16 Noncommutative Mathematics for Quantum Systems
Note thatψdoes not depend on the direction that we choose for
our coordinate system, if we take
u(θ,φ) = cos
θ
2
| 0 〉+eiφsin
θ
2
| 1 〉,
u⊥(θ,φ) = sin
θ
2
| 0 〉−eiφcos
θ
2
| 1 〉,
instead of the vectors| 1 〉and 0〉, withθ∈[0,π],φ∈[0, 2π), then
we have
ψ(θ,φ) =
1
√
2
(
|u(θ,φ)〉⊗|u⊥(θ,φ)〉−|u⊥(θ,φ)〉⊗|u(θ,φ)〉
)
=
1
√
2
(
cos
θ
2
sin
θ
2
| 00 〉−eiφcos^2
θ
2
| 01 〉+eiφsin^2
θ
2
| 10 〉
−e^2 iφcos
θ
2
sin
θ
2
| 11 〉−cos
θ
2
sin
θ
2
| 00 〉+eiφcos^2
θ
2
| 10 〉
−eiφsin^2
θ
2
| 01 〉+e^2 iφcos
θ
2
sin
θ
2
| 11 〉
)
=
−eiφ
√
2
(
| 01 〉−| 10 〉
)
=−eiφψ, (1.3.1)
that is, the vectorψ(θ,φ)implements the same state asψ,
We suppose that each of two physicists, called Alice and Bob,
receives one of the two particles. Let
SA(θ,φ) =S(θ,φ)⊗id and SB(θ,φ) =id⊗S(θ,φ)
where S(θ,φ) denotes the observable corresponding to a spin
measurement on an electron defined in Example 1.2.7. Then
SA(θ,φ)corresponds to a spin measurement on Alice’ particle, and
SB(θ,φ)corresponds to a spin measurement on Bob’s particle. The
spectral decompositions of these observables are given by
SA(θ,φ) = |u〉〈u|⊗id−|u⊥〉〈u⊥|⊗id,
SB(θ,φ) = id⊗|u〉〈u|−id⊗|u⊥〉〈u⊥|,
and it is straight forward to check that we have
P
(
SA(θ,φ) =± 1
)
=
1
2
=P
(
SB(θ,φ) =± 1
)