However, the converse is not true. In fact, the largest ball inscribed inDNis the
Gurvits-Barnum ball^3 [6]:
Bgb={
ρ∣
∣
∣
∣
∣
Tr(
ρ−1
N
) 2
≤
1
N− 1
−
1
N
}
(1.4)
It follows that:
- dimDN=N^2 − 1
- Since^4
radius of bounding ball ofDN
radius of inscribed ball ofDN
=N− 1 (1.5)
onlyD 2 is a ball andDNgets increasingly far from a ball whenN is large,
Fig. 3.Figure 3: The inscribed, Gurvits-Barnum, ball is represented by the small circle
and the bounding sphere by the large circle. The green area representsDN.
Another significant difference between a single qubit and the general case is
thatDNis inversion symmetric only forN= 2. Indeed, inversion with respect to
the fully mixed state is defined by
I(ρ) =1
N
−
(
ρ−1
N
)
=
2
N
1 −ρ (1.6)EvidentlyIis trace preserving andI^2 = 1. However, it is not positivity preserving
in general. I(ρ)≥0 implies that
0 ≤Tr(
ρI(ρ))
=
2
N
−Tr(ρ^2 ) (1.7)(^3) Gurvits and Barnum define the radius of the ball by its purity, rather than the distance from
the maximally mixed states.
(^4) The balls are centered at the mixed state which is the natural “center of mass” of all states.