∎

11institutetext: G.C. Knee 22institutetext: Department of Physics, University
of Warwick, Coventry CV4 7AL, UK
Department of Materials, University of Oxford, Oxford OX1 3PH, UK
Tel.: +44(0)24 76150943
https://orcid.org/0000-0002-7203-0271
22email<EMAIL_ADDRESS>33institutetext: G.A.D. Briggs 44institutetext:
Department of Materials, University of Oxford, Oxford OX1 3PH, UK

# Seeing opportunity in every difficulty

Protecting information with weak value techniques

George C. Knee  G. Andrew D. Briggs

(Received: 31 March 2018 / Accepted: 23 April 2018)

###### Abstract

A weak value is an effective description of the influence of a pre and
postselected ‘principal’ system on another ‘meter’ system to which it is
weakly coupled. Weak values can describe anomalously large deflections of the
meter, and deflections in otherwise unperturbed variables: this motivates
investigation of the potential benefits of the protocol in precision
metrology. We present a visual interpretation of weak value experiments in
phase space, enabling an evaluation of the effects of three types of detector
noise as ‘Fisher information efficiency’ functions. These functions depend on
the marginal distribution of the Wigner function of the ‘meter’, and give a
unified view of the weak value protocol as a way of protecting Fisher
information from detector imperfections. This approach explains why weak value
techniques are more effective for avoiding detector saturation than for
mitigating detector jitter or pixelation.

###### Keywords:

Wigner functions Weak values Technical noise

##  1 Introduction

Professor Izumi Tsutsui has identified three categories of research into weak
values Tsutsui2015 . The first works towards an understanding of the formal
and mathematical aspects of the theory. Taking $\hbar=1$ , a derivation
similar to the original one by Aharonov, Albert and Vaidman (AAV)
AharonovAlbertVaidman1988 :

$\displaystyle\langle f|e^{-ig\hat{A}\hat{k}}|i\rangle|m\rangle\approx\langle
f|i\rangle e^{-igA_{w}\hat{k}}|m\rangle;\qquad A_{w}=\frac{\langle
f|\hat{A}|i\rangle}{\langle f|i\rangle}$ (1)

involves the definition of a new quantity $A_{w}$, the weak value. It may be
thought of as a generalisation of the eigenvalue of $\hat{A}$ Vaidman2017 . As
can be seen from the above expression, it emerges only through a weak coupling
($g\ll 1$) of a principal system, preselected into $|i\rangle$ and preselected
into $|f\rangle$, with a ‘meter’ degree of freedom, initialised in
$|m\rangle$. This generalisation leaves us with an economical, effective
description of meter solely in terms of the generally complex and unbounded
quantity $A_{w}$. The physical effect of the imaginary component Jozsa2007 ,
the breakdown in the aforementioned approximations DuckStevensonSudarshan1989
; KofmanAshhabNori2012 , and the full-order, exact evolution of the meter
WuLi2011  all fall within the purview of Tsutsui’s first category and have
been the subject of many studies. Ref KofmanAshhabNori2012  is a comprehensive
treatment of nonperturbative analogues of Equation (1) featuring various
coupling Hamiltonians and meter states. Insofar as research in this category
is mathematical it is uncontroversial, and the discussion mainly focusses on
the approximations to be made and the range of their validity.

The fruits of this first category of research feed into the other two. The
second category considers the foundational aspects of weak values. They have
been associated with quantum paradoxes AharonovRohrlich2005 , with negative
probabilities, and with macrosopic realism GogginAlmeidaBarbieri2011  (as has
the wider topic of pre and post-selected measurements AharonovRohrlich2005 ;
GeorgeRobledoMaroney2013 ). Weak values have been argued to provide an
operational definition of the quantum wavefunction LundeenSutherlandPatel2011
, and to reveal particle trajectories of Bohmian mechanics
KocsisBravermanRavets2011 . The question of whether weak values are quantum
mechanical has also been debated FerrieCombes2014a . It is perhaps
unsurprising that this second category of research has enjoyed a large deal of
disagreement and controversy, since it involves speculation into the meaning
of weak values, and indeed into the meaning of quantum mechanics itself (where
there remain open questions on the interpretation of the formalism and the
peculiar role of measurement BriggsButterfieldZeilinger2013 ).

The third category ought to be far more clear-cut, for it deals with the use
of weak values and associated techniques in technology, specifically for
parameter estimation. Here, one might expect, the question of whether the
technique is genuinely useful or not would be a matter of fact: unforgiving
and unquestionable. Why then, has this line of research also been afflicted by
apparent contradictions and disagreements? The third category was ignited by
an experiment in which a weak value technique was used to measure the spin-
Hall effect of light HostenKwiat2008 . The interaction between the
polarisation and transverse momentum of a light beam at an interface may be
modelled using Equation (1), where the coupling constant $g$ is considered an
unknown parameter to be investigated. Using clever pre and postselected
polarization states the lateral displacements could be measured with a
sensitivity of around 1 Angstrom. Other impressive experiments include the
estimation of the angular tilt of a mirror in a Sagnac interferometer to a
precision of a few hundred femtoradians DixonStarlingJordan2009 . From 2013
onwards, however, theoretical treatments of the weak value technique called
into doubt whether there was any true advantage over standard strategies
KneeBriggsBenjamin2013 ; TanakaYamamoto2013 ; KneeGauger2014 ;
FerrieCombes2014 , especially from the point of view of parameter estimation.
There were also a slew of theoretical LeeTsutsui2014 ; AlvesEscherMatos-
Filho2015  and experimental VizaMartinez-RinconAlves2015  papers arguing for
the merits of weak value techniques. A review of these results is given in Ref
KneeCombesFerrie2016 .

A central tool in our approach is to use Fisher information, rather than the
more commonly employed signal-to-noise ratio. Fisher information is a way of
measuring the amount of information that an observable random variable such as
$x$ or $k$ carries about an unknown parameter upon which the probability of
$x$ or $k$ depends. In AAV’s original work AharonovAlbertVaidman1988 , the
subject of investigation is the operator $\hat{A}$ itself. The canonical
example they gave was of a spin 1/2 particle traversing a magnetic field
gradient, as in the famous Stern-Gerlach experiment (where the coupling
between the spin and the field gradient causes a bream of such particles to be
deflected ‘up’ or ‘down’ depending on their spin state). The title of the
paper trumpeted the astonishing result that $A_{w}$ could take very large
values (say 100), where usually an eigenvalue bounded by $\pm 1/2$ would
appear. From the parameter estimation perspective, however, the emphasis is
rather more on the coupling constant $g$: in the Stern-Gerlach setup this
corresponds to the magnitude of the magnetic field gradient, and generally
controls the ‘strength’ of the measurement via the degree of correlation that
is built up between principal system and meter. Since the weak value $A_{w}$
multiplies $g$ in Equation (1), it controls the information about $g$ that can
be extracted in the subsequent detection of the meter.

The lively debate in the third category has left open the question whether and
how weak value techniques can offer precision advantages. The arguments
against, in a nutshell, state that the postselected meters carry an amplified
amount of Fisher information per measured data point, but this is more than
cancelled by the low probability of the signal for each potential data point
surviving the post-selection process. So the arguments seem quite clear-cut
after all – but such a summary belies some of the subtleties in evaluating
metrological merit. Firstly, there is the issue of technical noise: as long as
postselection is performed in the experiment before any ultimate detection
(say with a polariser), rather than merely implemented as voluntary data loss,
the weak value technique has characteristics that physically distinguish it
from regular approaches. This means that certain types of detector noise may
favour one approach over the other, potentially reversing the conclusions
reached for ideal detection: although it is then difficult to say in full
generality when advantages may be had, and the situation must be judged mostly
on a case by case basis. Secondly, there is the issue of resource-counting.
This is an ever present question in metrology, and is informed by the ‘cost’
that is associated with each component of an experiment. The figure of merit
(e.g. the precision of the method) is often given ‘per unit cost’, as in
‘Fisher information per photon’. Since the constraints on the cost depend on
the nature of the experiment, and can even vary from laboratory to laboratory,
any general claims will necessarily depend on the details. Nevertheless we
will discuss a scenario that will end up favouring weak value techniques:
namely, when system-meter pairs (e.g. photons) are more costly to detect than
to create.

In this paper we present a new, visual way of intuiting the utility of weak
values. The main tool we use is the Wigner function: a quasi-probability
distribution over phase space. We hope our approach may help bring clarity to
an otherwise confusing field. We will not provide a comprehensive treatment of
all possible implementations of the weak value protocol, but instead make
choices that simplify the arguments while remaining faithful to the original
approach and also representing most experiments so far. In our treatment of
three types of technical noise – namely transverse jitter, pixelation and
saturation – we rely heavily on theoretical results that have been previously
reported KneeGauger2014 ; KofmanAshhabNori2012 ; HarrisBoydLundeen2017  but
require some re-application in our new setting.

##  2 A simple model

We will consider a simple model of a qudit (the principal system) coupled to a
single continuously varying degree of freedom (the meter), prepared in a real-
valued Gaussian state:

$\displaystyle|i\rangle\otimes|m\rangle=|i\rangle\otimes\int
dx\psi(x)|x\rangle=|i\rangle\otimes\int dk\phi(k)|k\rangle,$
$\displaystyle\psi(x)=\frac{1}{(2\pi)^{\frac{1}{4}}\sqrt{\sigma}}e^{-\frac{x^{2}}{4\sigma^{2}}},$
$\displaystyle\phi(k)=\left(\frac{2}{\pi}\right)^{\frac{1}{4}}\sqrt{\sigma}e^{-k^{2}\sigma^{2}}.$
(2)

$\psi(x)$ represents the position wavefunction of the meter, while $\phi(k)$
describes the distribution over values for the conjugate momentum. $\sigma$ is
a parameter controlling the uncertainty in $x$; and since Gaussian states
saturate the Heisenberg uncertainty principle it also controls the uncertainty
in $k$, but with an inverse relationship. To develop our visual depiction of
weak-value experiments, we make use of the Wigner function description
Case2008  of $\psi$:

$\displaystyle W_{m}(x,k):=$
$\displaystyle\frac{1}{2\pi}\int\psi\left(x+\frac{y}{2}\right)\psi^{*}\left(x-\frac{y}{2}\right)e^{-iky}dy$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\phi\left(k+\frac{u}{2}\right)\phi^{*}\left(k-\frac{u}{2}\right)e^{ixu}du$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}e^{-2\sigma^{2}k^{2}}e^{-\frac{x^{2}}{2\sigma^{2}}}.$
(3)

The Wigner function may be thought of as a representation of the wave-function
that allows one to simultaneously visualise both position ($x$) and momentum
($k$) variables. When $x$ is a component of the transverse position of a
quantum particle (e.g. a photon or spin) propagating along $z$, $k$ is the
corresponding component of the transverse wavevector (with the other
transverse coordinate integrated out). Despite the possibility of negative
regions (for a general Wigner function) precluding the interpretation $W(x,k)$
as a genuine probability distribution, it has the attractive property that its
marginal distributions $\tilde{p}(k)=\int W(x,k)dx$ and $p(x)=\int W(x,k)dk$
correspond to those derived in the usual fashion, via the Born rule, for
measurements of $k$ and $x$ respectively. They are therefore bona fide
probability distributions, in the sense that they are positive and integrate
to unity. Since the initial state of the meter is normalised, we thus
naturally have that the volume of the initial Wigner function $\int\int
W(x,k)dxdk=1$. See Figure 1.

Figure 1: We represent the initial quantum state of the meter via a Wigner
function. A standard measurement always succeeds and results in a small shift
in the $x$ direction (black arrow, final state show in Figure 4). The weak
value technique is postselected, and allows for larger shifts in any direction
$\phi$ in phase space. For the linear regime considered by AAV to hold, the
weak value shift must lie well within the red curve, which is a parametric
ellipse traced out by the maximum of the Wigner function when
$g|A_{w}|=\sigma$ is substituted into Equation (4) and $\phi$ is allowed to
vary. The characteristics of weak value experiments offer no fundamental
advantage under perfect detection, but can alleviate certain types of detector
imperfection. In this figure we take $\sigma=1,g=0.1,\lambda_{*}=1$.

###  2.1 Weak value technique

By applying equation (1) to (2), the Wigner function of the meter after the
weak interaction and conditioned on successful postselection of the system
into $|f\rangle$ will be:

$\displaystyle W_{A_{w}}(x,k)=|\langle
f|i\rangle|^{2}W_{m}(x-g\mathfrak{Re}A_{w},k-\frac{g}{2\sigma^{2}}\mathfrak{Im}A_{w}).$
(4)

This can be realised by spotting that the real and imaginary components of the
weak value have actions $e^{-ig\mathfrak{Re}A_{w}\hat{k}}$ (unitary) and
$e^{g\mathfrak{Im}A_{w}\hat{k}}$ (non-unitary) which mutually commute. This
expression extends beyond the simple case of a real Gaussian function that we
consider here but not to arbitrary complex valued functions: for a
comprehensive treatment, see Ref KofmanAshhabNori2012 .

The approximation leading to Equation (4) requires $g|A_{w}|\ll\sigma$
DuckStevensonSudarshan1989 ; a condition that is depicted as a red ellipse in
Figure 1. The probabilistic nature of the protocol is reflected in $W_{A_{w}}$
being sub-normalised: the volume under the post-selected Wigner function is
less than one and equal to the success probability.

The Wigner function representation implies that a generally complex weak value
will introduce a shift in an oblique direction in phase space. This raises the
question of using ‘rotated’ phase space observables to detect the shift.
Define

$\hat{s}_{\theta}:=\hat{x}\cos\theta+\hat{k}\sin\theta.$

The distribution for this rotated observable may be extracted from the Wigner
function by integrating over its conjugate
$\hat{t}_{\theta}:=-\hat{x}\sin\theta+\hat{k}\cos\theta$
BreitenbachSchillerMlynek1997 :

$\displaystyle p_{\theta}(s_{\theta})$
$\displaystyle=\int_{-\infty}^{\infty}dt_{\theta}W_{m}(s_{\theta}\cos\theta-
t_{\theta}\sin\theta,s_{\theta}\sin\theta+t_{\theta}\cos\theta)$ (5)
$\displaystyle=\frac{\sqrt{2}\sigma}{\sqrt{4\pi\sigma^{4}\cos^{2}\theta+\pi\sin^{2}\theta}}\exp\left\\{-\frac{s_{\theta}^{2}}{2\sigma^{2}(2\sigma^{2}\cos^{2}\theta+\frac{1}{2\sigma^{2}}\sin^{2}\theta)}\right\\}.$

This is once more a Gaussian distribution, which can be thought of as a ‘view’
of $W(x,k)$ from an oblique angle. The variance $\sigma_{\theta}^{2}$ is now a
function of the $x$ and $k$ variances. One can show that after the weak value
technique one is left with once again a simple shift:

$\displaystyle p_{\theta}(s_{\theta})\rightarrow|\langle
f|i\rangle|^{2}p_{\theta}(s_{\theta}-[g|A_{w}|\cos\phi\cos\theta+\frac{g}{2\sigma^{2}}|A_{w}|\sin\phi\sin\theta]).$
(6)

where we have written the weak value in polar form:

$\displaystyle\mathfrak{Re}A_{w}=|A_{w}|\cos\phi$
$\displaystyle\mathfrak{Im}A_{w}=|A_{w}|\sin\phi.$ (7)

###  2.2 Standard measurement

We will define a _standard measurement_ by setting
$|i\rangle=|f\rangle=|i_{*}\rangle$ to the eigenstate of $A$ with eigenvalue
of greatest magnitude. This scheme succeeds deterministically, and so results
in a Wigner function of unit volume:

$\displaystyle W_{\textrm{std}}(x,k)=W_{m}(x-g\lambda_{*},k)$
$\displaystyle\Rightarrow p_{\textrm{std}}(x)=p(x-g\lambda_{*}).$ (8)

Here
$\lambda_{*}=\textrm{arg}\,\textrm{max}_{\lambda}\\{|\lambda|:\hat{A}|i\rangle=\lambda|i\rangle\\}$.
The two strategies we consider can thus both be thought of as _restricted_
shifts of the initial Wigner function in phase space. Figure 1 shows the shift
corresponding to the case where $\lambda_{*}=1$ with a black arrow.

In depicting both standard and weak value techniques in the same plots (below
in Figures 2 and 4 in particular), we have avoided dramatic examples such as
$A_{w}=100$ for several reasons. Firstly, it is somewhat of a distraction that
we wish to demote in favour of the more subtle but ultimately more useful
properties of weak value techniques. Secondly, it is difficult to show such
examples without adversely affecting our visual picture. Thirdly, and most
importantly, our intuition is actually much better served by realising that
the red ellipse in Figure 1, which represents the very limit of validity of
AAV’s approach, is fixed by $\sigma$. Very large weak values are of course
still possible, but only by ensuring $g$ is smaller and smaller. In this
scenario, the black arrow shrinks while the red ellipse remains constant,
meaning that the ‘amplification factor’ $|A_{w}|/\lambda_{*}$ becomes very
large, _while the size of the shift itself must actually remain small_. This
has previously been described as the requirement that the signal-to-noise
ratio per data point must be low Kedem2012 .

##  3 Precision of estimating $g$

Whenever a quantity of interest is not directly measured, it is necessary to
process the raw data in order to ‘measure’ – or more strictly ‘estimate’ –
that quantity. For instance, $g$ is not directly measured, but inferred from
repeated measurements of $x$. The data are subsequently processed thus:

$\displaystyle\tilde{g}:=\frac{\langle x\rangle}{\lambda_{*}}.$ (9)

Here angled brackets stand for the sample average. The rules governing the
data processing are known as the ‘estimator’: above we used a tilde to denote
the estimator for $g$ in the standard technique. For the weak value technique,
the procedure is similar, but $x$ is replaced by $s_{\theta}$ and
$\lambda_{*}$ is replaced by the coefficient of $g$ in the argument of
$p_{\theta}$ of Equation (6). Because the data (the input to the estimator
function, $x$) are random variables, the estimates (the output of the
estimator function, $\tilde{g}$) are also random variables. We will assume
that all estimators under consideration are unbiased, meaning that they give
the correct answer on average111It is clear that our simple estimator is
exactly unbiased for the standard strategy, but only approximately unbiased
for the weak value technique. The latter fact is commonly overlooked., i.e.
$\langle\tilde{g}\rangle=g$.

As alluded above, the classical Fisher information about the coupling constant
$g$:

$\displaystyle
F_{g}:=\int\frac{(\partial_{g}p_{\theta}(s_{\theta}))^{2}}{p_{\theta}(s_{\theta})}ds_{\theta}$
(10)

is the central figure of merit of classical parameter estimation. The Cramér
Rao bound states that in the limit of many trials, the _precision_ (or
_uncertainty_ , or _standard deviation_) of an unbiased estimator for $g$ will
be lower-bounded by the reciprocal of $\sqrt{F_{g}}$ Van-Trees1968 . The
higher $F_{g}$, the better the precision, and the Cramér Rao bound can be
saturated by _efficient_ estimation strategies: maximum likelihood, for
example. Owing to the simplicity of the model considered here, the estimator
given in Equation (9) is indeed efficient for ideal detection, but requires
modification when detector imperfections are included KneeMunro2015b .

The Fisher information measures the information content of a probability
distribution $p_{\theta}(s_{\theta})$, which can be derived from the Wigner
function upon fixing a measurement. Under our formalism, any phase space
quadrature may be considered, or indeed noisy implementations thereof. The
choice of measurement thus influences $p_{\theta}(s_{\theta})$ and thereby
also influences $F_{g}$: so we may use the latter to evaluate the suitability
of different measurement schemes.

Because we consider a single parameter, the maximum classical Fisher
information about $g$ (when considering all possible measurements) of the
joint-system meter state immediately after the interaction is given by the
quantum Fisher information BraunsteinCaves1994 ; Helstrom1969 ; Paris2009 :

$\displaystyle
H_{g}[|i\rangle\otimes|m\rangle]=\textrm{Var}(A\otimes\hat{k})=\langle\hat{A}^{2}\rangle\langle\hat{k}^{2}\rangle-\langle\hat{A}\rangle^{2}\langle\hat{k}\rangle^{2}=\frac{\langle\hat{A}^{2}\rangle}{\sigma^{2}}=\frac{\langle
i|\hat{A}^{2}|i\rangle}{\sigma^{2}}.$ (11)

$H_{g}$ is therefore a property of the initial quantum states alone, once the
Hamiltonian has been fixed. It is clear that the smaller $\sigma$ – the
narrower the distribution in space – the more information can be extracted. We
will consider different measurement schemes that can harvest this maximal
information in different ways, by channelling it into different parts of
parameter space. That is, we will consider how close one can get to saturating

$\displaystyle F_{g}\leq H_{g}.$ (12)

In general, if we consider system and meter to be a single indivisible quantum
system, harvesting all of the information would require entangled preparation
and entangled measurement. However, for the case where the average momentum of
the meter is zero as it enters the weak interaction (i.e we take
$\langle\hat{k}\rangle=0$ in accordance with Equation (2)), factorable
preparations and measurements will suffice.222A notable departure from this
paradigm is when $\hat{k}$ is replaced with the particle number operator,
which clearly will take nonzero values in general. This can unlock Heisenberg
scaling in the Fisher information contained in the postselection probability
$|\langle f|i\rangle|^{2}$ itself ZhangDattaWalmsley2015 ;
JordanTollaksenTroupe2015 ; ChenAharonSun2016 . It follows from the convexity
of the QFI TothPetz2013 :

$\displaystyle
H_{g}\left[\sum_{j}p_{j}|i_{j}\rangle\otimes|m_{j}\rangle\right]\leq\sum_{j}p_{j}H_{g}\Huge[|i_{j}\rangle\otimes|m_{j}\rangle\Huge]$
(13)

that using mixed states cannot increase the Fisher information.

The chain rule of differentiation is a particularly effective tool in relating
the Fisher information of a shifted distribution (which does not change shape)
to properties of the shape of the distribution itself KneeGauger2014 . Namely:

$\displaystyle F_{g}[p(s-\nu g)]=\nu^{2}F_{s}[p(s)]$ $\displaystyle\Rightarrow
F_{g}[p_{\theta}(s_{\theta}-\nu g)]=\frac{\nu^{2}}{\sigma_{\theta}^{2}}$ (14)

where the last step follows from the Gaussian shape of the marginal
distribution.

###  3.1 Optimal protocols

It is straightforward to see that our _standard measurement_ sets a high
Quantum Fisher information, and then extracts all of it. Recall that we set
$|i\rangle=|f\rangle$ to the eigenstate of $A$ with highest magnitude
eigenvalue $\lambda_{*}$, and then measure $x$:

$\displaystyle F_{g}[p_{std}(x)]$
$\displaystyle=\frac{\lambda_{*}^{2}}{\sigma^{2}}.$ (15)

Full information is harvested, but one is ‘stuck’ with measuring $x$ and with
having a high flux onto the detector. Our weak value technique, on the other
hand, uses arbitrary initial and final states. Rotated quadratures have

$\displaystyle
F_{s}[p_{\theta}(s_{\theta})]=\frac{1}{\sigma_{\theta}^{2}}=\frac{1}{\sigma^{2}\cos^{2}\theta+\frac{1}{4\sigma^{2}}\sin^{2}\theta}.$
(16)

We may use this formula, along with Equations (4) and (14) to evaluate a
corrected Fisher information for a weak value shift for in an arbitrary
direction $\phi$ and arbitrary measurement direction $\theta$ in phase space.
To perform this calculation, we will normalize the distribution before
calculating the Fisher Information and then correct it by the postselection
probability. This is appropriate given the additivity of Fisher information:
$N$ independent experiments enjoy a total Fisher information of $NF$, or in
the case of weak value experiments a total of $|\langle f|i\rangle|^{2}NF$. We
have

$\displaystyle|\langle
f|i\rangle|^{2}F_{g}[p_{\textrm{wv}}(s_{\theta})]=|\langle
f|i\rangle|^{2}F_{g}[p_{\theta}(s_{\theta}-[g\mathfrak{Re}A_{w}\cos\theta+\frac{g}{2\sigma^{2}}\mathfrak{Im}A_{w}\sin\theta])$
$\displaystyle=|\langle
f|i\rangle|^{2}|A_{w}|^{2}\left(\cos\theta\cos\phi+\frac{\sin\theta\sin\phi}{2\sigma^{2}}\right)^{2}F_{s}[p(s_{\theta})].$
(17)

We may now show that this information is always lower than than of the
standard measurement:

$\displaystyle|\langle
f|i\rangle|^{2}F_{g}[p_{\textrm{wv}}(s_{\theta})]\leq\frac{|\langle
f|i\rangle|^{2}|A_{w}|^{2}}{\sigma^{2}}=\frac{|\langle
f|A|i\rangle|^{2}}{\sigma^{2}}$ (18)
$\displaystyle\leq\frac{\langle\hat{A}^{2}\rangle}{\sigma^{2}}=H_{g}[|i\rangle\otimes|m\rangle]$
(19)
$\displaystyle\leq\frac{\lambda_{*}^{2}}{\sigma^{2}}=F_{g}[p_{\textrm{std}}(x)]=H_{g}[|i_{*}\rangle\otimes|m\rangle].$
(20)

The first inequality is proved in the appendix, along with a prescription for
saturating it. The second inequality follows from the Cauchy-Schwarz
inequality KneeGauger2014 . The third inequality follows from the selection of
the optimal initial state $|i_{*}\rangle$ by the standard measurement
strategy. It is possible to get close to saturating these inequalities whilst
maintaining a large weak value AlvesEscherMatos-Filho2015 ; VizaMartinez-
RinconAlves2015 . This means that almost all of the Fisher information may be
concentrated into an unlikely or ‘dark’ detector mode. We will not focus so
much here on what combination of initial and final states, coupling parameters
and meter states are necessary to approach equality in Equations (18-20), save
to note that if $\hat{A}^{2}=\mathbb{I}$ then $|f\rangle=\hat{A}|i\rangle$
will saturate the second condition and implies a real weak value. The question
has been considered outside of the linear regime we consider KneeGauger2014 ;
Knee2014 . Instead the important point is that one may sacrifice only a small
amount of information at this stage of the discussion: the sacrifice may well
be considered negligible in comparison to other benefits that we describe
below.

##  4 Protecting information by alleviating technical noise

So far we have considered ‘ideal’ detection. This is to be understood as a
projective and orthogonal measurement that one would typically find in a
quantum mechanics text book: an Hermitian observable $\hat{x}$ or $\hat{k}$,
for example. Such an idealised measurement offers unlimited precision when the
number of repetitions $N$ tends to infinity. For finite trials, we get a
finite precision; but this is a consequence of the quantum noise in the
prepared state $|m\rangle$ (often referred to as ‘shot noise’) rather than the
detector. If the initial state was noise-free (which would correspond to the
limit of a highly ‘squeezed’ Wigner function with $x$ Dirac-delta distributed)
an ideal detector would offer ‘perfect precision’ – an exact estimate with
zero uncertainty after a single trial. Otherwise, non-ideal detectors are
modelled by different kinds of coarse-grained approximations to ideal
detectors, and imply a loss of information.

We will consider the following examples of technical noise: (1) transverse
jitter (2) pixelation and (3) saturation. Each will be defined by a
transformation which relates the ideal distribution $p(s)$ to the imperfect
one $p^{\prime}(s)$. The results of this paper are to present the information
obscuring effect of these transformations graphically, as Fisher information
efficiency functions $\eta$, defined by:

$\displaystyle F_{g}[p^{\prime}(s)]=\eta F_{g}[p(s)].$ (21)

These functions allow one to describe different imperfect measurements as
$\eta\times 100\%$ Fisher efficient, where $\eta=1$ recovers the ideal
information. Let us consider the relative Fisher information for the weak
value technique compared to the standard technique, both under noisy
detection:

$\displaystyle\frac{|\langle
f|i\rangle|^{2}N}{N}\frac{F_{g}[p^{\prime}_{\textrm{wv}}(s)]}{F_{g}[p^{\prime}_{\textrm{std}}(x)]}=\left(\frac{\eta^{\textrm{wv}}}{\eta^{\textrm{std}}}\right)\frac{|\langle
f|i\rangle|^{2}F_{g}[p_{\textrm{wv}}(s)]}{F_{g}[p_{\textrm{std}}(x)]}.$ (22)

At best $F_{g}[p_{\textrm{wv}}(s)]$ for the weak value technique may be
comparable to that of the standard measurement: then the second factor of this
equation (the relative Fisher information under ideal detection) will be close
to unity. If $\eta$ favours the weak value technique, however, the first
factor (the relative Fisher information efficiency) can be greater than unity
and there is scope for practical advantages to be had.

Our central idea is that one can guide the Fisher information around in phase
space. In particular one can usher it away from danger; from regions of phase
space where the information would be lost or degraded. One can concentrate a
large proportion (near one hundred percent) of the available information into
a low number of events, or even into a conjugate variable for less-noisy
detection. One can guide the detector distribution away from a faulty or noisy
part of the detector.

Figure 2: The Fisher information efficiency $\eta_{\textrm{jitter}}$ can
differ for measurements in position ($x$, black) and momentum ($k$, red). Each
panel shows the shifted Wigner distribution (wire mesh), along with the
marginal distributions (solid black $p(x)$ and red $p(k)$) of both the initial
and shifted meter states, drawn on the corresponding walls of the plot. Note
how the standard technique is limited to changes in $x$ while $k$ remains
unperturbed. In the weak value technique, $k$ carries Fisher information if
pre and postselection are appropriately chosen (here $A_{w}=3i$ is pure
imaginary). The Wigner function is attenuated due to the low success rate of
postselection, and a renormalised marginal distribution is shown as a dashed
line as a guide for the eye. Transverse jitter reduces the Fisher information
of a shifted Gaussian, but in a way that is independent of the magnitude of
the shift and the intensity of the beam. It can be more or less severe in
position or momentum space, however: in this example one could use a weak
value technique to swap detection in $x$ (90% Fisher efficient) for detection
in $k$ (99% Fisher efficient).

###  4.1 Transverse jitter

Consider that the detector is in a random motion, so that the actual
distribution at the detector is

$\displaystyle
p^{\prime}(s_{\theta})=\int_{-\infty}^{\infty}p(s_{\theta}-y)\frac{e^{-\frac{1}{2}\frac{y^{2}}{2\zeta_{\theta}^{2}}}}{\sqrt{2\pi}\zeta}dy.$
(23)

Under this model it is possible to show that KneeGauger2014 :

$\displaystyle\eta_{\textrm{jitter}}:=\frac{F_{g}[p^{\prime}(s_{\theta})]}{F_{g}[p(s_{\theta})]}=\left(1+\frac{\zeta_{\theta}^{2}}{\sigma_{\theta}^{2}}\right)^{-1}.$
(24)

Note that there is no dependence on $s_{\theta}$, but only on
$\sigma_{\theta}$ and $\zeta_{\theta}$. Since weak values may channel
information into the $S_{\theta}$ variable (with $\theta$ not necessarily
zero, which is impossible in standard measurements), this can be useful in the
case where a more stable measurement of $s_{\theta}$ (than of $x$) is
available Kedem2012 . See Figure (2).

###  4.2 Pixelation

Detection of continuous variables is often performed in a discretized fashion,
such as with the pixel arrays found in modern cameras. We can model this by

$\displaystyle\textrm{Pr}(n;\nu g)$ $\displaystyle=\int_{r(n)}^{r(n+1)}p(s-\nu
g)ds$ (25)
$\displaystyle=\frac{1}{2}\left(\text{erf}\left(\frac{\left(n+1\right)r-g\nu}{\sqrt{2}\sigma}\right)-\text{erf}\left(\frac{nr-g\nu}{\sqrt{2}\sigma}\right)\right),$

where $r$ is the width of each pixel, labelled by integers $n$ and centred at
$(n+\frac{1}{2})r$. In practice, this one-dimensional distribution is the
result of summing over the $y$ direction of a two-dimensional pixel array.
Here we have fixed the detector so that the pixel boundaries are aligned with
the centroid of the initial meter state – for the more general case of free
alignment, see Ref.KneeGauger2014 . We can numerically compute

$\displaystyle\eta_{\textrm{pixel}}(\nu g)=\frac{F_{g}[\textrm{Pr}(n;\nu
g)]}{F_{g}[p(s-\nu g)]};$ (26)

a few examples are shown in Figure (3). The efficiency is surprisingly high
and only very weakly dependent on the parameters that distinguish standard
measurements from weak value techniques: there is therefore likely that
$\eta^{\textrm{wv}}_{\textrm{pixel}}\approx\eta^{\textrm{std}}_{\textrm{pixel}}\approx
1$.

There are therefore two points to be made. Firstly, since
$\eta_{\textrm{pixel}}$ is so robust to changing $r$, we conclude that
pixelation does not represent much of a difficulty: there is limited scope for
protecting information from this kind of imperfection. Secondly,
$\eta_{\textrm{pixel}}$ does not depend very strongly on the size of the shift
$\nu g$. Recall that the distinguishing property of the weak value technique
is the ability to organise for anomalously large shits: so that even if
$\eta_{\textrm{pixel}}$ is low for the standard technique (say around 84% as
in Figure 3), it will be very similar for weak value technique. Therefore
there is not much of an opportunity in this particular difficulty.

Figure 3: Plots of $\eta_{\textrm{pixel}}(\nu g)$ (blue) for pixel widths of
$r=0.5,1,1.5$ respectively. This plot uses $\sigma=1$, with the continuous
distribution before pixelation ($p(x)$, black) and the pixel boundaries (at
$(n+\frac{1}{2})r$, vertical grey bars) shown as a guide for the eye. The
dependence on $s$ is weakly periodic and becomes weaker as the pixel size
reduces. Even for ostensibly very large pixel sizes, the detection scheme is
surprisingly effective: in these examples being at least 84% Fisher efficient.
For pixel sizes much smaller than $\sigma$, the detector is almost $100\%$
Fisher efficient. The bottom panel is a zoom of the $r=1.5$ example, and shows
the shifts under a standard measurement (black) and weak value measurement
with $A_{w}=3$ (red). Pixelation has mostly a negligible effect in general,
but can even have a more detrimental effect on larger shifts (as in this
example) depending on overall alignment of the detector KneeGauger2014 ;
KneeMunro2015b .

###  4.3 Saturation

Figure 4: The problem of detector saturation is shown here in red as a maximum
Wigner function volume (in our formalism this is equivalent to a maximum
Wigner function height). When the Wigner function pokes through the red
surface, this implies that the distribution at the detector will be ‘clipped’,
resulting in a loss of Fisher information. The weak value technique allows
almost all of the available Fisher information (which is mostly concentrated
into a distribution conditioned on an unlikely postselection event) to
continue to a detector which would otherwise saturate. This figure uses
$|\lambda_{*}|=1,A_{w}=3$. The black arrow and the red ellipse have the same
meaning as in Figure 1.

Avoiding detector saturation is perhaps the most promising application of weak
values in metrology. An information-theoretical advantage has been conjectured
several times HostenKwiat2008 ; Kedem2012 ; KneeGauger2014 ; Vaidman2014 : and
a recent study by Harris, Boyd and Lundeen HarrisBoydLundeen2017  gives these
conjectures a theoretical underpinning. We refer the reader to
HarrisBoydLundeen2017  for a full analysis, but the rough argument is as
follows: for a deterministic strategy, there will be some maximum photon flux
for which the information will be degraded due to ‘bleaching’ or saturation of
the photodetectors. On the other hand, if $|\langle f|i\rangle|^{2}$ is small
enough, then saturation will not be an issue and $\eta_{\textrm{sat}}\approx
1$.

In practice, it is likely that saturation problems will be avoided altogether
(by reducing the intensity of the laser source, for example with a neutral
density filter). It will therefore serve our purposes adequately to define

$\displaystyle\eta_{\textrm{sat}}=\Theta(P_{\textrm{sat}}-|\langle
f|i\rangle|^{2})$ (27)

for $\Theta$ the Heaviside step function defined by

$\displaystyle\Theta(z)=\begin{cases}0&z<0\\\ 1&z\geq 0.\end{cases}$ (28)

Of course in reality $\eta_{\textrm{sat}}$ may not fall to zero and may or may
not feature a discontinuity at $P_{\textrm{sat}}$, but will monotonically
decrease with $|\langle f|i\rangle|^{2}$ for a high enough input photon flux.
We show our rough model pictorially in Figure 4333Harris, Boyd and Lundeen
report the surprising claim that saturation alone is not sufficient for an
advantage with weak value techniques HarrisBoydLundeen2017 . In their model,
pixelation or generic detector noise are required, too. This is a consequence
of their ‘soft’ model of saturation which does not feature a discontinuity and
maintains a one-one relationship between incident photon number and output
photoelectron number in each pixel of the detector.

##  5 Conclusion

By using the Wigner representation of the meter to describe weak value
experiments, we have provided a simple picture of the flexibility of the
method and how it compares to standard measurements. Working with real
Gaussian meter states and in the linear regime considered by AAV, we have
extended previous arguments KneeGauger2014  concerning the Fisher information
about the coupling parameter between system and meter: allowing for arbitrary
complex weak values and measurements at oblique angles in phase space. Under
ideal detection, the Fisher information of the weak value technique will never
exceed the Fisher information of the standard strategy without post-selection.
We then considered the impact of three types of technical noise, namely (1)
transverse jitter, (2) pixelation and (3) photo-saturation of the detector.
The Fisher information efficiency that characterises these imperfections may
in some cases be higher for the weak value technique than for the standard
strategy, in which case weak value techniques would give improved precision.
Whilst this might conceivably be the case for (1) and (2), (3) is the killer
application of weak value techniques. By concentrating almost all of the
available quantum Fisher information into a low number of events, it can
amplify the parameter to be measured when detector saturation would otherwise
be a limitation. As long as the cost of generating a photon is considered
lower than the cost of detecting one, weak values therefore offer greater
precision per unit cost than standard measurements. This conclusion invites
experimental exploration and implementation.

##  6 Appendix

Let
$F=F_{g}[p_{\theta}(s_{\theta}-[g\mathfrak{Re}A_{w}\cos\theta+\frac{g}{2\sigma^{2}}\mathfrak{Im}A_{w}\sin\theta])$.
Assume for contradiction that $\frac{\sigma^{2}F}{|A_{w}|^{2}}>1$. By
algebraic manipulations, this condition is equivalent to

$\displaystyle\left(2\sigma\tan\phi-\frac{1}{\sigma}\tan\theta\right)^{2}<0\Rightarrow\perp.$
(29)

hence we must conclude that $F\leq|A_{w}|^{2}/\sigma^{2}$. Furthermore, the
optimum measurement to saturate this inequality is simply deduced:

$\displaystyle\theta\rightarrow\arctan(2\sigma^{2}\tan\phi),$ (30)

###### Acknowledgements.

GCK was supported by the Royal Commission for the Exhibition of 1851. We thank
Natalia Ares for helpful comments on this manuscript.

## References

  * [1] Izumi Tsutsui.  International workshop on weak value and weak measurement.  In International Workshop on Weak Value and Weak Measurement, 2015\.  http://qm.ims.ac.jp/wmwv2015/
  * [2] Yakir Aharonov, David Z. Albert, and Lev Vaidman.  How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100.  Phys. Rev. Lett., 60(14):1351–1354, Apr 1988.
  * [3] L. Vaidman.  Weak value controversy.  Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 375(2106), 2017.
  * [4] Richard Jozsa.  Complex weak values in quantum measurement.  Phys. Rev. A, 76(4):044103, Oct 2007.
  * [5] I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan.  The sense in which a “weak measurement” of a spin-1/2 particle’s spin component yields a value 100.  Phys. Rev. D, 40(6):2112–2117, Sep 1989.
  * [6] Abraham G. Kofman, Sahel Ashhab, and Franco Nori.  Nonperturbative theory of weak pre- and post-selected measurements.  Physics Reports, 520(2):43 – 133, 2012.
  * [7] Shengjun Wu and Yang Li.  Weak measurements beyond the Aharonov-Albert-Vaidman formalism.  Phys. Rev. A, 83(5):052106, May 2011.
  * [8] Yakir Aharonov and Daniel Rohrlich.  Quantum paradoxes: quantum theory for the perplexed.  Wiley, Weinheim, 2005.
  * [9] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde.  Violation of the Leggett-Garg inequality with weak measurements of photons.  Proceedings of the National Academy of Sciences, 108(4):1256–1261, 2011.
  * [10] Richard E. George, Lucio M. Robledo, Owen J. E. Maroney, Machiel S. Blok, Hannes Bernien, Matthew L. Markham, Daniel J. Twitchen, John J. L. Morton, G. Andrew D. Briggs, and Ronald Hanson.  Opening up three quantum boxes causes classically undetectable wavefunction collapse.  Proceedings of the National Academy of Sciences, 110(10):3777–3781, 2013.
  * [11] Jeff S. Lundeen, Brandon Sutherland, Aabid Patel, Corey Stewart, and Charles Bamber.  Direct measurement of the quantum wavefunction.  Nature, 474(7350):188–191, 06 2011.
  * [12] Sacha Kocsis, Boris Braverman, Sylvain Ravets, Martin J. Stevens, Richard P. Mirin, L. Krister Shalm, and Aephraim M. Steinberg.  Observing the average trajectories of single photons in a two-slit interferometer.  Science, 332(6034):1170–1173, 2011.
  * [13] Christopher Ferrie and Joshua Combes.  How the result of a single coin toss can turn out to be 100 heads.  Phys. Rev. Lett., 113:120404, Sep 2014.
  * [14] G. A. D. Briggs, J. N. Butterfield, and A. Zeilinger.  The oxford questions on the foundations of quantum physics.  Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 469(2157), 2013.
  * [15] Onur Hosten and Paul Kwiat.  Observation of the spin Hall effect of light via weak measurements.  Science, 319(5864):787–790, 2008.
  * [16] P. Ben Dixon, David J. Starling, Andrew N. Jordan, and John C. Howell.  Ultrasensitive beam deflection measurement via interferometric weak value amplification.  Phys. Rev. Lett., 102:173601, Apr 2009.
  * [17] George C. Knee, G. Andrew D. Briggs, Simon C. Benjamin, and Erik M. Gauger.  Quantum sensors based on weak-value amplification cannot overcome decoherence.  Phys. Rev. A, 87:012115, January 2013.
  * [18] Saki Tanaka and Naoki Yamamoto.  Information amplification via postselection: A parameter-estimation perspective.  Phys. Rev. A, 88:042116, Oct 2013.
  * [19] George C. Knee and Erik M. Gauger.  When amplification with weak values fails to suppress technical noise.  Phys. Rev. X, 4:011032, March 2014.
  * [20] Christopher Ferrie and Joshua Combes.  Weak value amplification is suboptimal for estimation and detection.  Phys. Rev. Lett., 112:040406, Jan 2014.
  * [21] Jaeha Lee and Izumi Tsutsui.  Merit of amplification by weak measurement in view of measurement uncertainty.  Quantum Studies: Mathematics and Foundations, pages 1–14, 2014\.
  * [22] G. Bié Alves, B. M. Escher, R. L. de Matos Filho, N. Zagury, and L. Davidovich.  Weak-value amplification as an optimal metrological protocol.  Phys. Rev. A, 91:062107, Jun 2015.
  * [23] Gerardo I. Viza, Julián Martínez-Rincón, Gabriel B. Alves, Andrew N. Jordan, and John C. Howell.  Experimentally quantifying the advantages of weak-value-based metrology.  Phys. Rev. A, 92:032127, Sep 2015.
  * [24] George C. Knee, Joshua Combes, Christopher Ferrie, and Erik M. Gauger.  Weak-value amplification: state of play.  Quantum Measurements and Quantum Metrology, 3(1):32–37, May 2016\.
  * [25] Jérémie Harris, Robert W. Boyd, and Jeff S. Lundeen.  Weak value amplification can outperform conventional measurement in the presence of detector saturation.  Phys. Rev. Lett., 118:070802, Feb 2017.
  * [26] William B. Case.  Wigner functions and Weyl transforms for pedestrians.  American Journal of Physics, 76(10):937–946, 2017/12/10 2008.
  * [27] G. Breitenbach, S. Schiller, and J. Mlynek.  Measurement of the quantum states of squeezed light.  Nature, 387:471 EP –, 05 1997.
  * [28] Y. Kedem.  Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values.  Phys. Rev. A, 85:060102, Jun 2012.
  * [29] H.L. Van Trees.  Detection, Estimation, and Modulation Theory.  Number pt. 1 in Detection, Estimation, and Modulation Theory. Wiley, 1968\.
  * [30] George C. Knee and William J. Munro.  Fisher information versus signal-to-noise ratio for a split detector.  Phys. Rev. A, 92:012130, Jul 2015.
  * [31] Samuel L. Braunstein and Carlton M. Caves.  Statistical distance and the geometry of quantum states.  Phys. Rev. Lett., 72:3439–3443, May 1994.
  * [32] Carl W. Helstrom.  Quantum detection and estimation theory.  Journal of Statistical Physics, 1:231–252, 1969.
  * [33] Matteo G. A. Paris.  Quantum estimation for quantum technology.  Int. J. Quant. Inf., 07:125, 2009.
  * [34] Lijian Zhang, Animesh Datta, and Ian A. Walmsley.  Precision metrology using weak measurements.  Phys. Rev. Lett., 114:210801, May 2015.
  * [35] Andrew N. Jordan, Jeff Tollaksen, James E. Troupe, Justin Dressel, and Yakir Aharonov.  Heisenberg scaling with weak measurement: a quantum state discrimination point of view.  Quantum Studies: Mathematics and Foundations, 2(1):5–15, Apr 2015\.
  * [36] Geng Chen, Nati Aharon, Yong-Nan Sun, Zi-Huai Zhang, Wen-Hao Zhang, De-Yong He, Jian-Shun Tang, Yaron Kedem, Chuan-Feng Li, and Guang-Can Guo.  Scalable Heisenberg-limited metrology using mixed states.  12 2016.
  * [37] Géza Tóth and Dénes Petz.  Extremal properties of the variance and the quantum Fisher information.  Phys. Rev. A, 87:032324, Mar 2013.
  * [38] George C. Knee.  Concepts and applications of quantum measurement.  PhD thesis, University of Oxford, 2014.
  * [39] L. Vaidman.  Comment on ”Weak value amplification is suboptimal for estimation and detection”.  arXiv:1402.0199, Feb 2014.