Noncommutative Mathematics for Quantum Systems

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

)