17 Entanglement, EPR, and Bell’s inequalities
Quantum systems exhibit correlations (entanglement) thatappear counter-intuitive from the clas-
sical viewpoint. The examination of these questions can be traced back to a 1932 book by von
Neumann, but especially to a 1935 paper by Einstein-Podolsky-Rosen (EPR). Both addressed the
question as to whether quantum mechanics can be a complete theory of physical phenomena or
whether there exists a full theory with extrahidden variables. These investigations remained some-
what philosophical until John Bell (1964) derived concreteand experimentally testable predictions
(Bell’s inequalities) of hidden variables. In the early 1980’s, various experiments on the correlations
of photon systems by Aspect, Gragnier, and Roger have shown experimental agreement with quan-
tum mechanics, and contradictory to hidden variable theory. Since then, however, entanglement
has been explored as a fundamental property of quantum theory, and applied to physical processes,
such as quantum computation. Here, we shall concentrate on the basics.
17.1 Entangled States for two spin 1/2
Consider the spin degrees of freedom of a two-electron system. In fact, the analysis applies to the
tensor productHab=Ha⊗Hbof any two-state systemsaandb, with respective Hilbert spaces
HaandHb, in the notation of the preceding section. For any unit vectorn, we shall denote the
two eigenstates ofn·Swith eigenvalue± ̄h/2 by|n+〉and|n−〉. We use the notationn=x,y,z
for the unit vectors in the direction of the coordinate axes. The tensor product states for the two
electrons will be denoted by
|naα〉⊗|nbβ〉=|naα;nbβ〉 α,β=± (17.1)The state of 0 total spin, or singlet, corresponds to the following combination of the two electron
states,
|Φ〉=1
√
2
(
|z+;z−〉−|z−;z+〉)
(17.2)The state|Φ〉has been expressed here with respect to the basis in whichSzis diagonal. Since|Φ〉
has total spin zero, however, it is invariant under arbitrary rotations of the full system, so that
(Sa+Sb)|Φ〉= 0. As a result, the state|Φ〉may be expressed in the same way in terms of the
eigenstates ofn·Sfor an arbitrary directionn,
|Φ〉=1
√
2
(
|n+;n−〉−|n−;n+〉)
(17.3)Thus, the preparation of the state|Φ〉could have been made in terms of eigenstates along any
directionn. The three remaining states of the two-electron system are the triplet states
|T+〉 = |z+;z+〉
|T^0 〉 =√^1
2
(
|z+;z−〉+|z−;z+〉)|T−〉 = |z−;z−〉 (17.4)