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fact that 1 ≤m<d2, together with the fact that d>2 means
4
d<4d2+2m(d−2)
d3<2 (245)31
Ifd= 3 or 4 there are no integers in this interval, which gives us a contrad iction
straight away. If, on the other hand, d≥5 there is the possibility
4d2+2m(d−2)
d3= 1 (246)
implying
m=d2(d−4)
2(d−2)(247)
This equationhasthe solution d= 6,m= 9(this is in fact the only integersolution,
ascanbe seenfrom ananalysisofthe possible primefactorizationso fthe numerator
and denominator on the right hand side). To eliminate this possibility de fine
L′′
r=2
dI−L′
d2+1−r (248)
for allr. It is easily verified that
Tr(L′′
rL′′
s) =dδrs+1
d+1(249)
d2/summationdisplay
r=1L′′
r=dI (250)
and
L′′
r=/braceleftigg
Πr r≤d2−m
2
dI−Πrr>d2−m(251)
So we can go through the same argument as before to deduce
d2−m=d2(d−4)
2(d−2)(252)
Eqs. (247) and (252) have no joint solutions at all with d/ne}ationslash= 0, integer or otherwise.
/square
To complete the proof of Theorem 7observe that Eqs. ( 210) and (227) imply
Tr(ΠrΠs) =dδrs+1
d+1(253)
So the Π rare a SIC-set. Moreover
Lr=ǫr(Πr+αI) (254)
whereǫr=ǫǫ′
randα= (ǫl−1)/d.
6.The Algebra sl(d,C)
The motivation for this paper is the hope that a Lie algebraic perspec tive may
cast some light on the SIC-existence problem, and on the mathemat ics of SIC-
POVMs generally. We have focused on gl( d,C) as that is the case where the con-
nection with Lie algebras seems most straightforward. However, it may be worth
mentioning that a SIC-POVM also gives rise to an interesting geometr ical structure
in sl(d,C) (the Lie algebra consisting of all trace-zero d×dcomplex matrices).32
Let Πrbe a SIC-set and define
Br=/radicaligg
d+1
2(d2−1)/parenleftbigg
Πr−1
dI/parenrightbigg
(255)
SoBr∈sl(d,C). Let
/an}bracketle{tA,A′/an}bracketri}ht= Tr(ad AadA′) = 2dTr(AA′) (256)
be the Killing form [ 55] on sl(d,C). Then
/an}bracketle{tBr,Bs/an}bracketri}ht=/braceleftigg
1 r=s
−1
d2−1r/ne}ationslash=s(257)
So theBrform a regular simplex in sl( d,C). Since sl( d,C) isd2−1 dimensional
theBrare an overcomplete set. However, the fact that
d2/summationdisplay
r=1Br= 0 (258)
means that for each A∈sl(d,C) there is a unique set of numbers arsuch that
A=d2/summationdisplay
r=1arBr (259)
and
d2/summationdisplay
r=1ar= 0 (260)
Thearcan be calculated using
ar=d2−1
d2/an}bracketle{tA,Br/an}bracketri}ht (261)
Similarly, given any linear transformation M: sl(d,C)→sl(d,C), there is a unique
set of numbers Mrssuch that
MBr=d2/summationdisplay
s=1MrsBs (262)
and
d2/summationdisplay
s=1Mrs=d2/summationdisplay
s=1Msr= 0 (263)
for allr. TheMrscan be calculated using
Mrs=d2−1
d2/an}bracketle{tBs,MBr/an}bracketri}ht (264)
In short, the Brretain many analogous properties of, and can be used in much the
same way as, a basis. It could be said that they form a simplicial basis.33
7.Further Identities
In the preceding pages we have seen that there are five different f amilies of ma-
trices naturally associated with a SIC-POVM: namely, the projecto rsQrtogether
with the matrices
Jr=Qr−QT
r (265)
¯Rr=Qr+QT
r (266)
Rr=Qr+QT