Fidelity of quantum interferometers

Thomas B. Bahder*Weapons Sciences Directorate, Army Aviation and Missile Research, Development, and Engineering Center,

Redstone Arsenal, Alabama 35898-5000, USA

Paul A. LopataArmy Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783-1197, USA

�Received 15 February 2006; revised manuscript received 21 August 2006; published 15 November 2006�

For a generic interferometer, the conditional probability density distribution p�� �m�, for the phase � givenmeasurement outcome m will generally have multiple peaks. Therefore, the phase sensitivity of an interferom-eter cannot be adequately characterized by the standard deviation, such as ���1/�N �the standard limit�, or���1/N �the Heisenberg limit�. We propose an alternative measure of phase sensitivity—the fidelity of aninterferometer—defined as the Shannon mutual information between the phase shift � and the measurementoutcomes m. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnderinterferometer, used as a sensor of a classical field. For the case where there exists no a priori information onthe phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity thanFock state input, for the same photon number.

DOI: 10.1103/PhysRevA.74.051801 PACS number�s�: 42.50.Dv, 07.60.Ly, 03.67.�a

I. INTRODUCTION

The phase sensitivity of interferometers has been a topicof research for many years because of interest in the funda-mental limitations of measurement �1,2�, gravitational-wavedetection �3�, and optical- �4,5�, atom- �6�, and Bose-Einstein-condensate- �BEC-�based gyroscopes �7–9�. Re-cently, applications to sensors are being explored �10,11�.The phase sensitivity of interferometers is believed to belimited by quantum fluctuations �12�, and the phase sensitiv-ity of various interferometers has been explored for differenttypes of input states, such as squeezed states �12,13� andnumber states �14–22�. In all the above cases, the phase sen-sitivity �� has been discussed in terms of two limits, knownas the standard limit ��SL=1/�N and the Heisenberg limit�23�, ��HL= 1/N, where N is the number of particles thatenter the interferometer during each measurement cycle.These arguments are based on results of standard estimationtheory �24� which connects an ensemble of measurementoutcomes mi, i=1,2 , . . . ,M, with corresponding phases �i,through a theoretical relation m=m���. An example of thetheoretical relation associated with some quantum observ-able is m���= �� � m ��, where the state is parametrized by asingle parameter �. Standard estimation theory predicts thatthe standard deviation �� of the probability distribution forthe phase � is related to the standard deviation in themeasurements �m by �24�

�� = dm���d�

−1

�m . �1�

Equation �1� assumes that there is a single peak in the phaseprobability density distribution p���, whose width can be

characterized by the standard deviation ��. In general, aBayesian analysis of measurement outcomes m for an inter-ferometer can lead to a conditional probability density distri-bution for the phase p�� �m� that has multiple peaks. Indeed,multiple peaks have been observed by Pezze and Smerzi�21,22� in the context of interferometry described by angularmomentum algebra �15,19�. Therefore, the standard devia-tion �� is not an adequate metric to characterize the phasesensitivity of an interferometer when multiple peaks arepresent in the phase probability distribution.

In this Rapid Communication, we propose to characterizethe phase sensitivity of an interferometer by an alternativemetric—the fidelity—which is the Shannon mutual informa-tion �25,26� H�� :M� between the phase shift � and the mea-surement outcomes m. To give physical insight into the useof Shannon mutual information to characterize a sensor, weconsider an example of an optical Mach-Zehnder interferom-eter in a sensor configuration �see Fig. 1�. We compare theShannon mutual information H�� :M� for two types of inputstates, Fock states and maximally entangled states �MESs�,which have been of great interest �11,27�.

*Electronic address: thomas.bahder@us.army.mil

medium

Beamsplitter

mirror

Beamsplitter

mirror

L1

L2

Fa

b d

c

φ

FIG. 1. �Color� A Mach-Zehnder interferometer is shown con-sisting of two 50-50 beam splitters and two mirrors. The two inputports and two output ports are shown along with the field F thatinduces the phase shift � in a nonlinear medium.

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II. PHASE SENSITIVITY

A quantum interferometer can act as a sensor of anexternal field F. A quantum state ��in is input into theinterferometer and, through an interaction HamiltonianHI�F�, the state interacts with an external classical field F,leading to a phase-shifted output state ���F� that is param-etrized by the field F. We assume that a single parameter, thephase shift �, is sufficient to describe the physics of theinteraction process. A general description of such a sensorcan then be given in terms of the scattering matrix Sij��� thatconnects the Np input-mode field operators ai to the Np

output-mode field operators bi,

bi = �j=1

Np

Sij���aj , �2�

where i , j=1,2 , . . . ,Np and � is the phase shift of the scat-tered �output� state. The field F leads to a phase shift � of thescattered state, whose detailed relation is determined by theinteraction Hamiltonian HI�F�, which we do not consider anyfurther here. The input state evolves through the interferom-

eter according to the unitary time-evolution operator U���,which relates the input state at t=−� to the output state att= +�,

��out = U�����in . �3�

Measurements are described by a set of positive operator-

valued measure �POVM� operators �E1 , E2 , . . . , ENm , where

each operator Em corresponds to a measurement outcome m.The conditional probability of a given measurement outcomem for a given phase shift � is the expectation value

P�m ���= ��in � Em��� ��in, where the Heisenberg operators

Em��� evolve in time and the states are constant. FromBayes’ rule, we find the conditional probability densityp�� �m� for the phase shift � for a given measurementoutcome m is

p���m� =P�m���p���

�−�

+�

P�m����p����d��

, �4�

where p��� is the a priori probability density for the phaseshift �, over the interval −����. In order to have a goodsensor, the distribution p�� �m� should have a narrow peakthat is centered about some value of the phase, for eachmeasurement outcome m. The phase sensitivity of an inter-ferometer, or quantum interferometric sensor, is usuallytaken to be the width of the single peak of the probabilitydensity p�� �m�.

A careful analysis of the probabilities P�m ��� as func-tions of the scattering matrix Sij��� shows that in general theprobabilities P�m ��� are oscillatory functions of �. Conse-quently, the probability density for the phase p�� �m�will have multiple peaks. The physics responsible for thisis due to the mutual symmetry of the quantum state and the

measuring apparatus �described by operators Em����. Since

the probability density p�� �m� has multiple peaks, thestandard deviation �� is not an adequate measure of theinterferometer’s phase sensitivity.

We propose a new metric for interferometer phasesensitivity—the fidelity—defined as the Shannon mutualinformation between the set of possible phase values �,and the possible measurement outcomes m. For convenience,we discretize the phase shift into values �k=�k /N�, fork= ±1, ±2, ¯ ± ,N�, and consider the mutual informationbetween the 2N�-dimensional alphabet of input phases�k and the Nm-dimensional alphabet of output measurementoutcomes m=0,1 ,2 , . . . ,Nm. In the limit N�→�, theShannon mutual information between the phase shift and themeasurements outcomes m is given by

H��:M� = �m�

−�

+�

d� P�m���p���

log2� P�m���

�−�

+�

P�m����p����d��� . �5�

The mutual information H�� :M� describes the amount ofinformation, on average, that an experimenter gains aboutthe phase � on each use of the interferometer. The mutualinformation depends on the type of input state, on the type ofmeasurement �POVM� performed, and on the a priori prob-ability density p���. The Shannon mutual information can beused by an experimentalist to optimize �28,29� the phasesensitivity of an interferometer by choosing input states thatmaximize H�� :M�.

III. MACH-ZEHNDER SENSOR

As a specific example of the above discussion, weconsider a generic optical Mach-Zehnder interferometer�see Fig. 1�. The interferometer can be characterized by ascattering matrix

Sij��� =1

2�ei�eikL1 − eikL2��z −

i

2�ei�eikL1 + eikL2��x, �6�

where

�x = �0 1

1 0�, �z = �1 0

0 − 1� , �7�

L1 �L2� is the upper �lower� path length through the interfer-ometer, k=� /c, � is the angular frequency of the photons,and c is the speed of light in vacuum. For any input state, theconditional probability for an outcome of observing nc andnd photons in output ports c and d, respectively, for a givenphase shift �, is

P�nc,nd��� = ��in��nc,nd�����in , �8�

where �nc,nd���= �nc ,nd�nc ,nd� and �nc ,nd is the output state

in the Schrödinger picture. For an N-photon Fock state inputin port a, ��in= �Na ,0b, we find �taking L1=L2�

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PN�nc,nd��� =N!

nc ! nd! N,nc+nd

sin2nc��

2�cos2nd��

2� . �9�

Similarly, for a MES input

��MES =1�2

��Na,0b + �0a,Nb� �10�

the conditional probability is

PMES�nc,nd��� =1

2

N!

nc ! nd! N,nc+nd

�sinnc��

2�cosnd��

2���

+ �− 1�ncsinnd��

2�cosnc��

2��2

. �11�

It is clear that the probabilities PN�nc ,nd ��� andPMES�nc ,nd ��� have multiple peaks, and, for the case of no apriori information, so that p���=1/ �2��, the resulting prob-ability densities for the phase p�� �nc ,nd� given by Eq. �4�,also have multiple peaks. For a given N-photon Fock stateinput into port a and vacuum input into port b, the probabil-ity distribution p�� �nc ,nd� has either one or two peaks, de-pending on the measurement outcome. For the N-photonMES input, the probability distribution p�� �nc ,nd� has one,two, three, or four peaks, depending on the measurementoutcome. See Fig. 2 for an example plot of p�� �nc ,nd� forFock state and MES input for measurement outcome�4c ,21d . There is more ambiguity in estimating the phasefrom the phase probability density for MES input than forFock state input, because there are more peaks.

For input states with increasing photon number, the prob-ability densities p�� �nc ,nd� have narrower peaks, but thenumber of peaks remains the same: one or two for Fock stateinput, and one, two, three, or four peaks for MES input.

When the interferometer is used as a sensor, it can bethought of as transmitting information about the phase to theexperimenter via each measurement outcome. As describedabove, due to multiple peaks in the phase distribution, we do

not attempt to use the width of the probability distributionsto describe the quality of this sensor. Instead, we use theShannon mutual information, given in Eq. �5�, as a measureof the fidelity of an interferometric sensor. For the case ofFock state and MES input, and assuming no prior informa-tion, the mutual information H�� :M� is plotted in Fig. 3. Inboth cases, the fidelity of the interferometer, acting like asensor, increases with increasing photon number due to theincreased information carrying capacity of a higher-dimensional output alphabet associated with the N+1 mea-surement outcomes �nc ,nd . However, for the same photonnumber input, and no prior information, the fidelity of theinterferometer is clearly greater for Fock state input than forMES input. This shows how the mutual information is sen-sitive to the number of peaks and not just to the width �� ofa single peak. Clearly, for the case of no prior information,Fock states can carry more information about the phase tothe measurement outcomes than MESs. This striking resultdemonstrates that the use of entangled input states does notlead to improvement over Fock state input �11�.

In the example that we used above, we assumed no priorinformation on the phase probability density distribution,taking p���=1/2�. We have also computed numerically themutual information in Eq. �5� assuming a narrow Gaussiandistribution for p��� for MES and for Fock state input. Forthis case, we find that input of a MES does not improve theinterferometer mutual information over input of Fock states.

In order to optimize the Mach-Zehnder sensor, we canconsider a more general class of states

��in = �n=0

N

cn�na,�N − n�b ,

where cn are complex coefficients. The sensor can be opti-mized by finding the N+1 coefficients that maximize thefidelity H�� :M� subject to the normalization constraint��in ��in=1.

–3 –2 –1 0 1 2 30

0.2

0.4

0.6

0.8

1

φ [radians]

|p

(n c, n

d)φ

FIG. 2. Probability density of the phase p�� �m� for Fock stateinput �solid curve� and MES input �dashed curve� �see Eq. �10��, forN=25 photons for measurement outcome m= �4c ,21d .

2 4 6 8 10 12 14N

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2 4 6 8 10 12 14N

0

0.25

0.5

0.75

1

1.25

1.5

1.75

MES input

Fock state input

H (

Φ :

M)

FIG. 3. �Color� The mutual information H�� :M� is plotted vsphoton number for the case of N-photon Fock state input into port a�blue dots� and for an N-photon MES input �green dots�. Linesconnect successive photon number points.

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IV. SUMMARY

We have considered a generic Mach-Zehnder optical in-terferometer operating as a sensor of a classical field. Usinga Bayesian analysis, and assuming no a priori informationon the phase, we have shown that the conditional probabilitydistribution for the phase shift, p�� �nc ,nd�, has multiplepeaks and is not adequately described by the standarddeviation ��, which has been used in discussion of thestandard limit ���SL�1/�N� and the Heisenberg limit���HL�1/N�.

We proposed an alternative metric—called the fidelity ofthe interferometer—which is the Shannon mutual informa-tion H�� :M� between the phase shift � and the possiblemeasurement outcomes m. For an interferometer used as aquantum sensor, we have shown that the fidelity is a measureof the quality of a sensor to detect external classical fields.

For the case of a generic Mach-Zehnder optical interfer-ometer and no a priori information, we found the surprisingresult that entangled MES input leads to a lower fidelity than

Fock state input, for the same photon number. This result isintuitively understood because there are a larger number ofpeaks �bigger ambiguity in phase� in p�� �nc ,nd� for MESinput than for Fock state input.

The interferometer fidelity that we proposed is applicableto a wide variety of optical and matter-wave interferometers,with arbitrary number of input and output ports. This mea-sure of interferometer fidelity can be used as a metric forquantum interferometric sensors of classical fields, such asgravitational wave sensors, as well as optical gyroscopes andmatter-wave gyroscopes based on BECs.

ACKNOWLEDGMENTS

This work was sponsored by the Disruptive TechnologyOffice �DTO� and the Army Research Office �ARO�. Por-tions of this work were done while T.B.B. was at Army Re-search Laboratory. This research was performed while P.L.held a the National Research Council Research AssociateshipAward at the Army Research Laboratory.

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