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

|β〉〈β|

ρpt

ρ

WS

Figure 15: The figure shows the maximally entangled state |β〉〈β|on the unit
circle, its witnessρpt, the entangled stateρand the normalized swap witness,WS.
ρandρptlie close to the boundary of the Gurvits-Barnum ball of separable states.
The dashed line represent the reflection plane associated with partial transposition.

Its radius is larger than the radiusr^20 of the Gurvits-Barnum ball, but it lives in a
lower dimension.
A standard construction of 2−1 anti-commuting Pauli matrices, acting on CN, N=M^2 , fromanti-commuting matrices acting onCMis

σμ⊗σ`, 1 ⊗σν, μ= 1...`, ν= 1...`− 1 (7.32)

Consider the 2`−1 dimensional family of quantum states inCN, parametrized by
a,b^16

ρ=

(^1) M⊗ (^1) M+a·σ⊗σ+ 1 ⊗b·σ M^2 ≥ 0 , a,b∈R, b= 0 (7.33) whereσ= (σ 1 ,...,σ) is a vector ofM×M(generalized, anti-commuting) Pauli
matrices. We shall show that fora^2 +b^2 ≤1,ρis essentially equivalent to a family
of 2-qubit states, which are manifestly separable. Re-scaling thea,bcoordinates
to fit with the convention in Eq.(3.1) gives the radiusr 0.
Note first that since
(a·σ)^2 =a^21 M, σ^2 = (^1) M, (b·σ)^2 =b^21 M, {σ,b·σ}= 0 (7.34)
we may write
a·σ= (^1) M/ 2 ⊗aZ σ`=Z⊗ (^1) M/ 2 , b·σ=bX⊗ (^1) M/ 2 (7.35)
(^16) Note the change of normalization relative to Eq. (3.1).