An elementary introduction to the geometry of quantum states with a picture book

The projection on|β〉can be written in terms ofσμas:

|β〉〈β|=

1

M

M∑− 1

jk=0

|jj〉〈kk|

=

1

M^3

M∑− 1

jk=0

M∑^2 − 1

μ,ν=0

〈kk|σμ⊗σνt|jj〉σμ⊗σtν

=

1

M^3

M∑^2 − 1

μ,ν=0

Tr(σμσν)σμ⊗σtν

=

1

M^2

M∑^2 − 1

μ=0

σμ⊗σtμ (7.8)

In the case of qubits andN=M^2 , a complete set of mutually orthogonal projec-
tions on theN maximally entangled states is:

Pα=

1

N

N∑^2 − 1

μ=0

σασμσα⊗σμt, α∈ 0 ,...,N− 1 (7.9)

This is a natural generalization of the Bell basis of two qubits, to 2nqubits.
In the two qubits case, M = 2, an equal mixture of two Bell states, is a
separable state:

Pα+P 0 =

1

4

∑^3

μ=0

(σασμσα+σμ)⊗σtμ

=

1

2

(σ 0 ⊗σ 0 +σα⊗σαt)

=

σ 0 +σα

2

⊗

σ 0 +σtα

2

+

σ 0 −σα

2

⊗

σ 0 −σtα

2

(7.10)

The two terms on the last line are products of one dimensional projections, and
represent together a mixture of pure product states.

7.4 Two types of entangled states

Choosing the basisσαmade with either symmetric real or anti-symmetric imag-
inary matrices, makes partial transposition a reflection in the anti-symmetric co-
ordinates (
σα⊗σβ

)pt

=σα⊗σβt=±σα⊗σβ (7.11)

Partial transposition [27, 1] distinguishes between two types of entangled states: