\left(\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{K}^{\dagger}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}-\frac{1}{2}\frac{\bra{\psi_{\bm{\theta}}}\hat{K}^{\dagger}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{\frac{3}{2}}}\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}\right)\cdot\left(\frac{\hat{K}\ket{\partial_{j}\psi_{\bm{\theta}}}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}-\frac{1}{2}\frac{\hat{K}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{\frac{3}{2}}}\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}\right)$
(19) $\displaystyle=$
$\displaystyle\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}$
$\displaystyle-\frac{1}{2}\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}+\frac{1}{4}\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{3}}\left(\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}\right)\left(\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}\right)$
(20) $\displaystyle=$
$\displaystyle\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}$
$\displaystyle-\frac{1}{2}\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}+\frac{1}{4}\frac{\left(\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}\right)\left(\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}\right)}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}.$
(21)

The last equality follows from
$p_{\bm{\theta}}^{\mathrm{ps}}=\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}$.

The second inner product is given by

$\displaystyle\left(\partial_{i}\frac{\bra{\psi_{\bm{\theta}}}\hat{K}^{\dagger}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)\cdot\left(\frac{\hat{K}\ket{\psi_{\bm{\theta}}}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)=$
$\displaystyle\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\left(\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}\right)=\frac{\bra{\partial_{i}\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\left(\partial_{i}p_{\bm{\theta}}^{\mathrm{ps}}\right)}{p_{\bm{\theta}}^{\mathrm{ps}}}.$
(22)

The third inner product is given by

$\displaystyle\left(\frac{\bra{\psi_{\bm{\theta}}}\hat{K}^{\dagger}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)\cdot\left(\partial_{j}\frac{\hat{K}\ket{\psi_{\bm{\theta}}}}{\sqrt{p_{\bm{\theta}}^{\mathrm{ps}}}}\right)=$
$\displaystyle\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\psi_{\bm{\theta}}}}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\left(\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}\right)=\frac{\bra{\psi_{\bm{\theta}}}\hat{F}\ket{\partial_{j}\psi_{\bm{\theta}}}}{p_{\bm{\theta}}^{\mathrm{ps}}}-\frac{1}{2}\frac{\left(\partial_{j}p_{\bm{\theta}}^{\mathrm{ps}}\right)}{p_{\bm{\theta}}^{\mathrm{ps}}}.$
(23)

Combining these expressions:

$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=4\Re\left[\frac{1}{p_{\bm{\theta}}^{\mathrm{ps}}}\braket{\partial_{i}\psi_{\bm{\theta}}}{\hat{F}}{\partial_{j}\psi_{\bm{\theta}}}-\frac{1}{\left(p_{\bm{\theta}}^{\mathrm{ps}}\right)^{2}}\braket{\partial_{i}\psi_{\bm{\theta}}}{\hat{F}}{\psi_{\bm{\theta}}}\braket{\psi_{\bm{\theta}}}{\hat{F}}{\partial_{j}\psi_{\bm{\theta}}}\right].$
(24)

Transforming the inner products to traces, we retrieve Eq. (9):

$\displaystyle\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=$
$\displaystyle
4\Re\bigg{\\{}\frac{1}{p_{\bm{\theta}}^{\mathrm{ps}}}\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]-\frac{1}{(p_{\bm{\theta}}^{\mathrm{ps}})^{2}}\operatorname{Tr}\left[\hat{F}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\bigg{\\}}.$
(25)

Here, we have used that

$\displaystyle\ket{\partial_{j}\psi_{\bm{\theta}}}=$
$\displaystyle\partial_{j}\hat{U}(\bm{\theta})\ket{\psi_{0}}$ (26)
$\displaystyle=$
$\displaystyle\hat{U}_{M}(\theta_{M})\cdots\hat{U}_{j+1}(\theta_{j+1})\hat{A}_{j}\hat{U}_{j}(\theta_{j})\cdots\hat{U}_{1}(\theta_{1})\ket{\psi_{0}}$
(27) $\displaystyle=$
$\displaystyle\hat{U}_{M}(\theta_{M})\cdots\hat{U}_{j+1}(\theta_{j+1})\hat{A}_{j}\hat{U}_{j+1}^{\dagger}(\theta_{j+1})\cdots\hat{U}_{M}^{\dagger}(\theta_{M})\hat{U}_{M}(\theta_{M})\cdots\hat{U}_{j+1}(\theta_{j+1})\hat{U}_{j}(\theta_{j})\cdots\hat{U}_{1}(\theta_{1})\ket{\psi_{0}}$
(28) $\displaystyle=$
$\displaystyle\hat{\tilde{A}}_{j}\hat{U}_{M}(\theta_{M})\cdots\hat{U}_{1}(\theta_{1})\ket{\psi_{0}}$
(29) $\displaystyle=$
$\displaystyle\hat{\tilde{A}}_{j}\hat{U}(\bm{\theta})\ket{\psi_{0}}$ (30)
$\displaystyle=$ $\displaystyle\hat{\tilde{A}}_{j}\ket{\psi_{\bm{\theta}}},$
(31)

where
$\hat{\tilde{A}}_{j}\equiv\hat{U}_{M}(\theta_{M})\cdots\hat{U}_{j+1}(\theta_{j+1})\hat{A}_{j}\hat{U}_{j+1}^{\dagger}(\theta_{j+1})\cdots\hat{U}_{M}^{\dagger}(\theta_{M})$
for $j<M$. For $j=M$, $\hat{\tilde{A}}_{M}=\hat{A}_{M}$.

##  Appendix B Derivation of Eq. (17)

This appendix evaluates the postselected quantum Fisher information matrix
$\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})$ [Eq. (9)]
for the choice of Kraus operator presented in Eq. (16):
$\hat{K}=(t-1)\hat{\rho}_{\bm{\theta}^{0}}+\hat{1}$. This Kraus operator
generates the postselection operator
$\hat{F}=(t^{2}-1)\hat{\rho}_{\bm{\theta}^{0}}+\hat{1}$. In the main text, we
defined
$\hat{\rho}_{\bm{\theta}^{0}}\equiv\hat{U}(\bm{\theta}^{0})\hat{\rho}_{0}\hat{U}^{\dagger}(\bm{\theta}^{0})=\hat{\rho}_{\bm{\theta}}+\left[\hat{\rho}_{\bm{\theta}}\,,\,\hat{D}\right]+\mathcal{O}\left(\delta^{2}\right)$.
The following calculations assume that $M\delta^{2}\ll t^{2}$. We define
$\hat{C}\equiv[\hat{\rho}_{\bm{\theta}}\,,\,\hat{D}]$ to simplify notation. We
begin by evaluating the individual terms of Eq. (9). Then, we combine these
terms.

First, we calculate the postselection probability
$p_{\bm{\theta}}^{\mathrm{ps}}$ in Eq. (9):

$\displaystyle p_{\bm{\theta}}^{\mathrm{ps}}=$
$\displaystyle\operatorname{Tr}\left[\hat{F}\hat{\rho}_{\bm{\theta}}\right]$
(32) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}^{0}}(t^{2}-1)+\hat{1}\right)\hat{\rho}_{\bm{\theta}}\right]$
(33) $\displaystyle=$
$\displaystyle(t^{2}-1)\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}^{0}}\hat{\rho}_{\bm{\theta}}\right]+1$
(34) $\displaystyle=$
$\displaystyle(t^{2}-1)\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}}+\hat{C}\right)\hat{\rho}_{\bm{\theta}}\right]+1+\mathcal{O}\left(\delta^{2}\right)$
(35) $\displaystyle=$
$\displaystyle(t^{2}-1)\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\right]+1+\mathcal{O}\left(\delta^{2}\right)$
(36) $\displaystyle=$ $\displaystyle
t^{2}+\mathcal{O}\left(\delta^{2}\right).$ (37)

Here, we have used that
$\operatorname{Tr}[\hat{\rho}_{\bm{\theta}}\hat{C}]=0$.

Second, we calculate the first trace in Eq. (9):

$\displaystyle\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]$
$\displaystyle=\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}^{0}}(t^{2}-1)+\hat{1}\right)\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]$
(38)
$\displaystyle=\operatorname{Tr}\left[\left((t^{2}-1)(\hat{\rho}_{\bm{\theta}}+\hat{C})+\hat{1}\right)\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta\right)$
(39)
$\displaystyle=(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta\right).$
(40)

Third, we calculate the second trace in Eq. (9):

$\displaystyle\operatorname{Tr}\left[\hat{F}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]=$
$\displaystyle\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}^{0}}(t^{2}-1)+\hat{1}\right)\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]]$
(41) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\left((t^{2}-1)(\hat{\rho}_{\bm{\theta}}+\hat{C})+\hat{1}\right)\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right)$
(42) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\operatorname{Tr}\left[(t^{2}-1)(\hat{\rho}_{\bm{\theta}}+\hat{C})\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right)$
(43) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right)$
(44) $\displaystyle=$ $\displaystyle
t^{2}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right).$
(45)

Fourth, we calculate the third trace in Eq. (9):

$\displaystyle\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]=$
$\displaystyle\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}^{0}}(t^{2}-1)+\hat{1}\right)\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]]$
(46) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\left((t^{2}-1)(\hat{\rho}_{\bm{\theta}}+\hat{C})+\hat{1}\right)\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\mathcal{O}\left(\delta^{2}\right)$
(47) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\operatorname{Tr}\left[(t^{2}-1)(\hat{\rho}_{\bm{\theta}}+\hat{C})\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\mathcal{O}\left(\delta^{2}\right)$
(48) $\displaystyle=$
$\displaystyle\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\mathcal{O}\left(\delta^{2}\right)$
(49) $\displaystyle=$ $\displaystyle
t^{2}\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\mathcal{O}\left(\delta^{2}\right).$
(50)

Fifth, we calculate the product of the second and third trace in Eq. (9):

$\displaystyle\operatorname{Tr}\left[\hat{F}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\times\operatorname{Tr}\left[\hat{F}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]=$
$\displaystyle
t^{4}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]$
$\displaystyle+t^{2}(t^{2}-1)\left\\{\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\operatorname{Tr}\left[\hat{C}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\right\\}+\mathcal{O}\left(\delta^{2}\right)$
(51) $\displaystyle=$ $\displaystyle
t^{4}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]$
$\displaystyle+t^{2}(t^{2}-1)\Big{\\{}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{D}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]-\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{D}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]$
$\displaystyle+\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{D}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]-\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\operatorname{Tr}\left[\hat{D}\hat{\rho}_{\bm{\theta}}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\Big{\\}}+\mathcal{O}\left(\delta^{2}\right)$
(52) $\displaystyle=$ $\displaystyle
t^{4}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]$
$\displaystyle+t^{2}(t^{2}-1)\Big{\\{}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{D}\hat{\tilde{A}}_{j}\right]-\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\operatorname{Tr}\left[\hat{D}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\Big{\\}}+\mathcal{O}\left(\delta^{2}\right)$
(53) $\displaystyle=$ $\displaystyle
t^{4}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+t^{2}(t^{2}-1)\operatorname{Tr}\left[\left(\hat{\rho}_{\bm{\theta}}\hat{D}-\hat{D}\hat{\rho}_{\bm{\theta}}\right)\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right)$
(54) $\displaystyle=$ $\displaystyle
t^{4}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+t^{2}(t^{2}-1)\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\mathcal{O}\left(\delta^{2}\right).$
(55)

Again, we have used the result that
$\operatorname{Tr}[\hat{\rho}_{\bm{\theta}}\hat{C}]=0$.

Finally, we combine the calculated expressions:

$\displaystyle\mathcal{I}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=$
$\displaystyle
4\Re\Bigg{\\{}\frac{(t^{2}-1)}{t^{2}}\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]+\frac{(t^{2}-1)}{t^{2}}\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]+\frac{1}{t^{2}}\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]$
$\displaystyle-\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]-\frac{(t^{2}-1)}{t^{2}}\operatorname{Tr}\left[\hat{C}\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\Bigg{\\}}+\mathcal{O}\left(\delta^{2}\right)$
(56) $\displaystyle=$
$\displaystyle\frac{4}{t^{2}}\Re\bigg{\\{}\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]-\operatorname{Tr}\left[\hat{\rho}_{\bm{\theta}}\hat{\tilde{A}}_{i}\right]\operatorname{Tr}\left[\hat{\tilde{A}}_{j}\hat{\rho}_{\bm{\theta}}\right]\bigg{\\}}$
(57) $\displaystyle=$
$\displaystyle\frac{1}{t^{2}}\mathcal{I}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})+\mathcal{O}\left(\delta^{2}\right).$
(58)

This is the expression given in Eq. (17).

##  Appendix C Postselected geometric quantumness

The geometric quantumness measure $\mathcal{Q}$ in Ineqs. (7) is given by

$\mathcal{Q}=||i\mathcal{I}^{-1}(\bm{\theta}|\psi_{\bm{\theta}})\mathcal{J}(\bm{\theta}|\psi_{\bm{\theta}})||_{\infty},$
(59)

where $||X||_{\infty}$ denotes the largest eigenvalue of $X$.
$\mathcal{J}(\bm{\theta}|\psi_{\bm{\theta}})$ is the Uhlmann
curvature101010For pure states, $\mathcal{J}(\bm{\theta}|\psi_{\bm{\theta}})$
is (four times) the imaginary part of the quantum geometric tensor.
$\mathcal{I}(\bm{\theta}|\psi_{\bm{\theta}})$ is (four times) the real part.
Carollo  _et al._ (2018) given by

$\mathcal{J}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})=4\Im\left[\braket{\partial_{i}\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}-\braket{\partial_{i}\psi_{\bm{\theta}}}{\psi_{\bm{\theta}}}\braket{\psi_{\bm{\theta}}}{\partial_{j}\psi_{\bm{\theta}}}\right].$
(60)

The same tricks used in Appendix B can be used to show that

$\mathcal{J}_{i,j}(\bm{\theta}|\psi^{\mathrm{ps}}_{\bm{\theta}})=\frac{1}{t^{2}}\mathcal{J}_{i,j}(\bm{\theta}|\psi_{\bm{\theta}})+\mathcal{O}\left(\delta^{2}\right).$
(61)

Thus, at least to $\mathcal{O}(\delta^{2})$, the geometric quantumness
$\mathcal{Q}$ is constant with respect to the postselection.

## References

  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nature photonics 5, 222 (2011).
  * Maccone (2013) L. Maccone, Phys. Rev. A 88, 042109 (2013).
  * Szczykulska  _et al._ (2016) M. Szczykulska, T. Baumgratz, and A. Datta, Advances in Physics: X 1, 621 (2016), https://doi.org/10.1080/23746149.2016.1230476 .
  * Kiani  _et al._ (2021) B. T. Kiani, G. De Palma, M. Marvian, Z.-W. Liu, and S. Lloyd, arXiv preprint arXiv:2101.03037 (2021).
  * Arvidsson-Shukur  _et al._ (2020a) D. R. Arvidsson-Shukur, J. C. Drori, and N. Y. Halpern, arXiv preprint arXiv:2009.04468 (2020a), https://arxiv.org/abs/2009.04468.
  * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * Liu  _et al._ (2019) J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).
  * Albarelli  _et al._ (2020) F. Albarelli, M. Barbieri, M. Genoni, and I. Gianani, Physics Letters A 384, 126311 (2020).
  * Rao (1992) C. R. Rao, in  _Breakthroughs in statistics_ (Springer, 1992) pp. 235–247.
  * Cramér (2016) H. Cramér,  _Mathematical methods of statistics (PMS-9)_ , Vol. 9 (Princeton University Press, 2016).
  * Ballester (2004) M. A. Ballester, Phys. Rev. A 69, 022303 (2004).
  * Imai and Fujiwara (2007) H. Imai and A. Fujiwara, Journal of Physics A: Mathematical and Theoretical 40, 4391 (2007).
  * Genoni  _et al._ (2013) M. G. Genoni, M. G. A. Paris, G. Adesso, H. Nha, P. L. Knight, and M. S. Kim, Phys. Rev. A 87, 012107 (2013).
  * Humphreys  _et al._ (2013) P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, Phys. Rev. Lett. 111, 070403 (2013).
  * 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).
  * Chen and Yuan (2017) Y. Chen and H. Yuan, New Journal of Physics 19, 063023 (2017).
  * Liuzzo-Scorpo  _et al._ (2018) P. Liuzzo-Scorpo, L. A. Correa, F. A. Pollock, A. Górecka, K. Modi, and G. Adesso, New Journal of Physics 20, 063009 (2018).
  * Lipka-Bartosik and Demkowicz-Dobrzański (2018) P. Lipka-Bartosik and R. Demkowicz-Dobrzański, Journal of Physics A: Mathematical and Theoretical 51, 474001 (2018).
  * Dressel  _et al._ (2014) J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014).
  * Harris  _et al._ (2017) J. Harris, R. W. Boyd, and J. S. Lundeen, Phys. Rev. Lett. 118, 070802 (2017).
  * Xu  _et al._ (2020) L. Xu, Z. Liu, A. Datta, G. C. Knee, J. S. Lundeen, Y.-q. Lu, and L. Zhang, Phys. Rev. Lett. 125, 080501 (2020).
  * Arvidsson-Shukur  _et al._ (2020b) D. R. M. Arvidsson-Shukur, N. Yunger Halpern, H. V. Lepage, A. A. Lasek, C. H. W. Barnes, and S. Lloyd, Nature Communications 11, 3775 (2020b).
  * Lupu-Gladstein et al. (prep) N. Lupu-Gladstein et al., “Experimental demonstration of postselection-enhanced metrology and its connection to quantum non-commutation,” (in prep).
  * Vella  _et al._ (2019) A. Vella, S. T. Head, T. G. Brown, and M. A. Alonso, Phys. Rev. Lett. 122, 123603 (2019).
  * Xia  _et al._ (2020) B. Xia, J. Huang, C. Fang, H. Li, and G. Zeng, Phys. Rev. Applied 13, 034023 (2020).
  * Ho and Kondo (2021) L. B. Ho and Y. Kondo, Journal of Mathematical Physics 62, 012102 (2021), https://doi.org/10.1063/5.0024555 .
  * Kirkwood (1933) J. G. Kirkwood, Phys. Rev. 44, 31 (1933).
  * Dirac (1945) P. A. M. Dirac, Rev. Mod. Phys. 17, 195 (1945).
  * Yunger Halpern  _et al._ (2018) N. Yunger Halpern, B. Swingle, and J. Dressel, Phys. Rev. A 97, 042105 (2018).
  * Cover and Thomas (2006) T. M. Cover and J. A. Thomas,  _Elements of Information Theory_ , 2nd ed. (John Wiley and Sons Inc., Hoboken, New Jersey, USA, 2006).
  * Lehmann and Casella (2006) E. L. Lehmann and G. Casella,  _Theory of point estimation_ (Springer Science & Business Media, 2006).
  * Helstrom (1967) C. Helstrom, Physics Letters A 25, 101 (1967).
  * Abbas  _et al._ (2020) A. Abbas, D. Sutter, C. Zoufal, A. Lucchi, A. Figalli, and S. Woerner, arXiv preprint arXiv:2011.00027 (2020).
  * Haug  _et al._ (2021) T. Haug, K. Bharti, and M. Kim, arXiv preprint arXiv:2102.01659 (2021).
  * Meyer (2021) J. J. Meyer, arXiv preprint arXiv:2103.15191 (2021).
  * Albarelli  _et al._ (2019) F. Albarelli, J. F. Friel, and A. Datta, Phys. Rev. Lett. 123, 200503 (2019).
  * Carollo  _et al._ (2019) A. Carollo, B. Spagnolo, A. A. Dubkov, and D. Valenti, Journal of Statistical Mechanics: Theory and Experiment 2019, 094010 (2019).
  * Holevo (1977) A. Holevo, Reports on Mathematical Physics 12, 251 (1977).
  * Matsumoto (2002) K. Matsumoto, Journal of Physics A: Mathematical and General 35, 3111 (2002).
  * Stone (1932) M. H. Stone, Ann. Math 33, 643 (1932).
  * Wigner (1932) E. Wigner, Phys. Rev. 40, 749 (1932).
  * Steinberg (1995) A. M. Steinberg, Phys. Rev. A 52, 32 (1995).
  * Dressel (2015) J. Dressel, Phys. Rev. A 91, 032116 (2015).
  * Hofmann (2011) H. F. Hofmann, New Journal of Physics 13, 103009 (2011).
  * Dressel and Jordan (2012) J. Dressel and A. N. Jordan, Phys. Rev. A 85, 012107 (2012).
  * Monroe  _et al._ (2020) J. T. Monroe, N. Y. Halpern, T. Lee, and K. W. Murch, arXiv preprint arXiv:2008.09131 (2020).
  * Johansen (2007) L. M. Johansen, Phys. Rev. A 76, 012119 (2007).
  * Lundeen  _et al._ (2011) J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011).
  * Lundeen and Bamber (2012) J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012).
  * Bamber and Lundeen (2014) C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014).
  * Thekkadath  _et al._ (2016) G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, Phys. Rev. Lett. 117, 120401 (2016).
  * Swingle  _et al._ (2016) B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Phys. Rev. A 94, 040302 (2016).
  * Halpern  _et al._ (2019) N. Y. Halpern, A. Bartolotta, and J. Pollack, Communications Physics 2, 1 (2019).
  * González Alonso  _et al._ (2019) J. R. González Alonso, N. Yunger Halpern, and J. Dressel, Phys. Rev. Lett. 122, 040404 (2019).
  * Landsman  _et al._ (2019) K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Nature 567, 61 (2019).
  * Mohseninia  _et al._ (2019) R. Mohseninia, J. R. G. Alonso, and J. Dressel, Phys. Rev. A 100, 062336 (2019).
  * Levy and Lostaglio (2019) A. Levy and M. Lostaglio, arXiv preprint arXiv:1909.11116 (2019).
  * Lostaglio (2020) M. Lostaglio, arXiv preprint arXiv:2004.01213 (2020).
  * Griffiths (1984) R. B. Griffiths, Journal of Statistical Physics 36, 219 (1984).
  * Goldstein and Page (1995) S. Goldstein and D. N. Page, Phys. Rev. Lett. 74, 3715 (1995).
  * Hartle (2004) J. B. Hartle, Phys. Rev. A 70, 022104 (2004).
  * Hofmann (2012) H. F. Hofmann, New Journal of Physics 14, 043031 (2012).
  * Hofmann (2014) H. F. Hofmann, Phys. Rev. A 89, 042115 (2014).
  * Hofmann (2015) H. F. Hofmann, Phys. Rev. A 91, 062123 (2015).
  * Hofmann (2016) H. F. Hofmann, The European Physical Journal D 70, 118 (2016).
  * Halliwell (2016) J. J. Halliwell, Phys. Rev. A 93, 022123 (2016).
  * Stacey (2019) B. C. Stacey, arXiv preprint arXiv:1907.02432 (2019).
  * Tang  _et al._ (2019) H. L. Tang, V. Shkolnikov, G. S. Barron, H. R. Grimsley, N. J. Mayhall, E. Barnes, and S. E. Economou, arXiv preprint arXiv:1911.10205 (2019).
  * Grimsley  _et al._ (2019) H. R. Grimsley, S. E. Economou, E. Barnes, and N. J. Mayhall, Nature communications 10, 1 (2019).
  * Yordanov  _et al._ (2020) Y. S. Yordanov, V. Armaos, C. H. Barnes, and D. R. Arvidsson-Shukur, arXiv preprint arXiv:2011.10540 (2020).
  * McArdle  _et al._ (2020) S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Rev. Mod. Phys. 92, 015003 (2020).
  * Lavrijsen  _et al._ (2020) W. Lavrijsen, A. Tudor, J. Müller, C. Iancu, and W. de Jong, in  _2020 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (IEEE, 2020) pp. 267–277.
  * Bittel and Kliesch (2021) L. Bittel and M. Kliesch, arXiv preprint arXiv:2101.07267 (2021).
  * Zarchan  _et al._ (2000) P. Zarchan, H. Musoff, A. I. of Aeronautics, and Astronautics,  _Fundamentals of Kalman Filtering: A Practical Approach_, Progress in astronautics and aeronautics (American Institute of Aeronautics and Astronautics, Incorporated, 2000).
  * Carollo  _et al._ (2018) A. Carollo, B. Spagnolo, and D. Valenti, Scientific reports 8, 1 (2018).