ARTICLE
Quantum advantage in postselected metrology
David R. M. Arvidsson-Shukur 1,2,3✉, Nicole Yunger Halpern 3,4,5, Hugo V. Lepage 1,
Aleksander A. Lasek 1, Crispin H. W. Barnes1 & Seth Lloyd2,3
In every parameter-estimation experiment, the final measurement or the postprocessing
incurs a cost. Postselection can improve the rate of Fisher information (the average infor-
mation learned about an unknown parameter from a trial) to cost. We show that this
improvement stems from the negativity of a particular quasiprobability distribution, a
quantum extension of a probability distribution. In a classical theory, in which all observables
commute, our quasiprobability distribution is real and nonnegative. In a quantum-
mechanically noncommuting theory, nonclassicality manifests in negative or nonreal quasi-
probabilities. Negative quasiprobabilities enable postselected experiments to outperform
optimal postselection-free experiments: postselected quantum experiments can yield
anomalously large information-cost rates. This advantage, we prove, is unrealizable in any
classically commuting theory. Finally, we construct a preparation-and-postselection proce-
dure that yields an arbitrarily large Fisher information. Our results establish the non-
classicality of a metrological advantage, leveraging our quasiprobability distribution as a
mathematical tool.
https://doi.org/10.1038/s41467-020-17559-w OPEN
1 Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge CB3 0HE, UK. 2 Department of Mechanical Engineering, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA. 3 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA. 4 ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA. 5Department of Physics, Harvard University, Cambridge, MA
02138, USA. ✉email: drma2@cam.ac.uk
NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications 1
12
34
56
78
9
0
()
:,;
Our ability to deliver new quantum-mechanical improve-ments to technologies relies on a better understanding ofthe foundation of quantum theory: When is a phenom-
enon truly nonclassical? We take noncommutation as our notion
of nonclassicality and we quantify this nonclassicality with
negativity: Quantum states can be represented by quasiprobability
distributions, extensions of classical probability distributions.
Whereas probabilities are real and nonnegative, quasiprobabilities
can assume negative and nonreal values. Quasiprobabilities’
negativity stems from the impossibility of representing quantum
states with joint probability distributions1–3. The distribution we
use, an extension of the Kirkwood–Dirac distribution4–6, signals
nonclassical noncommutation through the presence of negative
or nonreal quasiprobabilities.
One field advanced by quantum mechanics is metrology, which
concerns the statistical estimation of unknown physical para-
meters. Quantum metrology relies on quantum phenomena to
improve estimations beyond classical bounds7. A famous example
exploits entanglement8–10. Consider using N separable and dis-
tinguishable probe states to evaluate identical systems in parallel.
The best estimator’s error will scale as N−1/2. If the probes are
entangled, the error scaling improves to N−1 11. As Bell’s theorem
rules out classical (local realist) explanations of entanglement, the
improvement is genuinely quantum.
A central quantity in parameter estimation is the Fisher
information, IðθÞ. The Fisher information quantifies the
average information learned about an unknown parameter θ
from an experiment12–14. IðθÞ lower-bounds the variance of an
unbiased estimator θe via the Cramér–Rao inequality:
VarðθeÞ ≥ 1=IðθÞ15,16. A common metrological task concerns
optimally estimating a parameter that characterizes a physical
process. The experimental input and the final measurement are
optimized to maximize the Fisher information and to minimize
the estimator’s error.
Classical parameter estimation can benefit from postselecting
the output data before postprocessing. Postselection can raise the
Fisher information per final measurement or postprocessing
event. Postselection can also raise the rate of information per final
measurement in a quantum setting. But classical postselection is
intuitive, whereas an intense discussion surrounds postselected
quantum experiments17–29. The ontological nature of post-
selected quantum states, and the extent to which they exhibit
nonclassical behavior, is subject to an ongoing debate. Particular
interest has been aimed at pre- and postselected averages of
observables. These “weak values” can lie outside an observable’s
eigenspectrum when measured via a weak coupling to a pointer
particle17,30. Such values offer metrological advantages in esti-
mations of weak-coupling strengths19,24,31–36.
In this article, we go beyond this restrictive setting and ask, can
postselection provide a nonclassical advantage in general quan-
tum parameter-estimation experiments? We conclude that it can.
We study metrology experiments for estimating an unknown
transformation parameter whose final measurement or post-
processing incurs an experimental cost37,38. Postselection allows
the experiment to incur that cost only when the postselected
measurement’s result reveals that the final measurement’s Fisher
information will be sufficiently large. We express the Fisher
information in terms of a quasiprobability distribution. Quantum
negativity in this distribution enables postselection to increase the
Fisher information above the values available from standard
input-and-measurement-optimized experiments. Such an anom-
alous Fisher information can improve the rate of information
gain to experimental cost, offering a genuine quantum advantage
in metrology. We show that, within a commuting theory, a theory
in which observables commute classically, postselection can
improve information-cost rates no more than a strategy that uses
an optimal input and final measurement can. We thus conclude
that experiments that generate anomalous Fisher-information
values require noncommutativity.
Results
Postselected quantum Fisher information. As aforementioned,
postselection can raise the Fisher information per final mea-
surement. Figure 1 outlines a classical experiment with such an
information enhancement. Below, we show how postselection
affects the Fisher information in a quantum setting.
Consider an experiment with outcomes i and associated
probabilities pi(θ), which depend on some unknown parameter
θ. The Fisher information about θ is14
IðθÞ ¼
X
i
piðθÞ½∂θln ðpiðθÞÞ2 ¼
X
i
1
piðθÞ
½∂θpiðθÞ2: ð1Þ
Repeating the experiment N≫ 1 times provides, on average, an
amount NIðθÞ of information about θ. The estimator’s variance
is bounded by VarðθeÞ≥ 1=½NIðθÞ.
Below, we define and compare two types of metrological
procedures. In both scenarios, we wish to estimate an unknown
parameter θ that governs a physical transformation.
Optimized prepare-measure experiment: An input system
undergoes the partially unknown transformation, after which
the system is measured. Both the input system and the
measurement are chosen to provide the largest possible Fisher
information.
Postselected prepare-measure experiment: An input system
undergoes, first, the partially unknown transformation and,
second, a postselection measurement. Conditioned on the
postselection’s yielding the desired outcome, the system under-
goes an information-optimized final measurement.
In quantum parameter estimation, a quantum state is
measured to reveal information about an unknown parameter
encoded in the state. We now compare, in this quantum setting,
the Fisher-information values generated from the two metrolo-
gical procedures described above. Consider a quantum experi-
ment that outputs a state ρ^θ ¼ U^ðθÞρ^0U^
yðθÞ, where ρ^0 is the
input state and U^ðθÞ represents a unitary evolution set by θ. The
quantum Fisher information is defined as the Fisher information
maximized over all possible generalized measurements7,13,39,40:
IQðθjρ^θÞ ¼ Tr ρ^θΛ^
2
ρ^θ
h i
: ð2Þ
Λ^ρ^θ is the symmetric logarithmic derivative, implicitly defined by
∂θρ^θ ¼ 12 ðΛ^ρ^θ ρ^θ þ ρ^θΛ^ρ^θ Þ12.
If ρ^θ is pure, such that ρ^θ ¼ Ψθj i Ψθh j, the quantum Fisher
information can be written as32,41
IQðθjρ^θÞ ¼ 4h _Ψθj _Ψθi  4jh _ΨθjΨθij2; ð3Þ
where _Ψθ
   ∂θ Ψθj i.
p
1–p
F ps ()
Γ
Fig. 1 Classical experiment with postselection. A nonoptimal input device
initializes a particle in one of two states, with probabilities p and 1− p,
respectively. The particle undergoes a transformation Γθ set by an unknown
parameter θ. Only the part of the transformation that acts on particles in
the lower path depends on θ. If the final measurement is expensive, the
particles in the upper path should be discarded: they possess no
information about θ.
ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w
2 NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications
We assume that the evolution can be represented in accordance
with Stone’s theorem42, by U^ðθÞ  eiA^θ , where A^ is a Hermitian
operator. We assume that A^ is not totally degenerate: If all the A^
eigenvalues were identical, U^ðθÞ would not imprint θ onto the
state in a relative phase. For a pure state, the quantum Fisher
information equals IQðθjρ^θÞ= 4VarðA^Þρ^0 7. Maximizing Eq. (1)
over all measurements gives IQðθjρ^θÞ. Similarly, IQðθjρ^θÞ can be
maximized over all input states. For a given unitary U^ðθÞ ¼ eiA^θ ,
the maximum quantum Fisher information is
maxρ^0 IQðθjρ^θÞ
  ¼ 4maxρ^0 Var ðA^Þρ^0
n o
¼ ðΔaÞ2; ð4Þ
where Δa is the difference between the maximum and minimum
eigenvalues of A^7. (The information-optimal input state is a pure
state in an equal superposition of one eigenvector associated with
the smallest eigenvalue and one associated with the largest.) To
summarize, in an optimized quantum prepare-measure experi-
ment, the quantum Fisher information is (Δa)2.
We now find an expression for the quantum Fisher informa-
tion in a postselected prepare-measure experiment. A projective
postselection occurs after U^ðθÞ but before the final measurement.
Figure 2 shows such a quantum circuit. The renormalized
quantum state that passes the postselection is
Ψpsθ
   ψpsθ = ffiffiffiffiffiffippsθq , where we have defined an unnormalized
state ψpsθ
   F^ Ψθj i and the postselection probability
ppsθ  TrðF^ρ^θÞ. As before, ρ^θ ¼ U^ðθÞρ^0U^
yðθÞ. F^ ¼Pf2F ps fj i fh j
is the postselecting projection operator, and F ps is a set of
orthonormal basis states allowed by the postselection. Finally, the
postselected state undergoes an information-optimal
measurement.
When Ψpsθ
   ψpsθ = ffiffiffiffiffiffippsθq is substituted into Eq. (3), the
derivatives of ppsθ cancel, such that
IQðθjΨpsθ Þ ¼ 4h _ψpsθ j _ψpsθ i
1
ppsθ
 4jh _ψpsθ jψpsθ ij2
1
ðppsθ Þ2
: ð5Þ
Equation (5) gives the quantum Fisher information available from
a quantum state after its postselection. Unsurprisingly, IQðθjΨpsθ Þ
can exceed IQðθjρ^θÞ, since ppsθ ≤ 1. Also classical systems can
achieve such postselected information amplification (see Fig. 1).
Unlike in the classical case, however, IQðθjΨpsθ Þ can also exceed
the Fisher information of an optimized prepare-measure experi-
ment, (Δa)2. We show how below.
Quasiprobability representation. In classical mechanics, our
knowledge of a point particle can be described by a probability
distribution for the particle’s position, x, and momentum, k: p
(x, k). In quantum mechanics, position and momentum do not
commute, and a state cannot generally be represented by a joint
probability distribution over observables’ eigenvalues. A quantum
state can, however, be represented by a quasiprobability
distribution. Many classes of quasiprobability distributions exist.
The most famous is the Wigner function43. Such a distribution
satisfies some, but not all, of Kolmogorov’s axioms for probability
distributions44: the entries sum to unity, and marginalizing over
the eigenvalues of every observable except one yields a probability
distribution over the remaining observable’s eigenvalues. A qua-
siprobability distribution can, however, have negative or nonreal
values. Such values signal nonclassical physics in, for example,
quantum computing and quantum chaos2,6,45–53.
A cousin of the Wigner function is the Kirkwood–Dirac
quasiprobability distribution4–6. This distribution, which has
been referred to by several names across the literature, resembles
the Wigner function for continuous systems. Unlike the Wigner
functions, however, the Kirkwood–Dirac distribution is well-
defined for discrete systems, even qubits. The Kirkwood–Dirac
distribution has been used in the study of weak-value
amplification6,48,54–57, information scrambling6,52,53,58, and
direct measurements of quantum wavefunctions59–61. Moreover,
negative and nonreal values of the distribution have been linked
to nonclassical phenomena6,48,52,53. We cast the quantum Fisher
information for a postselected prepare-measure experiment in
terms of a doubly extended Kirkwood–Dirac quasiprobability
distribution6. (The modifier “doubly extended” comes from the
experiment in which one would measure the distribution: One
would prepare ρ^, sequentially measure two observables weakly,
and measure one observable strongly. The number of weak
measurements equals the degree of the extension6.) We employ
this distribution due to its usefulness as a mathematical tool: This
distribution enables the proof that, in the presence of non-
commuting observables, postselection can give a metrological
protocol a nonclassical advantage.
Our distribution is defined in terms of eigenbases of A^ and F^.
Other quasiprobability distributions are defined in terms of bases
independent of the experiment. For example, the Wigner function
is often defined in the bases of the quadrature of the electric field
or the position and momentum bases. However, basis-
independent distributions can be problematic in the hunt for
nonclassicality2,49. Careful application, here, of the extended
Kirkwood–Dirac distribution ties its nonclassical values to the
operational specifics of the experiment.
To begin, we define the quasiprobability distribution of an
arbitrary quantum state ρ^:
qρ^a;a0;f  hf jai ah jρ^ a0j iha0jf i: ð6Þ
Here, f fj ig, f aj ig, and f a0j ig are bases for the Hilbert space on
which ρ^ is defined. We can expand ρ^59,60 as
ρ^ ¼
X
a;a0;f
aj i fh j
hf jai q
ρ^
a;a0;f : ð7Þ
If any hf jai= 0, we perturb one of the bases infinitesimally while
preserving its orthonormality.
Let f aj ig ¼ f a0j ig denote an eigenbasis of A^, and let f fj ig
denote an eigenbasis of F^. The reason for introducing a doubly
extended distribution, instead of the standard Kirkwood–Dirac
distribution qρ^a;f  hf jai ah jρ^ fj i, is that IQðθjΨpsθ Þ can be
expressed most concisely, naturally, and physically meaningfully
in terms of qρ^θa;a0;f . Later, we shall see how the nonclassical entries
in qρ^a;a0;f and q
ρ^
a;f are related. We now express the postselected
quantum Fisher information (Eq. (5)) in terms of the
U( )
F
0 

1–F
ps
Fig. 2 Preparation of postselected quantum state. First, an input quantum
state ρ^0 undergoes a unitary transformation U^ðθÞ ¼ eiθA^: ρ^0 ! ρ^θ . Second,
the quantum state is subject to a projective postselective measurement
fF^; 1^ F^g. The postselection is such that if the outcome related to the
operator F^ happens, then the quantum state is not destroyed. The
experiment outputs renormalized states ρ^psθ ¼ F^ρ^θ F^=TrðF^ρ^θÞ.
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w ARTICLE
NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications 3
quasiprobability values qρ^θa;a0;f (see Supplementary Note 1).
IQðθjΨpsθ Þ ¼ 4
X
a; a0; f2F ps
qρ^θa;a0;f
ppsθ
aa0  4
X
a; a0; f2F ps
qρ^θa;a0;f
ppsθ
a


2
;
ð8Þ
where a and a0 denote the eigenvalues associated with aj i and
a0j i, respectively. (We have suppressed degeneracy parameters γ
in our notation for the states, e.g., a; γj i  aj i.) Equation (8)
contains a conditional quasiprobability distribution, qρ^θa;a0;f =p
ps
θ . If
A^ commutes with F^, as they do classically, then they share an
eigenbasis for which qρ^θa;a0;f =p
ps
θ 2 ½0; 1, and the postselected
quantum Fisher information is bounded as IQðθjΨpsθ Þ≤ ðΔaÞ2:
Theorem 1 In a classically commuting theory, no postselected
prepare-measure experiment can generate more Fisher informa-
tion than the optimized prepare-measure experiment.
Proof: We upper-bound the right-hand side of Eq. (8). First, if
f aj ig ¼ f a0j ig ¼ f fj ig is a eigenbasis shared by A^ and F^, Eq. (6)
simplifies to a probability distribution:
qρ^θa;a0;f ¼ ah jρ^θ a0j i½ fj i ¼ aj i½ a0j i ¼ fj i 2 ½0; 1; ð9Þ
where [X] is the Iverson bracket, which equals 1 if X is true and
equals 0 otherwise. Second, summing qρ^θa;a0;f =p
ps
θ over f 2 F ps, we
find X
f2F ps
qρ^θa;a0;f =p
ps
θ ¼ ah jρ^θ a0j i a0h jF^ aj i=ppsθ : ð10Þ
By the eigenbasis shared by A^ and F^, the sum simplifies to
ah jρ^θF^ a0j i½ a0j i ¼ aj i=ppsθ . We can thus rewrite Eq. (8):
IQðθjΨpsθ Þ ¼ 4
X
a;a0
ah jρ^θF^ a0j i½ a0j i ¼ aj i
ppsθ
aa0
 4
X
a;a0
ah jρ^θF^ a0j i½ a0j i ¼ aj i
ppsθ
a


2
¼ 4
X
a
qaa
2  4
X
a
qaa
 !2
;
ð11Þ
where we have defined the probabilities
qa  ah jρ^θF^ aj i=ppsθ ¼
P
f2F ps ah jρ^θ aj i½ fj i ¼ aj i=ppsθ .
Apart from the multiplicative factor of 4, Eq. (11) is in the form
of a variance with respect to the observable’s eigenvalues a. Thus,
Eq. (11) is maximized when qamin ¼ qamax ¼ 12:
max
fqag
fIQðθjΨpsθ Þg ¼ ðΔaÞ2: ð12Þ
This Fisher-information bound must be independent of our
choice of eigenbases of A^ and F^. In summary, if A^ commutes with
F^, then all qρ^θa;a0;f =p
ps
θ can be expressed as real and nonnegative,
and IQðθjΨpsθ Þ≤ ðΔaÞ2.
In contrast, if the quasiprobability distribution contains
negative values, the postselected quantum Fisher information
can violate the bound: IQðθjΨpsθ Þ> ðΔaÞ2.
Theorem 2 An anomalous postselected Fisher information
implies that the quantum Fisher information cannot be expressed
in terms of a nonnegative doubly extended Kirkwood–Dirac
quasiprobability distribution.
Proof: See Supplementary Note 2 for a proof.
(The theorem’s converse is not generally true.) This inability to
express implies that A^ fails to commute with F^. However,
pairwise noncommutation of ρ^θ , A^, and F^ is insufficient to enable
anomalous values of IQðθjΨpsθ Þ. For example, noncommutation
could lead to a nonreal Kirkwood–Dirac distribution without any
negative real components. Such a distribution cannot improve
IQðθjΨpsθ Þ beyond classical values. Furthermore, the presence or
absence of commutation is a binary measure. In contrast, how
much postselection improves IQðθjΨpsθ Þ depends on how much
negativity qρ^θa;a0;f =p
ps
θ has. We build on this observation, and
propose two experiments that yield anomalous Fisher-
information values, in Supplementary Notes 3 and 4. (It remains
an open question to investigate the relationship between
Kirkwood–Dirac negativity in other metrology protocols with
noncommuting operators, e.g., ref. 62.)
As promised, we now address the relation between nonclassical
entries in qρ^a;a0;f and nonclassical entries in q
ρ^
a;f . For pure
states ρ^ ¼ Ψj i Ψh j, the doubly extended quasiprobability distribu-
tion can be expressed time symmetrically in terms of the standard
Kirkwood–Dirac distribution4–6,51–53: qρ^a;a0;f ¼ 1pf q
ρ^
a;f ðqρ^a0;f Þ

,
where qρ^a;f ¼ hf jai ah jρ^ fj i and pf ≡ ∣〈f ∣Ψ〉∣2. (See refs. 20,63 for
discussions about time-symmetric interpretations of quantum
mechanics.) Therefore, a negative qρ^a;a0;f implies negative or
nonreal values of qρ^a;f . Similarly, a negative q
ρ^
a;a0;f implies a
negative or nonreal weak value 〈f ∣a〉〈a∣Ψ〉/〈f ∣Ψ〉17, which
possesses interesting ontological features (see below). Thus, an
anomalous Fisher information is closely related to a negative or
nonreal weak value. Had we weakly measured the observable
aj i ah j of ρ^θ with a qubit or Gaussian pointer particle before the
postselection, and had we used a fine-grained postselection
f1^ F^; fj i fh j : f 2 F psg, the weak measurement would have
yielded a weak value outside the eigenspectrum of aj i ah j. It has
been shown that such an anomalous weak value proves that
quantum mechanics, unlike classical mechanics, is contextual:
quantum outcome probabilities can depend on more than a
unique set of underlying physical states24,35,64. If ρ^θ had
undergone the aforementioned weak measurement, instead of
the postselected prepare-measure experiment, the weak measure-
ment’s result would have signaled quantum contextuality.
Consequently, a counterfactual connects an anomalous Fisher
information and quantum contextuality. While counterfactuals
create no problems in classical physics, they can lead to logical
paradoxes in quantum mechanics64–67. Hence our counter-
factual’s implication for the ontological relation between an
anomalous Fisher information and contextuality offers an
opportunity for future investigation.
Improved metrology via postselection. In every real experiment,
the preparation and final measurement have costs, which we
denote CP and CM, respectively. For example, a particle-
number detector’s dead time, the time needed to reset after a
detection, associates a temporal cost with measurements68.
Reference37 concerns a two-level atom in a noisy environment.
Liuzzo et al. detail the tradeoff between frequency estimation’s
time and energy costs. Standard quantum-metrology techniques,
they show, do not necessarily improve metrology, if the experi-
ment’s energy is capped. Also, the cost of postprocessing can be
incorporated into CM. (In an experiment, these costs can
be multivariate functions that reflect the resources and constraints.
Such a function could combine a detector’s dead time with
the monetary cost of liquid helium and a graduate student’s
salary. However, presenting the costs in a general form benefits
this platform-independent work.) We define the information-cost
rate as RðθÞ :¼ NIðθÞ=ðNCP þ NCMÞ ¼ IðθÞ=ðCP þ CMÞ. If our
ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w
4 NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications
experiment conditions the execution of the final measurement on
successful postselection of a fraction ppsθ of the states, we include a
cost of postselection, Cps. We define the postselected experiment’s
information-cost rate as RpsðθÞ :¼ Nppsθ IpsðθÞ=ðNCP þ NCpsþ
Nppsθ CMÞ ¼ ppsθ IpsðθÞ=ðCP þ Cps þ ppsθ CMÞ, where IpsðθÞ is the
Fisher information conditioned on successful postselection. Gen-
eralizing the following arguments to preparation and measure-
ment costs that differ between the postselected and
nonpostselected experiments is straightforward.
In classical experiments, postselection can improve the
information-cost rate. See Fig. 1 for an example. But can
postselection improve the information-cost rate in a classical
experiment with information-optimized inputs? Theorem 1
answered this question in the negative. IpsðθÞ≤ maxfIðθÞg in
every classical experiment. The maximization is over all
physically accessible inputs and final measurements. A direct
implication is that RpsðθÞ≤ maxfRðθÞg.
In quantum mechanics, IQðθjΨpsθ Þ can exceed
maxρ^0fIQðθjρ^θÞg ¼ ðΔaÞ
2. This result would be impossible
classically. Anomalous Fisher-information values require quan-
tum negativity in the doubly extended Kirkwood–Dirac distribu-
tion. Consequently, even compared to quantum experiments with
optimized input states, postselection can raise information-cost
rates beyond classically possible rates: RpsðθÞ>maxfRðθÞg. This
result generalizes the metrological advantages observed in the
measurements of weak couplings, which also require noncom-
muting operators. References69–77 concern metrology that
involves weak measurements of the following form. The primary
system S and the pointer P begin in a pure product state
ΨSj i  ΨPj i; the coupling Hamiltonian is a product
H^ ¼ A^S  A^P; the unknown coupling strength θ is small; and
just the system is postselected. Our results govern arbitrary input
states, arbitrary Hamiltonians (that satisfy Stone’s theorem),
arbitrarily large coupling strengths θ, and arbitrary projective
postselections. Our result shows that postselection can improve
quantum parameter estimation in experiments where the final
measurement’s cost outweighs the combined costs of state
preparation and postselection: CM  CP þ Cps. Earlier works
identified that the Fisher information from nonrenormalized
trials that succeed in the postselection cannot exceed the Fisher
information averaged over all trials, including the trials in which
the postselection fails78,79. (Reference 41 considered squeezed
coherent states as metrological probes in specific weak-
measurement experiments. It is shown that postselection can
improve the signal-to-noise ratio, irrespectively of whether the
analysis includes the failed trials. However, this work concerned
nonpostselected experiments in which only the probe state was
measured. Had it been possible to successfully measure also the
target system, the advantage would have disappeared.) In
accordance with practical metrology, not only the Fisher
information, but also measurements’ experimental costs, underlie
our results.
So far, we have shown that IQðθjΨpsθ Þ can exceed (Δa)2. But
how large can IQðθjΨpsθ Þ grow? In Supplementary Note 3, we
show that, if the generator A^ has M ≥ 3 not-all-identical
eigenvalues, there is no upper bound on IQðθjΨpsθ Þ. If CP and
Cps are negligible compared with CM, then there is no theoretical
cap on how large Rps(θ) can grow. In general, when
IQðθjΨpsθ Þ ! 1, ppsθ ´ IQðθjΨpsθ Þ< ðΔaÞ2, such that information
is lost in the events discarded by postselection. But if A^ has
doubly degenerate minimum and maximum eigenvalues,
ppsθ ´ IQðθjΨpsθ Þ can approach (Δa)2 while IQðθjΨpsθ Þ approaches
infinity (see Supplementary Note 4). In such a scenario,
postselection can improve information-cost rates, as long as
Cps < ð1 ppsθ ÞCM—a significantly weaker requirement than
CM  CP þ Cps.
Discussion
From a practical perspective, our results highlight an important
quantum asset for parameter-estimation experiments with
expensive final measurements. In some scenarios, the postselec-
tion’s costs exceed the final measurement’s costs, as an unsuc-
cessful postselection might require fast feedforward to block
the final measurement. But in single-particle experiments, the
postselection can be virtually free and, indeed, unavoidable:
an unsuccessful postselection can destroy the particle, precluding
the triggering of the final measurement’s detection apparatus80.
Thus, current single-particle metrology could benefit from
postselected improvements of the Fisher information. A
photonic experimental test of our results is currently under
investigation.
From a fundamental perspective, our results highlight the
strangeness of quantum mechanics as a noncommuting theory.
Classically, an increase of the Fisher information via postselection
can be understood as the a posteriori selection of a better input
distribution. But it is nonintuitive that quantum-mechanical
postselection can enable a quantum state to carry more Fisher
information than the best possible input state could. Further-
more, it is surprising that noncommutation can be proved to
underlie the metrological advantage: Other nonclassical phe-
nomena, such as entanglement and discord, could be expected to
underlie a given nonclassical advantage; noncommutation does
not guarantee a metrological advantage; and the proof turns out
to involve considerable mathematical footwork. The optimized
Cramér–Rao bound, obtained from Eq. (4), can be written in the
form of an uncertainty relation:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
VarðθeÞ
p ðΔaÞ≥ 17. Our results
highlight the probabilistic possibility of violating this bound.
More generally, the information-cost rate’s ability to violate a
classical bound leverages negativity, a nonclassical resource in
quantum foundations, for metrological advantage.
Data availability
The authors declare that all data supporting the findings of this study are available within
the paper and its supplementary information files.
Received: 20 November 2019; Accepted: 8 July 2020;
References
1. Lütkenhaus, N. & Barnett, S. M. Nonclassical effects in phase space. Phys. Rev.
A 51, 3340–3342 (1995).
2. Spekkens, R. W. Negativity and contextuality are equivalent notions of
nonclassicality. Phys. Rev. Lett. 101, 020401 (2008).
3. Ferrie, C. & Emerson, J. Frame representations of quantum mechanics and the
necessity of negativity in quasi-probability representations. J. Phys. A: Math.
Theor. 41, 352001 (2008).
4. Kirkwood, J. G. Quantum statistics of almost classical assemblies. Phys. Rev.
44, 31–37 (1933).
5. Dirac, P. A. M. On the analogy between classical and quantum mechanics.
Rev. Mod. Phys. 17, 195–199 (1945).
6. Yunger Halpern, N., Swingle, B. & Dressel, J. Quasiprobability behind the out-
of-time-ordered correlator. Phys. Rev. A 97, 042105 (2018).
7. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology.
Nat. Photonics 5, 222–229 (2011).
8. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum metrology. Phys. Rev. Lett.
96, 010401 (2006).
9. Krischek, R. et al. Useful multiparticle entanglement and sub-shot-noise
sensitivity in experimental phase estimation. Phys. Rev. Lett. 107, 080504
(2011).
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w ARTICLE
NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications 5
10. Demkowicz-Dobrzański, R. & Maccone, L. Using entanglement against noise
in quantum metrology. Phys. Rev. Lett. 113, 250801 (2014).
11. Maccone, L. Intuitive reason for the usefulness of entanglement in quantum
metrology. Phys. Rev. A 88, 042109 (2013).
12. Helstrom, C. W. Quantum Detection and Estimation Theory. 1st edn
(Academic Press, New York, 1976).
13. Braunstein, S. L. & Caves, C. M. Statistical distance and the geometry of
quantum states. Phys. Rev. Lett. 72, 3439–3443 (1994).
14. Cover, T. M. & Thomas, J. A. Elements of Information Theory. 2nd edn (John
Wiley and Sons Inc., Hoboken, 2006).
15. Cramér, H. Mathematical Methods of Statistics (PMS-9), Vol. 9 (Princeton
University Press, 2016).
16. Rao, C. R. in Breakthroughs in Statistics 235–247 (Springer, 1992).
17. Aharonov, Y., Albert, D. Z. & Vaidman, L. 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, 1351–1354 (1988).
18. Leifer, M. S. & Spekkens, R. W. Pre- and post-selection paradoxes and
contextuality in quantum mechanics. Phys. Rev. Lett. 95, 200405 (2005).
19. Tollaksen, J. Pre- and post-selection, weak values and contextuality. J. Phys. A:
Math. Theor. 40, 9033–9066 (2007).
20. Aharonov, Y. & Vaidman, L. in Time in Quantum Mechanics 235–247
(Springer, 2008).
21. Dressel, J., Agarwal, S. & Jordan, A. N. Contextual values of observables in
quantum measurements. Phys. Rev. Lett. 104, 240401 (2010).
22. Vaidman, L. Past of a quantum particle. Phys. Rev. A 87, 052104 (2013).
23. Ferrie, C. & Combes, J. How the result of a single coin toss can turn out to be
100 heads. Phys. Rev. Lett. 113, 120404 (2014).
24. Pusey, M. F. Anomalous weak values are proofs of contextuality. Phys. Rev.
Lett. 113, 200401 (2014).
25. Pusey, M. F. & Leifer, M. S. Logical pre- and post-selection paradoxes are
proofs of contextuality, in Proceedings of the 12th International Workshop on
Quantum Physics and Logic, Vol. 195, 295–306 (Open Publishing Association,
2015).
26. Arvidsson-Shukur, D. R. M. & Barnes, C. H. W. Quantum counterfactual
communication without a weak trace. Phys. Rev. A 94, 062303 (2016).
27. Arvidsson-Shukur, D. R. M., Gottfries, A. N. O. & Barnes, C. H. W. Evaluation
of counterfactuality in counterfactual communication protocols. Phys. Rev. A
96, 062316 (2017).
28. Arvidsson-Shukur, D. R. M. & Barnes, C. H. W. Postselection and
counterfactual communication. Phys. Rev. A 99, 060102 (2019).
29. Cimini, V., Gianani, I., Piacentini, F., Degiovanni, I. P. & Barbieri, M.
Anomalous values, fisher information, and contextuality, in generalized
quantum measurements. Quantum Sci. Technol. 5, 025007 (2020).
30. Duck, I. M., Stevenson, P. M. & Sudarshan, E. C. G. The sense in which a
“weak measurement” of a spin-1/2 particle’s spin component yields a value
100. Phys. Rev. D. 40, 2112–2117 (1989).
31. Dressel, J., Malik, M., Miatto, F. M., Jordan, A. N. & Boyd, R. W. Colloquium:
understanding quantum weak values: Basics and applications. Rev. Mod. Phys.
86, 307–316 (2014).
32. Pang, S., Dressel, J. & Brun, T. A. Entanglement-assisted weak value
amplification. Phys. Rev. Lett. 113, 030401 (2014).
33. Jordan, A. N., Martínez-Rincón, J. & Howell, J. C. Technical advantages for
weak-value amplification: When less is more. Phys. Rev. X 4, 011031 (2014).
34. Harris, J., Boyd, R. W. & Lundeen, J. S. Weak value amplification can
outperform conventional measurement in the presence of detector saturation.
Phys. Rev. Lett. 118, 070802 (2017).
35. Kunjwal, R., Lostaglio, M. & Pusey, M. F. Anomalous weak values and
contextuality: robustness, tightness, and imaginary parts. Phys. Rev. A 100,
042116 (2019).
36. Xu, L. et al. Approaching quantum-limited metrology with imperfect detectors
by using weak-value amplification. Preprint at http://arXiv.org/abs/
2005.03629 (2020).
37. Liuzzo-Scorpo, P. et al. Energy-efficient quantum frequency estimation. N. J.
Phys. 20, 063009 (2018).
38. Lipka-Bartosik, P. & Demkowicz-Dobrzański, R. Thermodynamic work cost
of quantum estimation protocols. J. Phys. A: Math. Theor. 51, 474001 (2018).
39. Fujiwara, A. & Nagaoka, H. Quantum fisher metric and estimation for pure
state models. Phys. Lett. A 201, 119–124 (1995).
40. Petz, D. & Ghinea, C. Quantum Probability and Related Topics 261–281
(World Scientific, 2011).
41. Pang, S. & Brun, T. A. Improving the precision of weak measurements by
postselection measurement. Phys. Rev. Lett. 115, 120401 (2015).
42. Stone, M. H. On one-parameter unitary groups in hilbert space. Ann. Math.
33, 643–648 (1932).
43. Wigner, E. On the quantum correction for thermodynamic equilibrium. Phys.
Rev. 40, 749–759 (1932).
44. Kolmogorov, A. N. & Bharucha-Reid, A. T. Foundations of the Theory of
Probability 2nd edn (Courier Dover Publications, 2018).
45. Ferrie, C. Quasi-probability representations of quantum theory with
applications to quantum information science. Rep. Prog. Phys. 74, 116001
(2011).
46. Kofman, A. G., Ashhab, S. & Nori, F. Nonperturbative theory of weak pre-
and post-selected measurements. Phys. Rep. 520, 43–133 (2012).
47. Howard, M., Wallman, J., Veitch, V. & Emerson, J. Contextuality supplies the
magic for quantum computation. Nature 510, 351–355 (2014).
48. Dressel, J. Weak values as interference phenomena. Phys. Rev. A 91, 032116
(2015).
49. Delfosse, N., Allard Guerin, P., Bian, J. & Raussendorf, R. Wigner function
negativity and contextuality in quantum computation on rebits. Phys. Rev. X
5, 021003 (2015).
50. Delfosse, N., Okay, C., Bermejo-Vega, J., Browne, D. E. & Raussendorf, R.
Equivalence between contextuality and negativity of the wigner function for
qudits. N. J. Phys. 19, 123024 (2017).
51. Yunger Halpern, N. Jarzynski-like equality for the out-of-time-ordered
correlator. Phys. Rev. A 95, 012120 (2017).
52. Halpern, N. Y., Bartolotta, A. & Pollack, J. Entropic uncertainty relations for
quantum information scrambling. Commun. Phys. 2, 1–12 (2019).
53. González Alonso, J., Yunger Halpern, N. & Dressel, J. Out-of-time-ordered-
correlator quasiprobabilities robustly witness scrambling. Phys. Rev. Lett. 122,
040404 (2019).
54. Steinberg, A. M. Conditional probabilities in quantum theory and the
tunneling-time controversy. Phys. Rev. A 52, 32–42 (1995).
55. Johansen, L. M. Quantum theory of successive projective measurements. Phys.
Rev. A 76, 012119 (2007).
56. Hofmann, H. F. Complex joint probabilities as expressions of reversible
transformations in quantum mechanics. N. J. Phys. 14, 043031 (2012).
57. Piacentini, F. et al. Measuring incompatible observables by exploiting
sequential weak values. Phys. Rev. Lett. 117, 170402 (2016).
58. Mohseninia, R., Alonso, J. R. & Dressel, J. Optimizing measurement strengths
for qubit quasiprobabilities behind out-of-time-ordered correlators. Phys. Rev.
A 100, 062336 (2019).
59. Lundeen, J. S., Sutherland, B., Patel, A., Stewart, C. & Bamber, C.
Direct measurement of the quantum wavefunction. Nature 474, 188–191
(2011).
60. Lundeen, J. S. & Bamber, C. Procedure for direct measurement of general
quantum states using weak measurement. Phys. Rev. Lett. 108, 070402 (2012).
61. Thekkadath, G. S. et al. Direct measurement of the density matrix of a
quantum system. Phys. Rev. Lett. 117, 120401 (2016).
62. Sun, L. et al. Exponentially enhanced quantum metrology using resources of
linear complexity. Preprint at http://arXiv.org/abs/2004.01216 (2020).
63. Leifer, M. S. & Pusey, M. F. Is a time symmetric interpretation of quantum
theory possible without retrocausality? Proc. R. Soc. A: Math., Phys. Eng. Sci.
473, 20160607 (2017).
64. Spekkens, R. W. Contextuality for preparations, transformations, and unsharp
measurements. Phys. Rev. A 71, 052108 (2005).
65. Kochen, S. & Specker, E. P. in The Logico-algebraic Approach to Quantum
Mechanics 293–328 (Springer, 1975).
66. Hardy, L. Quantum mechanics, local realistic theories, and lorentz-invariant
realistic theories. Phys. Rev. Lett. 68, 2981–2984 (1992).
67. Penrose, R. Shadows of the Mind: A Search for the Missing Science of
Consciousness 240 (Oxford University Press, 1994).
68. Greganti, C. et al. Tuning single-photon sources for telecom multi-photon
experiments. Opt. Express 26, 3286–3302 (2018).
69. Hosten, O. & Kwiat, P. Observation of the spin hall effect of light via weak
measurements. Science 319, 787–790 (2008).
70. Starling, D. J., Dixon, P. B., Jordan, A. N. & Howell, J. C. Optimizing the
signal-to-noise ratio of a beam-deflection measurement with interferometric
weak values. Phys. Rev. A 80, 041803 (2009).
71. Brunner, N. & Simon, C. Measuring small longitudinal phase shifts: weak
measurements or standard interferometry? Phys. Rev. Lett. 105, 010405
(2010).
72. Starling, D. J., Dixon, P. B., Jordan, A. N. & Howell, J. C. Precision frequency
measurements with interferometric weak values. Phys. Rev. A 82, 063822
(2010).
73. Egan, P. & Stone, J. A. Weak-value thermostat with 0.2 mk precision. Opt.
Lett. 37, 4991–4993 (2012).
74. Hofmann, H. F., Goggin, M. E., Almeida, M. P. & Barbieri, M. Estimation of a
quantum interaction parameter using weak measurements: theory and
experiment. Phys. Rev. A 86, 040102 (2012).
75. Magaña Loaiza, O. S., Mirhosseini, M., Rodenburg, B. & Boyd, R. W.
Amplification of angular rotations using weak measurements. Phys. Rev. Lett.
112, 200401 (2014).
76. Lyons, K., Dressel, J., Jordan, A. N., Howell, J. C. & Kwiat, P. G. Power-
recycled weak-value-based metrology. Phys. Rev. Lett. 114, 170801 (2015).
77. Martínez-Rincón, J., Mullarkey, C. A., Viza, G. I., Liu, W.-T. & Howell, J. C.
Ultrasensitive inverse weak-value tilt meter. Opt. Lett. 42, 2479–2482 (2017).
ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w
6 NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications
78. Ferrie, C. & Combes, J. Weak value amplification is suboptimal for estimation
and detection. Phys. Rev. Lett. 112, 040406 (2014).
79. Combes, J., Ferrie, C., Jiang, Z. & Caves, C. M. Quantum limits on
postselected, probabilistic quantum metrology. Phys. Rev. A 89, 052117
(2014).
80. Calafell, I. A. et al. Trace-free counterfactual communication with a
nanophotonic processor. npj Quantum Inf. 5, 61 (2019).
Acknowledgements
The authors would like to thank Justin Dressel, Nicolas Delfosse, Matthew Pusey, Noah
Lupu-Gladstein, Aharon Brodutch and Jan-Åke Larsson for useful discussions. D.R.M.
A.-S. acknowledges support from the EPSRC, the Sweden-America Foundation, Hitachi
Ltd, Lars Hierta’s Memorial Foundation and Girton College. N.Y.H. was supported by an
NSF grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at
Harvard University and the Smithsonian Astrophysical Observatory. H.V.L. received
funding from the European Union’s Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie grant agreement No 642688. A.A.L. acknowledges
support from the EPSRC and Hitachi Ltd. S.L. was supported by NSF, AFOSR, and by
ARO under the Blue Sky Initiative.
Author contributions
D.R.M.A.-S., N.Y.H., H.V.L., A.A.L., C.H.W.B., and S.L. contributed extensively to the
paper and to the writing of the manuscript.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/s41467-
020-17559-w.
Correspondence and requests for materials should be addressed to D.R.M.A.-S.
Peer review information Nature Communications thanks Rafal Demkowicz-Dobrzanski
and the other anonymous reviewer(s) for their contribution to the peer review of
this work.
Reprints and permission information is available at http://www.nature.com/reprints
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made. The images or other third party
material in this article are included in the article’s Creative Commons license, unless
indicated otherwise in a credit line to the material. If material is not included in the
article’s Creative Commons license and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from
the copyright holder. To view a copy of this license, visit http://creativecommons.org/
licenses/by/4.0/.
© The Author(s) 2020
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-17559-w ARTICLE
NATURE COMMUNICATIONS |         (2020) 11:3775 | https://doi.org/10.1038/s41467-020-17559-w |www.nature.com/naturecommunications 7