3 Basic geometry of Quantum states
3.1 Choosing coordinates
Any HermitianN×N matrix with unit trace can be written as:
ρ(x) =(^1) N+
√
N− 1 x·σ
N, x∈RN(^2) − 1
(3.1)
(^1) N=σ 0 is the identity matrix andσ= (σ 1 ,...,σN^2 − 1 ) a vector ofN^2 −1 traceless,
Hermitian, mutually orthogonal,N×Nmatrices
Tr(σασβ) =δαβN, α,β∈{ 1 ,...,N^2 − 1 } (3.2)
This still leaves considerable freedom in choosing the coordinatesσαand one may
impose additional desiderata. For example:
- σαare either real symmetric or imaginary anti-symmetric
σtα=±σα (3.3)This requirement is motivated byρ≥ 0 ⇐⇒ρt≥ 0- σαforα 6 = 0 are unitarily equivalent, i.e. are iso-spectral.
A coordinate system that has these properties inN= 2ndimensions, is the (gen-
eralized) Pauli coordinates:
σμ=σμ 1 ⊗···⊗σμn, μj∈{ 0 ,..., 3 }, μ∈{ 1 ,...,N^2 − 1 } (3.4)σμare iso-spectral with eigenvalues±1. This follows from:
σμ^2 = (^1) N, Trσμ= 0 (3.5)
The Pauli coordinates behave nicely under transposition:
σtμ=±σμ (3.6)
In addition, they either commute or anti-commute
σμσβ=±σβσμ. (3.7)
This will prove handy in what follows. One drawback of the Pauli coordinates is
that they only apply to Hilbert spaces with special dimensions, namelyN= 2n.