Precision Quantum Metrology and Nonclassicality in Linear and Nonlinear Detection Schemes
A
´
ngel Rivas
1
and Alfredo Luis
2,
*
1
Institut fu
¨
r Theoretische Physik, Universita
¨
t Ulm, Ulm D-89069, Germany
2
Departamento de O
´
ptica, Facultad de Ciencias Fı
´
sicas, Universidad Complutense, 28040 Madrid, Spain
(Received 14 February 2010; published 30 June 2010)
We examine whether metrological resolution beyond coherent states is a nonclassical effect. We show
that this is true for linear detection schemes but false for nonlinear schemes, and propose a very simple
experimental setup to test it. We find a nonclassicality criterion derived from quantum Fisher information.
DOI: 10.1103/PhysRevLett.105.010403 PACS numbers: 03.65.Ca, 03.65.Ta, 42.50.Ar, 42.50.Dv
Nonclassicality is a key concept supporting the necessity
of the quantum theory. There is widespread consensus that
the coherent states ji are the classical side of the border-
line between the quantum and classical realms [1,2]. In
quantum metrology it is usually believed that resolution
beyond coherent states is a quantum effect, since this is
achieved by famous nonclassical probe states, such as
squeezed, number, or coherent superpositions of distin-
guishable states [3]. However, this does not mean that
every state providing larger resolution than coherent states
is nonclassical.
In this Letter we test this belief by examining whether
metrological resolution beyond coherent states is neces-
sarily a nonclassical effect or not [4]. To this end we find a
novel nonclassicality criterion derived from quantum
Fisher information. We demonstrate that the belief is true
for linear detection schemes but false for nonlinear
schemes. Nonlinear detection is a recently introduced
item in quantum metrology that has plenty of promising
possibilities and is being thoroughly studied and imple-
mented in different areas such as quantum optics [5,6],
Bose-Einstein condensates [7,8], nanomechanical resona-
tors [9], and atomic magnetometry [10].
Throughout we focus on single-mode quantum light
beams with complex amplitude operators a such that
½a; a
y
¼1 and aj i¼ji. Resolution provided by
different probe states is compared for the same mean
number of photons
n that represents the energy resources
available for the measurement. We examine the following
proposition.
Proposition.—A probe state providing larger resolu-
tion than coherent states j
i with the same mean number
of photons
n is nonclassical, where
n ¼h
ja
y
aj
i¼j
j
2
¼ trða
y
aÞ: (1)
A customary signature of nonclassical behavior is the fail-
ure of the Glauber-Sudarshan PðÞ phase-space represen-
tation to exhibit all the properties of a classical probability
density [1]. This occurs when PðÞ takes negative values,
or when it becomes more singular than a delta function. To
test the proposition we must specify how resolution is
assessed.
Resolution.—In a detection scheme the signal to be
detected is encoded in the input probe state by a
transformation !
. For definiteness, we focus on the
most common and practical case of unitary transformations
with constant generator G independent of the parameter
¼ expðiGÞ expðiGÞ: (2)
The value of is inferred from the outcomes of measure-
ments performed on
. The ultimate resolution of such
inference is given by the quantum Fisher information
F
Q
ð
Þ since the variance of any unbiased estimator
~
is
bounded from below in the form [11,12]
ð
~
Þ
2
1
NF
Q
ð
Þ
; (3)
where N is the number of independent repetitions of the
measurement.
Better resolution is equivalent to larger quantum Fisher
information, which can be expressed as [12,13]
F
Q
ð
Þ¼2
X
j;k
ðr
j
r
k
Þ
2
r
j
þ r
k
jhr
j
jGjr
k
ij
2
; (4)
where jr
j
iare the eigenvectors of with eigenvalues r
j
and
the sum includes all the cases with r
j
þ r
k
0. So for
uniparametric unitary transformations F
Q
is independent
of [13].
In order to reach ultimate sensitivity predicted by the
quantum Fisher information, an optimum measurement
and an efficient estimator are required [12]. If we consider
the maximum likelihood as estimator, the number of repe-
titions required to reach the efficient regime may depend
on the probe state [14]. In order to focus on the intrinsic
capabilities of different schemes, we will assume that N is
large enough so that optimum conditions are reached for all
cases, so that schemes are compared by comparing their
quantum Fisher information. Note also that resolution
depends also on the duration of the measurement.
Because of this any meaningful comparison between dif-
ferent schemes should be done on equal-time basis.
Let us show three useful properties of the quantum
Fisher information. (i) For pure states, such as coherent
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states ji, the quantum Fisher information becomes pro-
portional to the variance of the generator [12]
F
Q
ðji;GÞ¼4ð
GÞ
2
¼ 4ðhjG
2
jihjGji
2
Þ: (5)
(ii) The quantum Fisher information is convex. For a proof
based on the monotonocity of quantum Fisher information
under complete positive maps, see Ref. [15]. A much
simpler proof is given by a straightforward use of the
convexity of the Fisher information and the Braunstein-
Caves inequality [12]. Thus, for classical states
class
¼
Z
d
2
P
class
ðÞjihj; (6)
where P
class
ðÞis a non-negative function no more singular
than a delta function, convexity implies the following
bound for the quantum Fisher information of classical
states:
F
Q
ð
class
;GÞ
Z
d
2
P
class
ðÞF
Q
ðji;GÞ
¼ 4
Z
d
2
P
class
ðÞð
GÞ
2
: (7)
(iii) In most cases it is rather difficult to compute analyti-
cally F
Q
ð; GÞ, especially in infinite dimensional systems.
A similar but simpler performance measure is
2
ð; GÞ¼trð
2
G
2
ÞtrðGGÞ (8)
or, equivalently, in the same conditions of Eq. (4),
2
ð; GÞ¼
1
2
X
j;k
ðr
j
r
k
Þ
2
jhr
k
jGjr
j
ij
2
; (9)
which for pure states such as coherent states also becomes
the variance of the generator ðji;GÞ¼
G [16]. This
is derived from the Hilbert-Schmidt distance between
and in the same terms in which the quantum Fisher
information is derived from the Bures distance [12,17].
The useful point here is that from Eqs. (4) and (9) and given
that r
k
þ r
l
1 it holds that
F
Q
ð; GÞ4
2
ð; GÞ; (10)
the equality being reached for pure states.
Nonclassicality from quantum Fisher information.—For
the sake of convenience let us express the variance of G on
coherent states as a mean value
ð
GÞ
2
¼hjA
G
ji;A
G
¼ G
2
:G
2
:; (11)
where :: denotes normal order, and G in :G
2
: must be
expressed in its normally ordered form so that
hj:G
2
:ji¼hjGji
2
. A key point is that hjA
G
jigives
the quantum Fisher information of coherent states,
F
Q
ðji;GÞ¼4hjA
G
ji; (12)
so that the bound (7) for the quantum Fisher information of
classical states reads
F
Q
ð
class
;GÞ4
Z
d
2
P
class
ðÞhjA
G
ji
¼ 4trð
class
A
G
Þ: (13)
This relation is derived from the convexity of F
Q
ð; GÞ,so
it relies entirely on the classical nature of P
class
ðÞ.
Therefore its violation provides the following nonclassi-
cality criterion:
F
Q
ð; GÞ > 4trðA
G
Þ! is nonclassical: (14)
Since this criterion is formulated in terms of the quan-
tum Fisher information, it will be useful to discuss the
interplay between improved metrological resolution and
nonclassicality. The key point is to link trð A
G
Þ in the
nonclassical criterion (14) with the quantum Fisher infor-
mation of coherent states with the same mean number of
photons F
Q
ðj
i;GÞ¼4h
jA
G
j
i. This is straightfor-
ward when A
G
/ a
y
a. To study this in detail let us split the
analysis in linear and nonlinear schemes.
Linear schemes.—By linear schemes we mean that the
signal is encoded via input-output transformations where
the output complex amplitudes are linear functions of the
input ones and their conjugates. Their generators are poly-
nomials of a, a
y
up to second order, embracing all tradi-
tional interferometric techniques exemplified by the phase
shifts generated by the photon-number operator
G ¼ A
G
¼ a
y
a; (15)
so that G and A
G
coincide. In this case the resolution
(quantum Fisher information) provided by coherent probe
states is given by its mean number of photons
F
Q
ðj
i;a
y
aÞ¼4h
ja
y
aj
i¼4j
j
2
¼ 4trða
y
aÞ;
(16)
where we have used Eqs. (1), (12), and (15). The probe
states providing larger resolution than coherent states
j
i with the same mean number of photons satisfy
F
Q
ð; a
y
aÞ >F
Q
ðj
i;a
y
aÞ¼4trða
y
aÞ; (17)
so that from the nonclassical criterion (14) they are neces-
sarily nonclassical states and the proposition being tested is
true.
This result also holds for other generators of linear
transformations such as G ¼ a expðiÞþa
y
expðiÞ,
which generates displacements of the quadratures, and
G ¼ a
2
expðiÞþa
y2
expðiÞ, which generates quadra-
ture squeezing, where is an arbitrary phase [18]. This is
because A
G
¼ 1 and A
G
¼ 4a
y
a þ 2, respectively, so that
4trðA
G
Þ¼F
Q
ðj
i;GÞ.
This also holds for two-mode SU(2) generators
G ¼ u J;A
G
¼ a
y
1
a
1
þ a
y
2
a
2
; (18)
where u is a three-dimensional unit real vector and J are
the bosonic realization of the angular momentum operators
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that generate the SU(2) group
J
x
¼ a
y
1
a
2
þ a
1
a
y
2
;J
y
¼ iða
y
1
a
2
a
1
a
y
2
Þ;
J
z
¼ a
y
1
a
1
a
y
2
a
2
:
(19)
This describes all two-beam lossless optical devices,
such as beam splitters, phase plates, and two-beam inter-
ferometers. In this two-mode context the coherent states
ji refer to the product of single-mode coherent states
ji¼j
1
ij
2
iwith mean number of photons
n ¼j
1
j
2
þ
j
2
j
2
¼hjA
G
ji. For a simple derivation of A
G
in
Eq. (18), note that any u J is in normal order, normal
order commutes with SU(2) transformations, u J is
SU(2) equivalent to J
z
, with J
2
z
:J
2
z
:
¼ a
y
1
a
1
þ a
y
2
a
2
,
and a
y
1
a
1
þ a
y
2
a
2
is SU(2) invariant. This is A
UGU
y
¼
UA
G
U
y
if G is in normal order and U is a SU(2) unitary
transformation.
When the angular momentum J refers collectively to a
system of qubits, it has been demonstrated [19] that im-
proved resolution beyond coherent states implies entangle-
ment between qubits. We recover this result by noticing
that spin nonclassicality is equivalent to entanglement [20].
This equivalence no longer holds when entanglement re-
fers to the entanglement between field modes; this is to say
that nonclassical factorized states j
c
1
ij
c
2
i, where j
c
j
i is
in mode a
j
, can provide better resolution than coherent
states.
Nonlinear schemes.—By nonlinear detection schemes
we mean that the signal is encoded via input-output trans-
formations where the output complex amplitudes are not
linear functions of the input ones. A suitable example is
given by
G ¼ða
y
aÞ
2
;A
G
¼ 4a
y3
a
3
þ 6a
y2
a
2
þ a
y
a; (20)
and the key point is that A
G
is no longer proportional to the
number operator. In practical quantum-optical terms this
corresponds to light propagation through nonlinear Kerr
media [1].
Next we show that there are classical states that provide
larger resolution than coherent states with the same mean
number of photons, so that the proposition being tested is
false. To this end let us consider the mixed probe state
class
¼ pj=
ffiffiffiffi
p
p
ih=
ffiffiffiffi
p
p
jþð1 pÞj0ih0j; (21)
where j=
ffiffiffiffi
p
p
i is a coherent state, j0i is the vacuum, and
1 >p>0. The state
class
has the same mean number of
photons as the coherent state ji for every p.
Since in general F
Q
ð
class
;GÞ is difficult to compute
when
class
is mixed, we resort to Eq. (10) so that if
4
2
ð
class
;GÞ >F
Q
ðji;GÞ; (22)
then F
Q
ð
class
;GÞ >F
Q
ðji;GÞ and
class
provides larger
resolution than ji. Using Eq. (8) the condition (22)is
equivalent to the following relation between variances of G
in coherent states
p
2
ð
=
ffiffiffi
p
p
GÞ
2
> ð
GÞ
2
; (23)
where we have used j0i as an eigenstate of G with null
eigenvalue. After Eqs. (11) and (20)
ð
GÞ
2
¼ 4jj
6
þ 6jj
4
þjj
2
; (24)
and from Eq. (23) the state
class
provides larger resolution
than ji provided that jj
2
>
ffiffiffiffi
p
p
=2, which can be easily
fulfilled.
We are able to observe this improvement even with a
very simple and practical measuring scheme such as ho-
modyne detection illustrated in Fig. 1. For that we evaluate
the Fisher information F
C
ð
class
;GÞ of the measurement
for
class
in Eq. (21),
F
C
ð
class
;GÞ¼
Z
dx
1
PðxjÞ
@PðxjÞ
@
2
; (25)
where PðxjÞ¼hxj
jxi is the probability of the outcome
x of the X quadrature, with X ¼ a
y
þ a and Xjxi¼xjxi.
We consider very small so that the classical Fisher
information is evaluated at ¼ 0. We also assume an
optimum value for the phase of the coherent amplitude
¼ i
ffiffiffi
n
p
. Using the results in Ref. [6] we get for large
n
F
C
ð
class
;GÞ¼16
n
3
p
¼
1
p
F
C
ðji;GÞ: (26)
Thus, the Fisher information for the classical probe state
class
is above the value for the coherent states with the
same mean number of photons ji, especially when p !0.
Discussion.—To some extent this may be regarded as a
paradoxical result, especially in the limit p ! 0 where
class
tends to be the vacuum, h0j
class
j0i!1, since the
vacuum state is useless for detection. Nevertheless next we
show that this is a fully meaningful and worthy result. To
this end let us consider that we repeat the measurement N
times with the probe
class
in Eq. (21). That will be equiva-
lent to get Np times the result of the probe state j=
ffiffiffiffi
p
p
i
and Nð1 pÞ times the useless vacuum. Therefore the
useful resources are Njj
2
photons distributed in Np
runs of jj
2
=p photons. When the probe is ji (this is
the case p ¼ 1), all runs are useful and we get the same
resources Njj
2
distributed in N runs of jj
2
photons. For
linear detection schemes the two allocations of resources
phase
control
nonlinear
transformation
probe
preparation
local
oscillator
FIG. 1 (color online). Sketch of a homodyne measurement.
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provide essentially the same resolution for every p because
for large number of photons hj0i’0 it holds that
F
Q
ð
class
;a
y
aÞ’pF
Q
ðj=
ffiffiffiffi
p
p
i;a
y
aÞ¼F
Q
ðji;a
y
aÞ.
However, the nonlinearity greatly privileges large pho-
ton numbers so that the best strategy is to put as many
photons as possible in a single run, instead of splitting them
into several runs. More specifically, for large jj it holds
that hj0i’0 and F
Q
ð
class
;GÞ’16jj
6
=p
2
while
F
Q
ðji;GÞ’16jj
6
so that
class
provides much larger
resolution than ji as p ! 0.
Incidentally, the above calculus shows that when
hj0i’0 we get F
C
ð
class
;GÞ’pF
Q
ð
class
;GÞ. This is to
say that whereas both F
C;Q
increase when p decreases, it
holds that F
Q
increases faster than F
C
.
Finally, it might be argued that the improvement of
resolution in nonlinear schemes, and the differences be-
tween different classical input probes just discussed, may
be ascribed to nonclassicality induced by nonlinear trans-
formations. We can rule out this possibility. The quantum
Fisher information does not depend on the value of the
signal, so that the optimum sensitivity cannot depend on
the amount of nonclassicality induced by the transforma-
tion. In particular, for the usual case of small signals the
induced nonclassicalities will be negligible.
Conclusions.—We have obtained a general nonclassical
test derived from quantum Fisher information. For linear
detection schemes this test demonstrates that improved
resolution beyond coherent states is a nonclassical feature.
For nonlinear schemes the situation is different since
mixed classical states can provide better resolution than
coherent states. This result is very attractive since the key
point of classical states is that they are extremely robust
against experimental imperfections [6,8] and they are easy
to generate in labs.
A. R. acknowledges Susana F. Huelga for illuminating
comments and financial support from the EU Integrated
Projects QAP, QESSENCE, and the STREP action
CORNER. A. L. acknowledges support from official
Spanish projects No. FIS2008-01267 and QUITEMAD
S2009-ESP-1594.
*alluis@fis.ucm.es
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