A loophole in Cabello's proof to Bell's theorem without inequalities

2 Pages·0.06 MB·English

by  W. LiMing

A loophole in Cabello’s proof to Bell’s theorem without inequalities
∗
W. LiMing and Z. L. Tang
Department of Physics, South China Normal University, Guangzhou 510631,China.
C. Liao
School for Information and Optoelectronic Science and Engineering,
South China Normal University, Guangzhou 510631, China
(Dated: February 1, 2008)
AloopholeintheproofsofBell’s theoremwithoutinequalitiesispointedout. Theassumption in
theseproofs to EPR’s physical reality does not fit EPR’s original arguments.
PACSnumbers: 03.65.Ud,03.67.-a,03.65.Ta
3
0
0 Thenonlocalityofquantummechanicshasbeenproved The loophole comes from the fact that (3-6) do not
2 withoutusinginequalitiesbyafewauthors[1,2,3]. How- work together with (7). Consider (3) and (7). From (3),
n ever, there exists a common loophole in these proofs. Alice predicts B2 = B4 if she obtained A1A3 = 1 by
a Take Cabello’s proof as an example. In his proof[1], Ca- measuring σz1σz3 on her particles (1,3). However, this
J bello considered two pairs of maximally entangled parti- predictionwon’tbe agreedby Bob,becauseBobhadob-

4 cles, (1,2) and (3,4), tainedB2b4,b2B4bymeasuringσz2σx4 andσx2σz4 onhis
2 particles (2,4). Since [σz2,σx2σz4]=0, [σz4,σz2σx4]=0,
2 |Ψi1234 =|Ψ−i12⊗|Ψ−i34, (1) Bobhasnoideaforthevaluesofσz62 andσz4. Especi6ally,
it is impossible to test Alice’s prediction after Bob has
v
7 where measuredσz2σx4 andσx2σz4. Alice’spredictionhasbeen
6 1 disturbedbyBob’smeasurement. Therefore,Alice’spre-
−
0 Ψ ij = (0 i 1 j 1 i 0 j). (2) dictionisnottrueforthejointmeasurementofAliceand
| i √2 | i ⊗| i −| i ⊗| i
1 Bob.
0
Two observers, Alice and Bob, at a space-like separa- AccordingtoEPR[4],anelementofphysicalrealityex-
3
0 tion have access to particles (1,3) and particles (2,4), re- istsonlywhenaquantitycanbepredictedwithcertainty
/ spectively. From this state one can predict the values of and without disturbing the system. In Alice and Bob’s
h
measurementonσx and σz ofthe four particles with the joint measurement they in fact cannot predict anything
p
following probabilities consistently, or Alice’s prediction for particles (2,4) has
-
t been disturbed by Bob’s measurement, thus elements of
n
a P(B2 =B4|A1A3 =+1) = 1, (3) physical reality in the sense of EPR cannot be assumed.
u P(b2 =b4 a1a3 =+1) = 1, (4) Therefore, Cabello’s final contradiction does not make
|
q P(A1 =a3 B2b4 =+1) = 1, (5) sense. This is the loophole.
: |
v P(a1 = A3 b2B4 = 1) = 1, (6) If a contradiction for the elements of physical reality
i − | −
X P(A1A3 =+1,a1a3 =+1 , of B2,B4,b2,b4 from (3,4) and
ar B2b4 =+1,b2B4 =−1) = 1/8, (7)
P(A1A3 =+1,a1a3 =+1)=1/4, (8)
wherea,AarethevaluesmeasuredbyAliceonσ andσ ,
x z
respectively,b,BarethosebyBob,andsubscriptsdenote
particles. Then Cabello found a contradiction under the could be found, that would be a true disaster of EPR’s
conditionthatEPR’selementsofphysicalreality[4]exist. physical reality.
Sinceσz1σz3,σx1σx3,σz2σx4,σx2σz4 arecommutativeop- The same problem exists between (4) and (7), (5) and
erators, they can be measured simultaneously. Thus Al- (7), and (6) and (7).
ice and Bob make joint measurements on these opera-
ThisloopholealsoexistsinGHZnonlocality[3],Hardy’s
tors. Cabello found, for a run of measurement which
argument[2].
gives A1A3 = +1,a1a3 = +1,B2b4 = +1,b2B4 = 1,
−
the elements of physical reality of σ and σ of the four ThisworkwassupportedbytheNationalFundamental
x z
particles contradict each other. Research Programunder Grant No 2001CD309300.
∗Electronicaddress: [email protected]
[1] A. Cabello, Phys.Rev.Lett. 86, 1911(2001). [2] L. Hardy,Phys.Rev.Lett. 71, 1665(1993).
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[3] D.M.Greenberger,M.A.Horne,A.shimony,A.Zeilinger, 777(1935).
Am. J. Phys. 58, 1131(1990).
[4] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,