2.4 Composite Systems and Entanglement
17
2.4.1 The Schmidt Decomposition and Purifications
In Sect. 2.4 we observed that entanglement detection is not an easy task and that a
pure state is entangled if it cannot be expressed as a product state. A useful tool to
detect entanglement in bipartite pure states is given by the Schmidt decomposition.
Theorem 2.2 (Schmidt decomposition) Let |ψ AB ∈ H A ⊗ H B be the pure state of
a bipartite system. Then there exists an orthonormal basis {|α i }
d A
i=1 of H A and an
orthonormal basis {|β j }
d B
j=1 of H B such that:
|ψ AB =
R
k=1
λ k |α k , β k ,
(2.33)
where λ k are positive real coefficients called Schmidt coefficients and R ≤ min(d A ,
d B ) is the Schmidt rank.
Proof Let {|a i }
d A
i=1 and {|b j }
d B
j=1 two orthonormal bases of H A and H B , respectively.
Then the state |ψ AB can be expressed as:
|ψ AB =
d A ,d B
i, j=1
c i j |a i , b j ,
(2.34)
for some complex coefficients c i j which define the complex matrix C ∈ C
d A ×d B . From
the singular value decomposition of C we obtain: C = U DV , where U ∈ C
d A ×d A and
V ∈ C
d B ×d B are unitary matrices and D ∈ R
d A ×d B is a rectangular diagonal matrix
of non-negative numbers, the singular values of C. By substituting the expression
for C in (2.34) we get:
|ψ AB =
min(d A ,d B )
k=1
d A ,d B
i, j=1
u ik d kk v k j |a i , b j ,
(2.35)
where u ik , d kk and v k j are the matrix elements of U , D and V , respectively. We now
define new basis elements |α k =
d A
i=1 u ik |a i and |β k =
d B
j=1 v k j |b j . The newly
defined bases {α k }
d A
k=1 and {β k }
d B
k=1 are orthonormal since the starting ones were so.
By substituting the bases in (2.35) and by discarding the terms in the sum over k
where d kk = 0, we obtain the claim in (2.33).
Note that the Schmidt coefficients are given by the non-zero singular values of C,
which can be computed as the square roots of the non-zero eigenvalues of CC
† .
17
2.4.1 The Schmidt Decomposition and Purifications
In Sect. 2.4 we observed that entanglement detection is not an easy task and that a
pure state is entangled if it cannot be expressed as a product state. A useful tool to
detect entanglement in bipartite pure states is given by the Schmidt decomposition.
Theorem 2.2 (Schmidt decomposition) Let |ψ AB ∈ H A ⊗ H B be the pure state of
a bipartite system. Then there exists an orthonormal basis {|α i }
d A
i=1 of H A and an
orthonormal basis {|β j }
d B
j=1 of H B such that:
|ψ AB =
R
k=1
λ k |α k , β k ,
(2.33)
where λ k are positive real coefficients called Schmidt coefficients and R ≤ min(d A ,
d B ) is the Schmidt rank.
Proof Let {|a i }
d A
i=1 and {|b j }
d B
j=1 two orthonormal bases of H A and H B , respectively.
Then the state |ψ AB can be expressed as:
|ψ AB =
d A ,d B
i, j=1
c i j |a i , b j ,
(2.34)
for some complex coefficients c i j which define the complex matrix C ∈ C
d A ×d B . From
the singular value decomposition of C we obtain: C = U DV , where U ∈ C
d A ×d A and
V ∈ C
d B ×d B are unitary matrices and D ∈ R
d A ×d B is a rectangular diagonal matrix
of non-negative numbers, the singular values of C. By substituting the expression
for C in (2.34) we get:
|ψ AB =
min(d A ,d B )
k=1
d A ,d B
i, j=1
u ik d kk v k j |a i , b j ,
(2.35)
where u ik , d kk and v k j are the matrix elements of U , D and V , respectively. We now
define new basis elements |α k =
d A
i=1 u ik |a i and |β k =
d B
j=1 v k j |b j . The newly
defined bases {α k }
d A
k=1 and {β k }
d B
k=1 are orthonormal since the starting ones were so.
By substituting the bases in (2.35) and by discarding the terms in the sum over k
where d kk = 0, we obtain the claim in (2.33).
Note that the Schmidt coefficients are given by the non-zero singular values of C,
which can be computed as the square roots of the non-zero eigenvalues of CC
† .