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Fidelity of Quantum Interferometers
Thomas B. Bahder∗ and Paul A. Lopata
U. S. Army Research Laboratory
2800 Powder Mill Road
Adelphi, Maryland, USA 20783-1197
(Dated: February 14, 2006)
For a generic interferometer, the conditional probability density distribution, p(φ|m), for the
phase φ given measurement outcome m, will generally have multiple peaks. Therefore, the phase
sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such
as ∆φ ∼ 1/
√
N (the standard limit), or ∆φ ∼ 1/N (the Heisenberg limit). We propose an alterna-
tive measure of phase sensitivity–the fidelity of an interferometer–defined as the Shannon mutual
information between the phase shift φ and the measurement outcomes m. As an example applica-
tion of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a
sensor of a classical field. We find the surprising result that an entangled N00N state input leads to
a lower fidelity than a Fock state input, for the same photon number.
Introduction. Phase sensitivity of interferometers
has been a topic of research for many years because
of interest in the fundamental limitations of measure-
ment [1, 2], gravitational-wave detection[3], and opti-
cal [4, 5], atom [6], and Bose-Einstein condensate(BEC)-
based gyroscopes[7, 8, 9]. Recently, applications to sen-
sors are being explored[10, 11]. The phase sensitiv-
ity of interferometers is believed to be limited by quan-
tum fluctuations[12], and the phase sensitivity of various
interferometers has been explored for different types of
input states, such as squeezed states[12, 13], and num-
ber states[14, 15, 16, 17, 18, 19, 20, 21, 22]. In all
the above cases, the phase sensitivity ∆φ has been dis-
cussed in terms of two limits, known as the standard
limit, ∆φSL = 1/
√
N , and the Heisenberg limit[23],
∆φHL = 1/N , where N is the number of particles that
enter the interferometer during each measurement cycle.
These arguments are based on results of standard estima-
tion theory[24] which connects an ensemble of measure-
ment outcomes, mi, i = 1, 2, · · · ,M , with corresponding
phases, φi, through a theoretical relation m = m(φ).
An example of the theoretical relation associated with
some quantum observable is m(φ) = 〈φ|m̂|φ〉, where the
state is parameterized by a single parameter φ. Standard
estimation theory predicts that the standard deviation,
∆φ, of the probability distribution for the phase φ, is
related to the standard deviation in the measurements,
∆m, by [24]
∆φ =
∣∣∣∣
dm(φ)
dφ
∣∣∣∣
−1
∆m. (1)
Equation (1) assumes that there is a single peak in the
phase probability density distribution p(φ), whose width
can be characterized by the standard deviation ∆φ. In
general, a Bayesian analysis of measurement outcomes m
for an interferometer can lead to a conditional probability
density distribution for the phase, p(φ|m), that has mul-
tiple peaks. Indeed, multiple peaks have been observed
by Pezze and Smerzi [21, 22] in the context of interfer-
ometry described by angular momentum algebra [15, 19].
Therefore, the standard deviation ∆φ is not an adequate
metric to characterize the phase sensitivity of an inter-
ferometer when multiple peaks are present in the phase
probability distribution.
In this Letter, we propose to characterize the phase
sensitivity of an interferometer by an alternative metric—
the fidelity—which is the Shannon mutual informa-
tion [25, 26], H(Φ:M), between the phase shift φ and
the measurement outcomes, m. As an example, we con-
sider the specific case of an optical Mach-Zehnder inter-
ferometer in a sensor configuration, see Figure 1. We use
an exact Bayesian analysis to compute the conditional
probability density distribution for the phase φ, p(φ|m),
and we find that multiple peaks exist. We compute the
Shannon mutual information, H(Φ:M), for two types of
input states, Fock states and N00N states, which have
been of great interest [11, 27]. We find that the fidelity
associated with the Fock state input is greater than for
N00N state input.
Phase Sensitivity. A quantum interferometer can act
as a sensor of an external field F . A quantum state |Ψin〉
is input into the interferometer and, through an inter-
action Hamiltonian HI(F ), the state interacts with an
external classical field F , leading to a phase-shifted out-
put state |Ψ(F )〉 that is parameterized by the field F .
We assume that a single parameter, the phase shift φ, is
sufficient to describe the physics of the interaction pro-
cess. A general description of such a sensor can then
be given in terms of the scattering matrix, Sij(φ), that
connects the Np input-mode field operators âi to the Np
output-mode field operators b̂i,
b̂i =
Np∑
j=1
Sij(φ) âj , (2)
where i, j = 1, 2, · · · , Np and φ is the phase shift of the
scattered (output) state. The field F leads to a phase
shift φ of the scattered state, whose detailed relation is

2
phase shift
medium
a
b
c
d
Beam
splitter
mirror
Beam
splitter
mirror
L1
L2
FIG. 1: A Mach-Zehnder interferometer is shown consisting
of two 50-50 beam splitters and two mirrors. The two input
ports and two output ports are shown along with a medium
that induces the phase shift φ.
determined by the interaction Hamiltonian HI(F ), which
we do not consider any further here. The input state
evolves through the interferometer according to the uni-
tary time-evolution operator, Û(φ), which relates the
input state at t = −∞ to the output state at t = +∞,
|Ψout〉 = Û(φ) |Ψin〉. (3)
Measurements are described by a set of Pos-
itive Operator-Valued Measure (POVM) operators,
{Ê1, Ê2, · · · , ÊNm}, where each operator Êm corresponds
to a measurement outcome m. The conditional prob-
ability of a given measurement outcome m for a given
phase shift φ, is the expectation value P (m|φ) =
〈Ψin|Êm(φ)|Ψin〉, where the Heisenberg operators Êm(φ)
evolve in time and the states are constant. From Bayes’
rule, we find the conditional probability density, p(φ|m),
for the phase shift φ for a given measurement outcome
m is
p(φ|m) = P (m|φ)∫ +pi
−pi P (m|φ′) dφ′
, (4)
where we have assumed a uniform a priori probability
density for the phase shift φ.
In order to have a good sensor, the distribution p(φ|m)
should have a narrow peak that is centered about some
value of the phase, for each measurement outcome m.
The phase sensitivity of an interferometer, or quantum
interferometric sensor, is usually taken to be the width
of the single peak of the probability density p(φ|m).
A careful analysis of the probabilities P (m|φ) as func-
tions of the scattering matrix Sij(φ) shows that in gen-
eral the probabilities P (m|φ) are oscillatory functions of
φ. Consequently, the probability density for the phase,
p(φ|m), will have multiple peaks. The physics responsible
for this is due to the mutual symmetry of the quantum
state and the measuring apparatus (described by opera-
tors Êm(φ)). Since the probability density p(φ|m) has
multiple peaks, the standard deviation ∆φ is not an ad-
equate measure of the interferometer’s phase sensitivity.
We propose a new metric for interferometer phase
sensitivity–the fidelity–defined as the Shannon mutual in-
formation between the set of possible phase values φ, and
the possible measurement outcomes m. For convenience,
we discretize the phase shift into values φk = pik/Nφ,
for k = ±1,±2, · · ·±, Nφ, and consider the mutual infor-
mation between the 2Nφ dimensional alphabet of input
phases, φk, and the Nm-dimensional alphabet of output
measurement outcomes, m = 0, 1, 2, · · · , Nm. In the limit
Nφ → ∞, the Shannon mutual information between the
phase shift and the measurements outcomes m is given
by
H(Φ:M) =
1
2pi
∑
m
∫ +pi
−pi
dφ P (m|φ) log2
[
2pi P (m|φ)
∫ +pi
−pi P (m|φ′) dφ′
]
, (5)
where we have taken the a priori phase distribu-
tion p(φ) = 1/(2pi) to be uniform over the interval
−pi < φ ≤ pi. The mutual information H(Φ:M) describes
the amount of information, on average, that an experi-
menter gains about the phase φ on each use of the inter-
ferometer. The mutual information depends on the type
of input state and on the type of measurement (POVM)
performed.
Mach-Zehnder Sensor. As a specific example of the
above discussion, we consider a generic optical Mach-
Zehnder interferometer, see Figure 1. The interferometer
can be characterized by a scattering matrix
Sij(φ) =
1
2
(eiφeikL1 − eikL2)σz
− i
2
(eiφeikL1 + eikL2)σx, (6)
where
σx =
(
0 1
1 0
)
, σz =
(
1 0
0 −1
)
, (7)
L1 (L2) is the upper (lower) path length through the
interferometer, k = ω/c, ω is the angular frequency of the

3-3 -2 -1 0 1 2 3f @radiansD
0
0.2
0.4
0.6
0.8
1
p
Hf»
n c
,
n
d
L
FIG. 2: Probability density of the phase p(φ|m) for Fock state
input (solid curve) and N00N state input (dashed curve) for
N = 25 photons for measurement outcome m = {4c, 21d}.
photons and c is the speed of light in vacuum. For any
input state, the conditional probability for an outcome
of observing nc and nd photons in output ports “c” and
“d”, respectively, for a given phase shift φ, is
P (nc, nd | φ) = 〈Ψin |̂nc,nd(φ)|Ψin〉, (8)
where ̂nc,nd(φ) = |nc, nd〉〈nc, nd| and |nc, nd〉 is the out-
put state in the Schro¨dinger picture. For an N -photon
Fock state input in port “a”, |Ψin〉 = |Na, 0b〉, we find
(taking L1 = L2)
PN (nc, nd|φ) =
N !
nc! nd!
δN,nc +nd sin
2nc(
φ
2
) cos2nd(
φ
2
). (9)
Similarly, for a N00N -state input
|ΨN00N〉 =
1√
2
[ |Na, 0b〉+ |0a, Nb〉 ] (10)
the conditional probability is
PN00N (nc, nd|φ) =
1
2
N !
nc! nd!
δN,nc +nd
[
sinnc(
φ
2
) cosnd(
φ
2
) + (−1)nc sinnd(φ
2
) cosnc(
φ
2
)
]2
. (11)
It is clear that the probabilities PN (nc, nd|φ) and
PN00N (nc, nd|φ), have multiple peaks, and therefore the
resulting probability densities for the phase, p(φ|nc, nd)
given by Eq.(4), also have multiple peaks. For a given N -
photon Fock state input into port “a” and vacuum input
into port “b”, the probability distribution p(φ|nc, nd) has
either one or two peaks, depending on the measurement
outcome. For the N -photon (entangled) N00N state in-
put, the probability distribution p(φ|nc, nd) has one, two,
three, or four peaks, depending on the measurement out-
come. See Figure 2 for an example plot of p(φ|nc, nd) for
Fock state and N00N state input for measurement out-
come {4c, 21d}. There is more ambiguity in estimating
the phase from the phase probability density for N00N
state input than Fock state input, because there are more
peaks.
For input states with increasing photon number, the
probability densities, p(φ|nc, nd), have narrower peaks,
but the number of peaks remains the same: one or two
for Fock state input, and one, two, three, or four peaks
for N00N state input.
When the interferometer is used as a sensor, it can be
thought of as transmitting information about the phase
to the experimenter via each measurement outcome. As
described above, due to multiple peaks in the phase dis-
tribution, we do not attempt to use the width of the prob-
ability distributions to describe the quality of this sensor.
Instead, we use the Shannon mutual information, given in
Eq.(5), as a measure of the fidelity of an interferometric
sensor. For the case of Fock state and N00N state input,
the mutual information, H(Φ:M), is plotted in Figure 3.
In both cases, the fidelity of the interferometer, acting
like a sensor, increases with increasing photon number
due to the increased information carrying capacity of a
higher-dimensional output alphabet associated with the
N +1 measurement outcomes {nc, nd}. However, for the
same photon number input, the fidelity of the interfer-
ometer is clearly greater for Fock state input than for
N00N state input. This shows how the mutual infor-
mation is sensitive to the number of peaks and not just
the width ∆φ of a single peak. Clearly, Fock states can
carry more information about the phase to the measure-
ment outcomes than N00N states. This striking result
demonstrates that the use of entangled input states does
not lead to improvement over Fock state input[11].
In order to optimize the Mach-Zehnder sensor, we
can consider a more general class of states |Ψin〉 =
N∑
n=0
cn |na, (N −n)b 〉, where cn are complex coefficients.
The sensor can be optimized by finding the N + 1 coef-
ficients that maximize the fidelity, H(Φ:M), subject to
the normalization constraint 〈Ψin|Ψin〉 = 1.
Work is in progress to analyze the mutual information
for repeating the experiment N times for arbitrary input

42 4 6 8 10 12 14
N
0
0.25
0.5
0.75
1
1.25
1.5
1.75
H
HF:M
L
NOON state input
Fock state input
H
HF:M
L
FIG. 3: The mutual information H(Φ:M) is plotted vs. pho-
ton number for the case of N-photon Fock state input into
port “a” (blue dots) and for an N-photon N00N state input
(green dots). Lines connect successive photon number points.
states. An interesting example is the case when one-
photon is input into port “a” and vacuum is input into
port “b”. When the experiment is repeated N times,
with M0 outcomes {0c, 1d} and M1 outcomes {1c, 0d},
where N = M0 +M1 , the mutual information H(Φ:M)
vs. N is identical to that of Fock state input with the
same N .
Summary. We have considered a generic Mach-
Zehnder optical interferometer operating as a sensor of a
classical field. Using a Bayesian analysis, we have shown
that the conditional probability distribution for the phase
shift, p(φ|nc, nd), has multiple peaks and is not ade-
quately described by the standard deviation ∆φ, which
has been used in discussion of the the standard limit
(∆φSL ∼ 1/
√
N) and the Heisenberg limit (∆φHL ∼
1/N) .
We proposed an alternative metric–called the fidelity
of the interferometer–which is the Shannon mutual in-
formation, H(Φ:M), between the phase shift φ and the
possible measurement outcomes m. For an interferome-
ter used as a quantum sensor, we have shown that the
fidelity is a measure of the quality of a sensor to detect
external classical fields.
For the case of a generic Mach-Zehnder optical inter-
ferometer, we found the surprising result that entangled
N00N state input leads to a lower fidelity than Fock
state input, for the same photon number. This result is
intuitively understood because there are a larger number
of peaks (bigger ambiguity in phase) in p(φ|nc, nd) for
N00N state input than for Fock state input.
The interferometer fidelity that we proposed is applica-
ble to a wide variety of optical and matter wave interfer-
ometers, with arbitrary number of input/output ports.
This measure of interferometer fidelity can be used as
a metric for quantum interferometric sensors of classi-
cal fields, such as gravitational wave sensors, as well as
optical gyroscopes and matter-wave gyroscopes based on
BEC.
This work was sponsored by the Disruptive Technol-
ogy Office (DTO) and the Army Research Office (ARO).
This research was performed while P. L. held a National
Research Council Research Associateship Award at the
U. S. Army Research Laboratory.
∗ Electronic address: bahder@arl.army.mil
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