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eigenvalues = 1 and the rest all = 0. So Pis a rank-dprojector.
It remains to show that the POVM corresponding to a given rank- dprojector
is unique up to unitary equivalence. To prove this let Pbe a rank-dprojector, let
|ξr/an}bracketri}htbe the vectors defined by Eq. ( 40), and let |η1/an}bracketri}ht,...,|ηd2/an}bracketri}htbe any other set of
vectors such that
/an}bracketle{tηr|ηs/an}bracketri}ht=Prs (53)
for allr,s. Define
ηar=/an}bracketle{tηr|a/an}bracketri}ht (54)
Then
d2/summationdisplay
r=1η∗
arηbr=/an}bracketle{ta|
d2/summationdisplay
r=1|ηr/an}bracketri}ht/an}bracketle{tηr|
|b/an}bracketri}ht=δab (55)
(because |ηr/an}bracketri}ht/an}bracketle{tηr|is a POVM) and
d/summationdisplay
a=1ηarη∗
as=Prs (56)
(because the |ηr/an}bracketri}hthave Gram matrix P). So thedcolumn vectors
η11
η12
...
η1d2
,
η21
η22
...
η2d2
,...,
ηd1
ηd2
...
ηdd2
(57)10
are an orthonormal basis for the subspace onto which Pprojects. But the column
vectors 
ξ11
ξ12
...
ξ1d2
,
ξ21
ξ22
...
ξ2d2
,...,
ξd1
ξd2
...
ξdd2
(58)
are also an orthonormal basis for this subspace. So there must ex ist ad×dunitary
matrixUabsuch that
ηar=d/summationdisplay
b=1Uabξbr (59)
for alla,r. Define
U=d/summationdisplay
a,b=1U∗
ab|a/an}bracketri}ht/an}bracketle{tb| (60)
Then
|ηr/an}bracketri}ht=U|ξr/an}bracketri}ht (61)
for allr. /square
In the case of a SIC-POVM we have
|ξr/an}bracketri}ht=1√
d|ψr/an}bracketri}ht (62)
where the vectors |ψr/an}bracketri}htare normalized, and
Prs=1
dGrs=1
dKrseiθrs(63)
whereGis the Gram matrix of the vectors |ψr/an}bracketri}htandθrsis the order-2 angle tensor.
In the sequel we will distinguish these matrices by referring to Gas the Gram
matrix and Pas the Gram projector.
We have
Corollary 2. Letθrsbe a real anti-symmetric tensor. Then the following state-
ments are equivalent:
(1)θrsis an order- 2angle tensor corresponding to a SIC-family.
(2)θrssatisfies
d2/summationdisplay
t=1KrtKtsei(θrt+θts)=dKrseiθrs(64)
for allr,s.
(3)θrssatisfies
d2/summationdisplay
r,s,t=1KrsKstKtrei(θrs+θst+θtr)=d4(65)
and
d2/summationdisplay
r,s,t,u=1KrsKstKtuKurei(θrs+θst+θtu+θur)=d5(66)11
LetΠr,Π′
rbe two different SIC-sets, and let θrs,θ′
rsbe corresponding order- 2
angle tensors. Then there exists a unitary Usuch that
Π′
r=UΠrU†(67)
for allrif and only if
θ′
rs=θrs−φr+φs (68)
for some arbitrary set of phase angles φr(in other words two SIC-sets are unitarily
equivalent if and only if their order- 2angle tensors are gauge equivalent).
A SIC-family can be reconstructed from its order- 2angle tensor θrsby calculating