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

4.2 Cross sections that are balls

Suppose N = 2n. Consider the largest set of mutually anti-commuting matri-
ces among theN^2 −1 (generalized) Pauli matricesσα. Since the Pauli matrices
include the matrices that span a basis of a Clifford algebra we have at least`
anti-commuting matrices

{σj,σk}= 2δjk, j,k∈{ 1 ,...,`}, `=n+ 2

⌊

n+ 1

2

⌋

(4.4)

For the anti-commutingσjwe have
(∑
xjσj

) 2

=r^2 =

∑

j

x^2 j (4.5)

The positivity of

(^1) N+

√

N− 1

∑`

j=

xjσj≥ 0 (4.6)

holds iff

r≤r 0 =

1

√

N− 1

This means thatDNhas`dimensional cross sections that are perfect balls^8. This
result extends to 2n≤N < 2 n+1.

4.3 Cross sections that are polyhedra and hyper-octahedra

Consider the set of commuting matricesσα. Since the matrices can be simultane-
ously diagonalized, there areN−1 of them and the positivity condition on the
cross-section

1 +

√

N− 1

N∑− 1

α=

xασj≥ 0

reduces to a set of linear inequalities forxα. This defines a polyhedron.
In the case ofnqubits, a set ofncommutingσαmatrices with no relations is:
σα= 1 ⊗... 1 ⊗σx⊗ 1 ···⊗ 1 , j=α,...,n

The cross section is the intersection ofN half-spaces

πα·x≥−r 0 , α= 1,...,N, πα∈{− 1 , 1 }n

The corresponding cross section is a regularndimensional hyper-octahedron^9 : A
regular, convex polytope withnvertices andN= 2nhyper-planes. (The dual of
thendimensional cube.)

(^8) This and section 6 are reminiscent of Dvoretzki-Milmann theorem [10].
(^9) See footnote 1.