17.7 Entanglement in the EPR paradox
Einstein Podolsky and Rosen (EPR) proposed agedankenexperiment in 1935 the outcome of which
they viewed as a paradox, namely a contradiction between theprinciples of causality in relativity
and the principles of quantum mechanics.
The set-up (as re-interpreted by David Bohm) is as follows. Consider a spin 0 particle which
decays into two stable particlesaandb. To make this interesting, we consider a case where the
particlesaandbhave both spin 1 (typically photons) or spin 1/2 (electrons). Physical examples
are
π^0 → γ+γ
π^0 → e++e− (17.33)The first is the dominant decay mode of theπ^0 , whose life-time is 10−^16 s. The branching ratio for
the second decay is 10−^7. Let us concentrate on the example ofπ^0 →e++e−, so that the two
subsystems both correspond to spin 1/2.
Because theπ^0 has zero spin, and angular momentum is conserved, the spins of the electron
and positron must be opposite. This means that the electron/positron state produced by the decay
ofπ^0 must be
|Φ〉=1
√
2
(
|z+;z−〉−|z−;z+〉)
=1
√
2
(
|n+;n−〉−|n−;n+〉)
(17.34)following the notations of the preceding sections. Here, weignore all extra quantum numbers the
particles carry, such as energy, momentum, and electric charge.
Assuming that thee+ande−can travel a long distance without any interaction with other
particles or with electric and magnetic fields, this correlation between the spins ofe+ande−will
persist, and the quantum state of the spins will remain|Φ〉. We now imagine two observersAand
Bfar away in opposite directions from where theπ^0 decays. These observers can measure the spins
of thee+ande−.
e+ e−
A <−−−−−− π^0 −−−−−−> B (17.35)Assume thatAmeasures the spin of the electron along theSazdirection. The probability forA
to measure + ̄h/2 is 50%, and the probability forAto measure− ̄h/2 is also 50%. But now, if
Ameasures spin + ̄h/2 in the Saz direction, thenAknows for sure (with probability 1) that, if
Bmeasures spin in theSbzdirection as well,Bmust necessarily find− ̄h/2, since these spins are
opposite in the state|Φ〉. The same would be true ifAandBmeasured spin along any other
directionn.
At first sight, one may conclude that because of these correlations, there must be some infor-
mation that can travel fromBtoAinstantaneously. This would appear to be so because as soon
asAhas measured spinSza= + ̄h/2, then the outcome of the measurement thatB can perform
will be known toAinstantaneously. This issue, however, is not where the EPR paradox lies. After