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an orthonormal basis for the subspace onto which the Gram pro jector
Prs=1
dKrseiθrs(69)
projects, as described in Theorem 1.
Remark. The sense in which we areusing the term “gaugeequivalence”is explain ed
in the passage immediately following Eq. ( 21).
Note that condition (2) imposes d2(d2−1)/2 independent constraints (taking
account of the anti-symmetry of θrs). Condition (3), by contrast, only imposes 2
independent constraints. It is to be observed, however, that th e price we pay for
the reduction in the number of equations is that Eqs. ( 65) and (65) are respectively
cubic and quartic in the phases, whereas Eq. ( 64) is only quadratic.
Proof.Letθrsbe an arbitrary anti-symmetric tensor, and define
Prs=1
dKrseiθrs(70)
The anti-symmetry of θrsmeans that Pis automatically Hermitian. So it follows
from Theorem 1that a necessary and sufficient condition for Prsto be a rank- d
projector, and for θrsto be the order-2 angle tensor of a SIC-family, is that
d2/summationdisplay
t=1KrtKtsei(θrt+θts)=dKrseiθrs(71)
for allr,s.
To prove the equivalence of conditions (1) and (3) note that the co nditions
Tr(P) = Tr(P2) =dare an automatic consequence of Phaving the specified form.
So it follows from Theorem 1thatθrsis the order-2 angle tensor of a SIC-family if
and only if Eqs. ( 65) and (66) are satisfied.
Now let Πr, Π′
rbe two SIC-sets and let θrs,θ′
rsbe order-2 angle tensors corre-
sponding to them. Then there exist normalized vectors |ψr/an}bracketri}ht,|ψ′
r/an}bracketri}htsuch that
Πr=|ψr/an}bracketri}ht/an}bracketle{tψr| Π′
r=|ψ′
r/an}bracketri}ht/an}bracketle{tψ′
r| (72)
for allr, and
/an}bracketle{tψr|ψs/an}bracketri}ht=Krseiθrs/an}bracketle{tψ′
r|ψ′
s/an}bracketri}ht=Krseiθ′
rs (73)
for allr,s.
Suppose, first of all, that there exists a unitary Usuch that
Π′
r=UΠrU†(74)12
Then there exist phase angles φrsuch that
|ψ′
r/an}bracketri}ht=eiφrU|ψr/an}bracketri}ht (75)
for allr, which is easily seen to imply that
θ′
rs=θrs−φr+φs (76)
for allr,s. Soθrs,θ′
rsare gauge equivalent.
Conversely, suppose there exist phase angles φrsuch that
θ′
rs=θrs−φr+φs (77)
Define
|ψ′′
r/an}bracketri}ht=e−iφr|ψ′
r/an}bracketri}ht (78)
Then
/an}bracketle{tψ′′
r|ψ′′
s/an}bracketri}ht=Krseiθrs=/an}bracketle{tψr|ψs/an}bracketri}ht (79)
for allr,s. So it follows from Theorem 1that there exists a unitary Usuch that
|ψ′′
r/an}bracketri}ht=U|ψr/an}bracketri}ht (80)
for allr. Consequently
Π′
r=|ψ′′
r/an}bracketri}ht/an}bracketle{tψ′′
r|=UΠrU†(81)
for allr. So Πrand Π′
rare unitarily equivalent. /square
We now turn to the order-3 angle tensors. We have
Theorem 3. Letθrstbe a real completely anti-symmetric tensor. Then the follow -
ing conditions are equivalent:
(1)θrstis the order- 3angle tensor for a SIC-family
(2)For some fixed aand allr,s,t
θars+θast+θatr=θrst (82)
and for all r,s
d2/summationdisplay
t=1KrtKtseiθrst=dKrs (83)
(3)For some fixed aand allr,s,t
θars+θast+θatr=θrst (84)
and
d2/summationdisplay
r,s,t=1KrsKstKtreiθrst=d4(85)
d2/summationdisplay
r,s,t,u=1KrsKstKtuKurei(θrst+θtur)=d5(86)13
LetΠr,Π′
rbe two different SIC-sets and let θrst,θ′
rstbe the corresponding order-
3angle tensors. Then the necessary and sufficient condition fo r there to exist a
unitaryUsuch that
Π′
r=UΠrU†(87)
for allris thatθ′
rst=θrstfor allr,s,t(in other words two SIC-sets are unitarily
equivalent if and only if their order- 3angle tensors are identical).
Letθrstbe the order- 3angle tensor corresponding to a SIC-family. Then the
order-2angle tensor is given by (up to gauge freedom)
θrs=θars (88)