7.3 Entangled pure states
Pure bi-partite states can be put into equivalence classes labeled by the Schmidt
numbers, [1], leading to a simple geometric description.
Write the bipartite pure state inCN^1 ⊗CN^2 , (N 1 ≤N 2 ), in the Schmidt de-
composition [1],
|ψ〉=N∑ 1 − 1
j=0√
pj|φj〉⊗|χj〉, 〈φj|φk〉=〈χj|χk〉=δjk (7.2)withpj≥0 probabilities. The simplex
1 ≥p 0 ≥···≥pN 1 − 1 > 0 ,∑
pj= 1 (7.3)has the pure product state as the extreme point
(1, 0 ,...,0) (7.4)All other points of the simplex represent entangled states. The extreme point
1
N 1(1, 1 ,...,1) (7.5)
is the maximally entangled state. Most pure states are entangled. (In contrast to
the density matrix perspective, where by Eq. (1.11), the separable states are of
full dimension.)
The maximally entangled state is, (N 1 =M to simplify the notation):
|β〉=1
√
M
M∑− 1
j=0|j〉⊗|j〉 (7.6)LetσμbeM^2 hermitian and mutually orthogonalM×Mmatrices i.e.
σμ=σμ∗, Trσμσν=Mδμν, μ∈ 0 ,...,M^2 − 1 (7.7)