"Exercise 1. Verify equation (2)."

sigmax = ((0,1),(1,0))
sigmay = ((0,-i),(i,0))
sigmaz = ((1,0),(0,-1))

A0 = sigmaz
A1 = sigmax

B0 = -(sigmax + sigmaz) / sqrt(2)
B1 = (sigmax - sigmaz) / sqrt(2)

uu = (1,0,0,0)
ud = (0,1,0,0)
du = (0,0,1,0)
dd = (0,0,0,1)

s = (ud - du) / sqrt(2)

S = dot(conj(s),kronecker(A0,B0),s) +
    dot(conj(s),kronecker(A0,B1),s) +
    dot(conj(s),kronecker(A1,B0),s) -
    dot(conj(s),kronecker(A1,B1),s)

check(S == 2 sqrt(2))

"ok"

"Exercise 2. Verify spin expectation values."

I = ((1,0),(0,1))

check(dot(conj(s),kronecker(A0,I),s) == 0)
check(dot(conj(s),kronecker(A1,I),s) == 0)

check(dot(conj(s),kronecker(I,B0),s) == 0)
check(dot(conj(s),kronecker(I,B1),s) == 0)

"ok"

"Exercise 3. Verify that A and B are correlated for all entangled states."

s1 = (uu + dd) / sqrt(2)
s2 = (uu - dd) / sqrt(2)
s3 = (ud + du) / sqrt(2)

S1 = dot(conj(s1),kronecker(A0,B0),s1) +
     dot(conj(s1),kronecker(A0,B1),s1) +
     dot(conj(s1),kronecker(A1,B0),s1) -
     dot(conj(s1),kronecker(A1,B1),s1)

S2 = dot(conj(s2),kronecker(A0,B0),s2) +
     dot(conj(s2),kronecker(A0,B1),s2) -
     dot(conj(s2),kronecker(A1,B0),s2) +
     dot(conj(s2),kronecker(A1,B1),s2)

S3 = dot(conj(s3),kronecker(A0,B0),s3) +
     dot(conj(s3),kronecker(A0,B1),s3) -
     dot(conj(s3),kronecker(A1,B0),s3) +
     dot(conj(s3),kronecker(A1,B1),s3)

check(abs(S1) > 2)
check(abs(S2) > 2)
check(abs(S3) > 2)

"ok"