A The average purity of quantum states
In section 6.2 we quoted a result of [24], Eq. (6.8), which allows to explicitly
compute the radius of inertiareofDN as a rational function ofN. The aim of
this appendix is to give an elementary derivation of this formula.
The measuredμon the space of density matrices in section 6.2 is a special
case of a more general measuresdμN,KwhenN =K. The measuresdμN,Kare
the induced measure on density matrices acting onCN, obtained from the uniform
measure on pure states onCN⊗CK withK ≥ N, by partial tracing over the
second factor^17. They are all simply related [24]
dμN,K=1
ZN,K
(
det(ρ))K−N
dx 1 ...dxN (^2) − 1 , K≥N (A.1)
ZN,K is a normalization factor. The Euclidean measuredμcorresponds to the
caseN=K.
The derivation given below of Eq. (6.8) is simpler than the original derivation
in [24] in that it avoids the constraint associated with the normalization of the
wave functions. Using this observation,computing the second moment ofx^2 with
respect to the measuredμN,Kreduces to an exercise in Gaussian integration.
Let〈αj|ψ〉=ξαjbe the amplitudes of the pure state|ψ〉inCN⊗CK. The first
factor is the system and the second is the ancila. The density matrixρis obtained
by partial tracing the ancila.
〈α|ρ|β〉=〈α|
(
TrK|ψ〉〈ψ|)
|β〉=∑
jξαjξ ̄βj= (ξξ∗)αβ (A.2)whereξis anN×Kmatrix. The requirementK≥Nguarantees that (generically)
ρis full rank.
ChoosingReξαjandImξαjto be normally distributed i.i.d., gives a uniform
measure on pure states,dμψ, which is unitary invariant underU(NK). The in-
duced measure on the density matricesdμN,K
dμNK=dρ∫
δ(ρ−TrK|ψ〉〈ψ|)dμψ (A.3)Since the Gaussian measure forξ allows for states that are not normalized, the
measuredμN,Kallows for anyTrρ≥0. This means that to compute the moments
of normalized density matrices we need to compute
∫
dμN,K
Trρ^2
(Tr ρ)^2(A.4)
(^17) In the caseK=N−1 the measure is concentrated on the boundary ofDNbeing proportional
toδ(detρ).