Local-realism violations in two-particle interferometry

Xiao-hua Wu,1,* Rui-hua Xie,2,3 Xiao-dong Huang,1 and Yuan-fu Hsia11Department of Physics, Nanjing University, Nanjing, 210008, People’s Republic of China

2National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210008, People’s Republic of China3CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, People’s Republic of China

~Received 20 September 1995!

Using a quantum-optical setting, we demonstrate local-realism violation for a maximally entangled state oftwo particles without using inequalities. Our proof can be generalized to all arbitrary entangled states for twoparticles.

PACS number~s!: 03.65.Bz

Recently, Hardy@1# has shown that it is possible to dem-onstrate nonlocality for two particles without using inequali-ties for any entangled states except maximally entangledstates. Hardy’s proof is generalized by Goldstein@2# in thefollowing sense: while Hardy’s analysis concerns four ob-servables, the choice of each of which is very much con-strained by the quantum state, for Goldstein’s argument thechoice of one of the observables is almost arbitrary. Since amaximally entangled state, such as the singlet state for twospin-12 particles, will violate the Bell’s inequalities and admita nonlocality proof@3#, it is possible that Hardy’s argumentdoes not include the maximally entangled state just becauseof the mathematical difficulty. In this paper, we shall show anonlocality proof for two particles in a maximally entangledstate by using a quantum-optical setting, and our proof canbe run by all arbitrary entangled states for two particles with-out using inequalities.

A schematic of the apparatus for the present Gedankenexperiment is shown in Fig. 1. It represents little changefrom the two-particle interferometry suggested by Horne andco-workers@4#. An ensemble of particle pairs is emitted by asource into the beamsa, b, c, andd, and each pair in theensemble is in the quantum state

uc&51

A2~ u1&au1&b1u1&cu1&d). ~1!

Particle 1 propagates in the opposite direction of particle2 to make sure that the apparatus for measuring particle 1 isspace separated away from the other one for particle 2. Par-ticle 1 enters beama and is reflected from mirrorMa tophase shifterf1 en route to beam splitter BS1, from which itproceeds either to detectorE or to detectorF, while particle2 enters beamb and is reflected from mirrorMb to phaseshifterf2 en route to beam splitter BS2, from which it pro-ceeds to detectorsG or H. In the other pair of paths, particle1 enters beamc and proceeds either toK via BS3 or to beamc8, from which it proceeds toE or F via BS1, while particle2 enters beamd and proceeds either toL via BS4, or tod8,from which it proceeds toG or H via BS2. The detectors areassumed to have a quantum efficiency of 100%.

First, we shall derive some basic results for particles im-pinging on beam splitters. Ifa is the annihilation operator formodea, then we have the result

un&a5~ a†!n

An!u0&. ~2!

For the beam splitter BS1, the input and output annihilationoperators are related by

*FAX: 086-025-3326028. Electronic address:postphys@njueducn

PHYSICAL REVIEW AATOMIC, MOLECULAR, AND OPTICAL PHYSICS

THIRD SERIES, VOLUME 53, NUMBER 4 APRIL 1996

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S feD 5S AT1 iAR1

iAR1 AT1D S a8

c8 D , ~3!

wherea85exp(2if1)a. Then we have

S exp~2 if1!a

c8 D 5S AT1 2 iAR1

2 iAR1 AT1D S f

eD . ~4!

Similarly, we get

S ck8

D 5S AT3 2 iAR3

2 iAR3 AT3D S k

c8D , ~5!

whereR1 and T1 are the reflectance and transmittance ofBS1, whileR3 andT3 are the reflectance and transmittanceof BS3. Using~4! and ~5!, we have

a†5exp~2 if1!~AT1 f †1 iAR1e†!, ~6!

c†5AT3k†2AR1R3 f†1 iAT1R3e

†. ~7!

By an analogous argument, we can also have the followingresults for particle 2:

b†5exp~2 if2!~AT2h†1 iAR2g†!, ~8!

d†5AT4l †2AR2R4h†1 iAT2R4g

†, ~9!

whereR2 and T2 are the reflectance and transmittance ofBS2, whileR4 andT4 are the reflectance and transmittanceof BS4.

The quantum state can be written in such a form:

uc&51

A2~ a†b†1 c†d†!u0&. ~10!

Using Eqs.~6!–~9!, we get

uc&5uc0&1uc1&, ~11!

where

uc0&51

A2~AT3T4

k†l †2AR2T3R4k†h†1 iAT2T3R4k

†g†

2AR1R3T4 f†l †1 iAT1R3T4e

†l †!u0&, ~12!

uc1&51

A2„$exp@2 i ~f11f2!#AT1T21bAR1R2% f

†h†

1 i ~exp@2 i ~f11f2!#AT1R22bAR1T2! f†g†

1 i ~exp$2 i ~f11f2!%AR1T22bAT1R2!e†h†

2$exp@2 i ~f11f2!#AR1R21bAT1T2%e†g†…u0&,

~13!

with b5AR3R4 andb,1. The stateuc0& contains the termsin which at least one of the detectorsE andF is fired. Weshall only be interested in those runs of the experiment forK5L50, which means that particle 1 does not go to endk,while at the same time particle 2 does not go to endl. There-fore, we need not pay any special attention to the evolutionof the stateuc0&. Now, we shall give a nonlocality argumentbased on Eqs.~13!, which keepsK5L50. We choose

R15R251

11b, ~14!

T15T25b

11b. ~15!

Settingf15f25p/2, Eq. ~13! can be written as

uc1&521

A2@•••#~AT1R11bAR1T2!u1& f u1&g1 i ~AR1T2

1bAT1R2!u1&eu1&h2~AR1R22bAT1T2!u1&eu1&g .

~16!

Let F(T1 ,T2 ,f15p/2, f25p/2) denote the value of detec-tor F conditioned on the experiment settings: the transmit-tance of BS1 isT1 ~while R1512T1! and the transmittance

FIG. 1. An arrangement for two-particle interferometry with variable phase shifters. The sourceSemits two particles 1 and 2 into fourbeamsa, b, c, andd. The particles are registered in detectorsE, F, G, F, K, andL. The state of the pair is assumed to be given by Eqs.~1!,which is a superposition of two amplitudes:~I! particle 1 in beama and particle 2 in beamb and~II ! particle 1 in beamc and particle 2 inbeamd. The two beamsa andc8 of particle 1 are given a variable relative phase shiftf1 before recombination near the BS1. Likewise, thetwo beamsb andd8 of particle 2 are given a variable phase shiftf2 before recombination near BS2.

R1928 53WU, XIE, HUANG, AND HSIA

of BS2 isT2; the phase shifterf1 is set top/2 andf2 is setto p/2. Since there is nou1& f u1&h term in Eqs.~13!, we canget

~FH!~T1,T2,f15p/2, f25p/2!50, ~17!

where~FH! represents the probability of the appearance ofFandH. Then, we setf15p/2, f253p/2 and substituteT2with T28 , which satisfies

T285b3

11b3 , ~18!

R2851

11b3 . ~19!

Then we obtain

uc1&51

A2@~AT1T281bAR1R28!u1& f u1&h1 i ~AT1R28

2bAR1T28!u1& f u1&g2~AR1R281bAT1T28!u1&eu1&g].

~20!

Using it, we have the following quantum prediction:If

H~T1,T28 ,f15p/2, f253p/2!51,

thenF~T1,T28 ,f15p/2, f253p/2!51, ~21!

since there is only one termu1& f u1&h containingu1&h . Settingf153p/2, f25p/2 and choosing the transmittance of BS1to T18 and the transmittance of BS2 toT2,

T285b3

11b3 , ~22!

R1851

11b3 , ~23!

we arrive at another form of the quantum state

uc1&51

A2@•••#~AT18T21bAR18R2!u1& f u1&h1 i ~AR18T2

2bAT18R2!u1&eu1&h2~AR18R21bAT18T2!u1&eu1&g ,

~24!

and we can also obtain the following.If

F~T18 ,T2,f153p/2, f25p/2!51,

thenH~T18 ,T2,f153p/2, f25p/2!51. ~25!

Finally, letf153p/2,f253p/2 andT18 for BS1, while withT28 for BS2, it can be found that when

uc1&521

A2@~AT18T282bAR18R28!u1& f u1&h1 i ~AT18R28

1bAR18T28!u1& f u1&g1 i ~AR18T281bAT18R28!u1&eu1&h

2~AR18R282bAT18T28!u1&eu1&g], ~26!

we can have the following quantum prediction for the experi-ment:

~FH !~T18 ,T28 ,f153p/2, f253p/2!51

with a nonzero probability1

2 S b32b

11b3 D 2. ~27!

Using predictions~17!, ~21!, ~25!, and ~27!, which areconditioned onK5L50, we can prove that a realistic andlocal interpretation of quantum mechanics is impossible. Thenotion of realism is introduced by assuming that there existsome hidden variablesl that describe the state of each indi-vidual pair of particles. The assumption of locality is that theeffect of the choice of the measurement on particle 1 cannotinfluence the outcome of the measurement on the other par-ticle, which means that, whenl is specified,F should beconditioned on the transmittance of BS1 and the setting ofphase shifterf1, while H should be decided only by thetransmittance of BS2 andf2. We denote these byF(l,T1 ,f1) or F(l,T18 ,f1) and H(l,T2 ,f2) orH(l,T28,f2). Using ~27!, we can getF(l,T18 ,f153p/2)5H(l,T28 ,f253p/2)51 for some values of hidden vari-ablesl. From ~25!, we haveH(l,T2 ,f25p/2)51, sinceF(l,T18 ,f153p/2)51; and F(l,T1 ,f15p/2)51 forH(l,T28 ,f253p/2)51, according to ~21!. Finally, weshould have F(l,T1 ,f15p/2)5H(l,T2 ,f25p/2)51,which contradicts prediction~17!.

Our proof can be generalized to the arbitrary entangledstate for two particles like

uf&5au1&au1&b1r u1&cu1&d , ~28!

wherea and r are real constants, witha21r 251. The statecan be interpreted in the following manner: the pairs of par-ticles enter beamsa and b with probability a2, while theprobability of the pairs entering beamsc andd is r 2. uc1& canbe written in such a form:

uc1&5aF S exp@2 i ~f11f2!#AT1T21r

abAR1R2D f †h†1 i S exp@2 i ~f11f2!#AT1R22

r

abAR1T2D f †g†

1 i S exp@2 i ~f11f2!#AR1T22r

abAT1R2D e†h†2S exp@2 i ~f11f2!#AR1R21

r

abAT1T2D e†g†G u0&, ~29!

and a nonlocality proof can be given by following the argument we have presented. However, what we want to emphasize inthis paper is that even the maximally entangled states still permit a nonlocality proof without using inequalities.

This work was supported by the Tianma Foundation.

53 R1929LOCAL-REALISM VIOLATIONS IN TWO-PARTICLE INTERFEROMETRY

@1# L. Hardy, Phys. Rev. Lett.71, 1665~1993!.@2# S. Goldstein, Phys. Rev. Lett.72, 1951~1994!.@3# J. S. Bell, Physics~Long Island City, N.Y.! 1, 195 ~1964!.

@4# M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett.62, 2209~1989!.

R1930 53WU, XIE, HUANG, AND HSIA