Introduction to Quantum Physics and Information Processing

Mixed States, Open Systems, and the Density Operator 97

• |ψ

1

i =

1

√

2

(|00i + |01i). This is a Bell state and is already in Schmidt

form. It is clearly entangled: it has 2 terms.

• |ψ

2

i =

1

2

(|00i + |01i + |10i + |11i) =

1

√

2

(|0i + |1i)

1

√

2

(|0i + |1i)) has

only one term: it is a product state.

• |ψ

3

i =

1

√

3

(|00i + |01i + |10i). This and more general states are more

tricky. We would need to find the SVD of the matrix of coefficients C

ij

for the state represented in the computational basis.

In our example, C =

1

√

3

"

1 1

1 0

#

, which turns out to be directly diago-

nalizable. So the SVD reduces to diagonalization.

Eigenvalues are λ

1,2

=

1

2

√

3

(1 ±

√

5) and normalized eigenvectors ...