Measurement of Quantum Weak Values of Photon Polarization
G.J. Pryde,1,*J.L. O’Brien,1,*A.G. White,1,*T.C. Ralph,1and H.M. Wiseman2
1Centre for Quantum Computer Technology, Physics Department, The University of Queensland, Brisbane, QLD, 4072, Australia
2Centre for Quantum Computer Technology and Centre for Quantum Dynamics, Griffith University, Brisbane, QLD, 4111, Australia
(Received 27 December 2004; published 9 June 2005)
We experimentally determine weak values for a single photon’s polarization, obtained via a weak
measurement that employs a two-photon entangling operation, and postselection. The weak values cannot
be explained by a semiclassical wave theory, due to the two-photon entanglement. We observe the
variation in the size of the weak value with measurement strength, obtaining an average measurement of
the S1Stokes parameter more than an order of magnitude outside of the operator’s spectrum for the
smallest measurement strengths.
DOI: 10.1103/PhysRevLett.94.220405PACS numbers: 03.65.Ta, 03.65.Ud, 03.67.Hk, 42.50.Dv
It is commonly thought that the mean value of a
quantum-mechanical measurement must be bounded by
the extrema of a spectrum of eigenvalues, a consequence
of statistical mathematics and the measurement postulate
of quantum mechanics. However, there exist certain mea-
surement outcomes for which this is not the case—these
results are called weak values, since they arise as the out-
comes of weak measurements on certain preselected and
postselected quantum systems [1–11]. The canonical ex-
ample of weak values is the gedanken experiment of
Aharonov, Albert, and Vaidman , who described how
it wouldbe possible to use aweak measurement to measure
(say) the ?zeigenvalue of a spin-1=2 particle, and deter-
mine an average value h?zi ? 100.
Weak values are an important and interesting phenome-
non, because they assist us in understanding many couter-
intuitive results of quantum mechanics. For instance, weak
values form a language by which we can resolve certain
paradoxes and model strange quantum behavior. Important
examples include: Hardy’s paradox [12,13], in which two
particles that always annihilate upon meeting are some-
times paradoxically measured after this annihilation event;
the apparent superluminal transport of pulses in optical
fibres displaying polarization mode dispersion ; appar-
ently superluminal particles travelling in vacuum ; and
quantifying momentum transfer in twin-slit ‘‘which-path’’
experiments [14–16]. Weak values are useful in simplify-
ing calculations wherever a system is weakly coupled to a
monitored environment [7,9]. They also are an example of
a manifestly quantum phenomenon, in that the analysis of
weak values can lead to negative (pseudo-) probabilities
, an effect never observed in analogous classical
Here we present the first unambigously quantum-
mechanical experimental realization of weakvalues, where
we use a nondeterministic entangling circuit to enable one
single photon to make a weak measurement of the polar-
ization of another, subject to certain preselections and
postselections. Previous demonstrations of weak values
using electromagnetic radiation [17–20] have used coher-
ent states and weak measurements arising from the cou-
pling of two degrees of freedom of the photon. They can
thus be explained semiclassically using a wave equation
derived from Maxwell’s equations . A cavity QED
experiment  has been performed that was subsequently
analyzed in terms of weak values , but the continuous
spectrum precluded observations of anomalously large
average values. By using two single photons, and realizing
the weak measurement with a two-particle entangling op-
eration, the weak values we measure (including extra-
spectral weak values) are not able to be described in
semiclassical terms—a crucial result in the experimental
verification and study of the phenomenon.
The observable we measure is the polarization of a
single photon in the horizontal-vertical (H ? V) basis,
i.e., the quantum operator corresponding to the S1Stokes
parameter ,^S1? jHihHj ? jVihVj, with expectation
value h^S1i ? h?j^S1j?i for some state j?i. According to
the standard quantum formulation of measurement ,
?1 ? h^S1i ? 1 for any single-photon polarization state.
We will find that it is possible, using weak measurements,
to obtain average values for S1far outside this range.
By analogy with the scheme of Aharonov, Albert, and
Vaidman, we prepare the polarization of a single photon in
j i ? ?jHi ? ?jVi;
where j?j2? j?j2? 1. Subsequently, we make a weak,
nondestructive measurement on the photon’s polarization
in the H ? V basis. The weak measurement is made using
a nondeterministic generalized photon polarization mea-
surement device , which is deemed to have worked
whenever a single photon is present at each of the signal
and meter outputs. The generalized measurement device
works by entangling the signal photon’s polarization with
that of a meter photon prepared in the state ?jHi ? ? ?jVi,
before measuring the meter photon’s polarization. Without
loss of generality we choose ? to be real; ?2? ? ?2? 1.
The experimental setup is shown in Fig. 1.
When operating with balanced modes , and after
signal and meter photons interact but before either is
PRL 94, 220405 (2005)
10 JUNE 2005
2005 The American Physical Society
measured, the state of the system in the two-qubit subspace
corresponding to successful operation is
j?i????jHis??? ?jVis?jHim???? ?jHis???jVis?jVim
where s, m denote signal and meter photons, respectively.
It follows that, with measurement of the meter photon in
the H ? V basis, the weak measurement device imple-
ments a positive-operator-valued measurement (POVM)
f^?H;^?Vg on the signal photon, with
where ? is the Kronecker delta. Equation (3) gives the
measurement strength as 2?2? 1, which is set by the
initial state of the meter photon. For a strong, projective
measurement, ? ? 1; weak measurement occurs when ? is
close to 1=
. A single weak measurement provides little
information about the signal photon’s polarization—the
result is dominated by the randomness of measuring a
meter state close to ?jHi ? jVi?=
However, for a sufficiently large number of measurements
on identically prepared photons, the average signal polar-
ization can be recovered with arbitrary precision. The
expectation value for^S1can be written in terms of prob-
abilities of measuring H or V in the meter output:
2?^1 ? ??iH? ?iV??2?2? 1?^S1?;
in the H ? V basis.
h^S1i ?h j^?Hj i ? h j^?Vj i
After making the weak measurement, we postselect the
signal in a basis mutually unbiased with respect to H ? V
(specifically, on the state jAi ?
selection of a subensemble of measurement results that can
lead to the strange results of weak values. This leads to an
expression for the postselected mean value:
?P?H? ? P?V?
pjVi). It is the
Ah^S1i ?P?HjA? ? P?VjA?
where the leading subscript represents the postselected
state and where, for example, P?HjA? denotes the proba-
bility of measuring H in the meter output given that post-
selection on signal state jAi was successful. The general
expression for the expected postselected value is then
??? ? ???
1 ? 4?? ?Re????:
Using Eq. (2), it can be shown that if ? ! 1=
Ah^S1i ? Re??? ? ??=?? ? ??? so that when ? ? ? ? 0,
the weak value of^S1can be arbitrarily large. In practice, it
is necessary to operate with nonzero measurement strength
and postselection probability, so that a precise experimen-
tal value forAh^S1i can be obtained in a finite acquisition
time. More detail on the theory of qubit weak values can be
found in Ref. .
Postselected weak values are an important indicator of
quantum behavior, since the bizarre results that can be
obtained are not paralleled inthe probabilities ofanalogous
classical measurements. Large weak values arise from a
quantum interference effect that results from the postse-
lection of the signal photon state, which can be seen from
the entangled state in Eq. (2). Consider the result when the
meter photon is detected in the state jHim, but no post-
selection is employed in the signal arm. The probability of
this event is given by the expectation value of the projector
FIG. 1 (color online).
experiment. A single photon is injected into the upper (signal)
input port, where it is prepared in the state of Eq. (1) using a
polarizing beam splitter (PBS) and half waveplate (HWP1;
OA ? optic axis). A weak measurement of the polarization is
made by interacting the photon with another (meter) photon in a
weak measurement device, which operates via measurement-
induced nonlinearity . The interaction of the two photons can
be controlled using HWP2. The signal photon is then postse-
lected in the state jAi ?
a photon counter. A coincidence count flags successful postse-
lection of the signal photon, and weak measurement with an
outcome corresponding to the final meter waveplate (HWP4)
setting. The signal and meter photons are produced in pairs by
spontaneous parametric downconversion from a beta-barium
borate (BBO) crystal, pumped at 1.2 W at 351.1 nm by an
Ar?laser. The mean coincidence rate without postselection on
jAi was 44:6 s?1in 0.36 nm bandwidth; the postselection
reduced this to 0:52 s?1across the measurement outcomes.
Photon counting occurred over 100 s when making measure-
ments in the absence of postselection on jAi. This was increased
to 1000 s when postselection on jAi was included. The coinci-
dence window was 1 ns. We delivered the photons to the
experiment through single mode optical fibers to provide
Gaussianspatial modes for
(b) Conceptual representation of the weak measurement device
[25,26]. The entangling operation occurs because of the non-
classical interference at the 1=3 beam splitter, and conditional on
obtaining one and only one photon at the meter output.
(a) Conceptual representation of the
p ?jHi ? jVi? using HWP3, a PBS and
PRL 94, 220405 (2005)
PHYSICAL REVIEW LETTERS
10 JUNE 2005
^1 ? jHimhHj, with the value j??j2? j?? ?j2corresponding
to the probability of measuring HsHmplus the probability
of measuring VsHm: the probabilities add, and there is no
quantum interference. If we postselect on jAi in the signal
arm, the probability of measuring H in the meter, condi-
tional onthe postselection,
?? ?j2?=?j?? ? ?? ?j2? j?? ? ? ??j2?. It can be seen in the
numerator that now the amplitudes add before squaring,
allowing the possibility of a quantum interference effect.
Combined with the similar expression for a V measure-
ment result, this leads to Eq. (6).
We measured the weak value of the single-photon po-
larization for a range of measurement strengths, with a
nominal input state j i ? cos?42??jHi ? sin?42??jVi ?
0:743jHi ? 0:669jVi. In principle, the experimental value
of ? can be determined from the meter input waveplate
settings. However, since the calculated values of Ah^S1i are
very sensitive to ?, it is desirable to obtain the actual
measurement strength from additional coincidence mea-
surements,to deal witherrorsinthe inputwaveplate setting
and the remainder of the setup. The measurement strength
is identical to the knowledge of the generalized measure-
ment device, K ? PHH? PVV? PHV? PVH? 2?2? 1
(Ref. ), where, e.g., PHVis the probability of observing
a horizontally and a vertically polarized photon at the
signal and meter outputs of the device, respectively, and
where these probabilities are measured with a signal input
state jDi ? ?jHi ? jVi?=
state jAi. Because of Poissonian counting statistics in the
measurement of K, the relative size of the error is quite
large when K is near zero.
The weak values for^S1were determined using Eq. (5)
over a range of measurement strengths (Fig. 2). P?HjA?
and P?VjA? were obtained from experimental coincidence
measurements. For the smallest measurement strength,
K ? 0:006, we observedAh^S1i ? 47, which is much larger
nondemolition measurement followed by postselection on
jAi, i.e.,Ah^S1i ? ??? ? ??? ? 0:1, and also well outside
the spectrum of^S1.
The errors in K ? 2?2? 1 of approximately ?0:015
lead to substantial errors in the largest weak values, due to
the form of Eq. (5). In fact, for the smallest measurement
strengths, the uncertainty in K encompasses K ? 0, and
the error in jAh^S1ij is unbounded above. The triangles in
Fig.2illustrate the ‘‘worstcase’’whereeach pointisvaried
by1?inthevalue of K,ina direction that reduces theweak
value. Even in this case, the smallest measurement strength
yields aweak value of19.Inprinciple, the errors,which are
all derived from Poissonian photon counting statistics,
could be reduced arbitrarily by collecting larger samples
of data. However, the low probability of the postselection,
along with the very small correlation between the signal
and meter photons, leads to very long collection times—a
practical restriction on the size of the data set .
is given by
, and without postselecting the
As the measurement strength is increased, we observe
that the weak value of S1decreases until it is no longer
greater than the strong value j?j2? j?j2? 0:1. As noted
in Ref. , the generalized measurement device does not
exhibit perfect correlations between signal and meter due
to imperfect mode matching. In the present case, this leads
to a systematic offset in the weak value for large K, so that
in fact it drops below this value.
The slight imperfections of the device mean that the
theoretical weak value of Eq. (6), which assumes no mix-
ture, does not completely describe the measurement.
Instead, we determine the actual transfer matrix of the
device—the process matrix—using quantum process to-
mography (in the manner of Ref. ). This provides an
independent means of obtaining a parametric model of the
postselected weak measurement process. As with Eq. (6),
the expression obtained is parametrized by ?, ?, ?, and ? ?,
although the slight mixture leads to a lengthier form .
The calculated curve forAh^S1i, plotted for our nominal
input state, is shown in Fig. 2.
From a classical point of view,or even a typical quantum
measurement point of view, it is quite strange that the
measured expectation value of the^S1Stokes operator lies
outside the interval ??1;1?. The strangeness is perhaps
more dramatic when we consider the results in terms of
mean photon number. We can think of the expectation
value of^S1as the difference in mode occupation between
the two (spatially degenerate) H and V polarization modes.
For instance, writing the strong measurement of^S1for a
^S1(circles) with measurement strength, K ? 2?2? 1. The error
bars plotted arise from the effect of Poissonian counting statistics
on P?HjA? and P?VjA?. Bars not shown are smaller than the
marker dimensions. In addition, errors in K of approximately
?0:015 (over all values of K) lead to correlated errors inAh^S1i
via Eq. (5)—i.e., a displacement in K due to error leads to a
displacement in the weak value such that a given data point
moves along a hyperbola. The triangles illustrate the worst case
where each point is varied by 1? in the value of K, in a direction
that reduces the weak value. The dashed line indicates the
predicted weak value, based on a model (with no free parame-
ters) of the generalized measurement device obtained from
quantum process tomography.
Variation of experimentally observed weak values of
PRL 94, 220405 (2005)
10 JUNE 2005
single photon in state j i in terms of mode occupation
h^S1i ? ???h1jHh0jV? ??h0jHh1jV??^ nH? ^ nV???j1iHj0iV
? ?j0iHj1iV? ? h^ nHi ? h^ nVi:
It follows that in the weak postselected case,?h^ nHi ?
?h^ nVi ??h^S1i, for postselection on the state j?i. That is
to say, we experimentally predict that conditional on pre-
paring a single photon superposed across two polarization
modes, and conditional on the measurement of jAi in the
signal arm, there is a difference of as many as 47 photons
between the two modes when we use the weakest general-
ized measurement. This seems nonsensical when we know
that there was one signal input photon .
The resolution to this problem is that the weak values
emerging from postselected weak measurements can be
combined with the those from the complementary postse-
lection to yield the expectation value of Eq. (4):
h^S1i ?Ah^SiP?A? ?Dh^SiP?D? ? ???? ???:
Again, the effects of mixture in the experimental process
mean that this relationship does not exactly hold for the
data—for instance, the weak measurement device slightly
decoheres the signal photon, resulting in P?A? ? 0:012
instead of the expected value, which is
0:0027 when 2?2? 1 ? 0. Using the expected value for
P?A?, and the measured weak valueAh^S1i ? 47, we obtain
h^S1i ? 0:25. The standard error for h^S1i is bounded below
by 0.10 and is unbounded above due to the error inAh^S1i.
Using the expression obtained from the tomographic re-
construction, we can determine an accurate expectation
value for^S1from the measured weak value, even in the
presence of mixture. We obtain h^S1i ? 0:08 ? 0:03 by this
method. From the input settings, we expect h^S1i ? 0:10.
Inconclusion, wehave demonstrated a completely quan-
tum realization of weak values. The weak measurement
step relies on nonclassical interference between signal and
meter photons, meaning that the results cannot be ex-
plained by Maxwell’s equations alone. Using this tech-
nique, we observe expectation values of quantum
observables far outside the range generally allowed by
quantum measurement theory, including mean values of
the single-photon S1Stokes parameter of up to 47.
We thank S.D. Bartlett for stimulating discussions. This
work was supported by the Australian Research Council
and the State of Queensland.
*Electronic address: www.quantinfo.org
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the online article’s HTML reference section. The docu-
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(http://www.aip.org/pubservs/epaps.html) or from ftp.
aip.org in the directory /epaps/. See the EPAPS homepage
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10 JUNE 2005