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Trst= Tr(Π rΠsΠt) (10)
To a large extent this paper consists in an exploration of the proper ties of these
important quantities, which we will refer to as the triple products. T hey are inti-
mately related to the geometric phase, in which context they are us ually referred
to as 3-vertex Bargmann invariants (see Mukunda et al[50], and references cited
therein). We have, as an immediate consequence of the definition,
Trst=Ttrs=Tstr=T∗
rts=T∗
tsr=T∗
srt (11)
It is convenient to define
Jrst=d+1
d(Trst−T∗
rst) (12)
Rrst=d+1
d(Trst+T∗
rst) (13)4
SoJrstis imaginary and completely anti-symmetric; Rrstis real and completely
symmetric. Both these quantities play a significant role in the theory . It follows
from Eq. ( 9) that
[Πr,Πs] =d2/summationdisplay
t=1JrstΠt (14)
So theJrstare structure constants for the Lie algebra gl( d,C). As an immediate
consequence of this they satisfy the Jacobi identity:
d2/summationdisplay
b=1/parenleftbig
JrsbJtba+JstbJrba+JtrbJsba/parenrightbig
= 0 (15)
for allr,s,t,a. The Jacobi identity holds for any representation of the structu re
constants. In the following sections we will derive many other identit ies which are
specific to this particular representation.
Turning to the quantities Rrst, it follows from Eq. ( 9) that they feature in the
expression for the anti-commutator
{Πr,Πs}=/summationdisplay
tRrstΠt−2(dδrs+1)
d+1I (16)
They also play an important role in the description of quantum state s pace. Let
ρbe any density matrix and let pr=1
dTr(Πrρ) be the probability of obtaining
outcomerin the measurement described by the POVM with elements1
dΠr. Then
it follows from Eq. ( 8) thatρcan be reconstructed from the probabilities by
ρ=d2/summationdisplay
r=1/parenleftbigg
(d+1)pr−1
d/parenrightbigg
Πr (17)
Suppose, now, that the prareanyset ofd2real numbers. So we do not assume
that theprare even probabilities, let alone the probabilities coming from a density
matrix according to the prescription pr=1
dTr(Πrρ). Then it is shown in ref. [ 34]
that theprare in fact the probabilities coming from a pure state if and only if they
satisfy the two conditions
d2/summationdisplay
r=1p2
r=2
d(d+1)(18)
d2/summationdisplay
r,s,t=1Rrstprpspt=2(d+7)
d(d+1)2(19)
Let us look at the quantities JrstandRrstin a little more detail. For each r
choose a unit vector |ψr/an}bracketri}htsuch that Π r=|ψr/an}bracketri}ht/an}bracketle{tψr|. Then the Gram matrix for these
vectors is of the form
Grs=/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs(20)
where the matrix θrsis anti-symmetric and
Krs=/radicalbigg
dδrs+1
d+1(21)
Note that the SIC-POVM does not determine the angles θrsuniquely since making
the replacements |ψr/an}bracketri}ht →eiφr|ψr/an}bracketri}htleaves the SIC-POVM unaltered, but changes5
the angles θrsaccording to the prescription θrs→θrs−φr+φs. This freedom
to rephase the vectors |ψr/an}bracketri}htis not usually important. However, it sometimes has
interesting consequences (see Section 9). It can be thought of as a kind of gauge
freedom.
The Gram matrix satisfies an important identity. Every SIC-POVM ha s the
2-design property [ 5,17]
d2/summationdisplay
r=1Πr⊗Πr=2d
d+1Psym (22)
wherePsymis the projector onto the symmetric subspace of Hd⊗Hd. Expressed
in terms of the Gram matrix this becomes
d2/summationdisplay
r=1Gs1rGs2rGrt1Grt2=d
d+1/parenleftbig
Gs1t1Gs2t2+Gs1t2Gs2t1/parenrightbig
(23)
Turning to the triple products we have
Trst=GrsGstGtr=KrsKstKtreiθrst(24)
where
θrst=θrs+θst+θtr (25)
Note that the tensor θrstis completely anti-symmetric. In particular θrst= 0 if any
two of the indices are the same. Note also that re-phasing the vect ors|ψr/an}bracketri}htleaves
the tensors Trstandθrstunchanged. They are in that sense gauge invariant.
Finally, we have the following expressions for JrstandRrst:
Jrst=2i
d√
d+1sinθrst (26)
Rrst=2(d+1)
dKrsKstKtrcosθrst (27)
Like the triple products, JrstandRrstare gauge invariant.