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for any fixed a, from which the SIC-family can be reconstructed using the me thod
described in Theorem 1.
Remark. Unlike the order-2tensor, the order-3angletensoris gaugeinvar iant. This
means that it provides what is, in many ways, a more useful charact erization of
the SIC-family. For that reason we will be almost exclusively concern ed with the
order-3 tensor in the remainder of this paper.
Proof.The fact that (1) = ⇒(2) is an immediate consequence of the definition of
theorder-3angletensorandcondition(2)ofCorollary 2. Toprovethat(2) = ⇒(1)
letθrstbe a completely anti-symmetric tensor such that condition (2) holds . Define
θrs=θars (89)
for allr,s. Then Eq. ( 83) implies
d2/summationdisplay
t=1KrtKtsei(θrt+θts)=eiθrs
d2/summationdisplay
t=1KrtKtseiθrst
∗
=dKrseiθrs(90)
for allr,s. It follows from Corollary 2thatθrsis the order-2 and θrstthe order-3
angle tensor of a SIC-family.
The equivalence of conditions (1) and (3) is proved similarly.
It remains to show that two SIC-sets are unitarily equivalent if and o nly if
their order-3 angle tensors are identical. To see this let Πr=|ψr/an}bracketri}ht/an}bracketle{tψr|and Π′
r=
|ψ′
r/an}bracketri}ht/an}bracketle{tψ′
r|be two different SIC-sets having the same order-3 angle tensor θrst. Let
θrs(respectively θ′
rs) be the order-2 angle tensor corresponding to the vectors |ψr/an}bracketri}ht
(respectively |ψ′
r/an}bracketri}ht). Choose some fixed index a. We have
θ′
ar+θ′
sa+θ′
rs=θar+θsa+θrs (91)
for allr,s. Consequently
θ′
rs=θrs+φr−φs (92)
for allr,s, where
φr=θar−θ′
ar (93)
Soθ′
rsandθrsare gauge equivalent. It follows from Corollary 2that Πrand Π′
rare
unitarily equivalent. Conversely, suppose that Πrand Π′
rare unitarily equivalent,
and letθrs,θ′
rsbe order-2 angle tensors corresponding to them. It follows from
Corollary 2thatθrsandθ′
rsare gauge equivalent. It is then immediate that the
order-3 angle tensors are identical. /square14
Finally, let us note that when expressed in terms of the triple produc ts Eq. (83)
reads
d2/summationdisplay
t=1Trst=dK2
rs (94)
while Eq. ( 85) reads
d2/summationdisplay
r,s,t=1Trst=d4(95)
For Eq. ( 86) we have to work a little harder. We have
d2/summationdisplay
r,s,t,u=11
K2
rtTrstTtur=d5(96)
from which it follows
d5=d2/summationdisplay
r,s,t,u=1/parenleftbig
−dδrt+d+1/parenrightbig
TrstTtur
= (d+1)d2/summationdisplay
r,s,t,u=1TrstTtur−dd2/summationdisplay
r,s,u=1K2
rsK2
ru
= (d+1)d2/summationdisplay
r,s,t,u=1TrstTtur−d5(97)
Consequently
d2/summationdisplay
s,u=1Tr/parenleftbig
TsTu/parenrightbig
=d2/summationdisplay
r,s,t,u=1TrstTtur=2d5
d+1(98)
This equation be alternatively written
d2/summationdisplay
r,s=1Tr/parenleftbig
TrTs/parenrightbig
=2d5
d+1(99)
whereTris the matrix with matrix elements ( Tr)uv=Truv.
When they are written like this, in terms of the triple products, the f act that
Eq. (94) implies Eqs. ( 95) and (98) becomes almost obvious. The reverse implica-
tion, by contrast, is rather less obvious.
3.Spectral Decompositions
LetTr,Jr,Rrbe thed2×d2matrices whose matrix elements are
(Tr)st=Trst (Jr)st=Jrst (Rr)st=Rrst(100)
whereJrst,RrstarethequantitiesdefinedbyEqs.( 12)and(13). SoJristheadjoint
representation matrix of Π r. In this section we derive the spectral decompositions
of these matrices. To avoid confusion we will use the notation |ψ/an}bracketri}htto denote a ket in
ddimensional Hilbert space Hd, and/bardblψ/an}bracketri}ht/an}bracketri}htto denote a ket in d2dimensional Hilbert15
spaceHd2. In terms of this notation the spectral decompositions will turn ou t to