expressed here with respect to thez-basis.
Basic observables inHaand inHbrespectively are the spinna·Saandnb·Sb. In addition, we
have observables defined only in the full Hilbert spaceHab, such as (na·Sa)⊗(nb·Sb).
To understand the concept of entanglement of two statesaandb, we begin by evaluating various
observables on the state|Φ〉. First,|Φ〉is an eigenstate of the observableSza⊗Sbz,
Sza⊗Sbz|Φ〉=−̄h^2
4|Φ〉 (17.5)
Thus, in the state|Φ〉, the eigenvalues ofSazand ofSbzare always opposite to one another (with
probability 1). Expressed alternatively, the spins of electronsaandbare perfectly correlated with
one another; whenSazis +, thenSzbis−and vice-versa. The same correlation exists in the triplet
states,
Sza⊗Sbz|T^0 〉 = −̄h^2
4|T^0 〉
Sza⊗Sbz|T±〉 = +
̄h^2
4|T±〉 (17.6)
The analysis could be repeated with respect to any orientationn.
The key distinction appears when we investigate observables measuring properties of only sub-
systema, but notb(or vice-versa). Take for example the operatorSaz =Sza⊗Ib. From the
construction of the states|T±〉, it is immediate that
Sza|T±〉=±
̄h
2|T±〉 (17.7)
When observed from the point of view of subsystemaalone, the states|T±〉behaves as pure states,
as if subsystembwere absent. These states are examples ofnon-entangled states(we shall present
a general definition of such states later on). On the other hand, applying the operatorSzato the
states|Φ〉and|T^0 〉, we find,
Saz|Φ〉 =̄h
2|T^0 〉
Saz|T^0 〉 =
̄h
2|Φ〉 (17.8)
Neither|Φ〉, nor|T^0 〉are eigenstates ofSaz. Hence, measurements ofSazin the state|Φ〉will give
both the results + ̄h/2 and− ̄h/2. By computing the expectation value ofSzain the state|Φ〉, we
gain information on the quantum mechanical probabilities with which either eigenvalues± ̄h/2 will
be measured. The expectation values are given by
〈Φ|Saz|Φ〉=〈T^0 |Saz|T^0 〉= 0 (17.9)