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r=4d2
(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (130)
and consequently
QrQT
r= 0 (131)
Eqs. (102) and (103) are immediate consequences of the results already proved
and the deﬁnitions of Jr,Rr.
We deﬁnedthe Jmatricestobe theadjointrepresentativesofthe SIC-projecto rs,
considered as a basis for the Lie algebra gl( d,C), and that is certainly a most
important fact about them. However, the results of this section s how that, along
with the vectors /bardbler/an}bracketri}ht/an}bracketri}ht, they actually determine the whole structure. Speciﬁcally,
we have
Qr=1
2/parenleftbig
Jr+J2
r/parenrightbig
(132)
Rr=J2
r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (133)
Tr=d
2(d+1)/parenleftig
Jr+J2
r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightig
(134)
Moreover, if we know the Tmatrices then we know the order-3 angle tensor, which
in view of Theorem 3means we can reconstruct the SIC-projectors. Since the
vectors/bardbler/an}bracketri}ht/an}bracketri}htare given, once and for all, this means that the problem of proving th e
existenceofa SIC-POVMin dimension dis equivalent to the problem ofprovingthe
existence of a certain remarkable structure in the adjoint repres entation of gl( d,C)
(as we will see in more detail in Section 5).
In the Introduction webegan with the concept ofa SIC-POVM,and then deﬁned
theJmatrices in terms of it. However, one could, if one wished, go in the op posite
direction, and take the Lie algebraic structure to be primary, with t he SIC-POVM
being the secondary, derivative entity.
4.TheQ-QTProperty
The next ﬁve sections are devoted to a study of the Jmatrices which, as we will
see, have numerous interesting properties. We begin our investiga tion by trying to
get some additional insight into what we will call the Q-QTproperty: namely, the
fact that the Jmatrices have the spectral decomposition
Jr=Qr−QT
r (135)
whereQrisarankd−1projectorwhichisorthogonaltoits owntranspose. We wish
to characterize the general class of matrices which are of this typ e. The following
theorem provides one such characterization.
Theorem 4. LetAbe a Hermitian matrix. Then the following statements are
equivalent:19
(1)Ahas the spectral decomposition
A=P−PT(136)
wherePis a projector which is orthogonal to its own transpose.
(2)Ais pure imaginary and A2is a projector.
Proof.To show that (1) = ⇒(2) observe that the fact that Pis Hermitian means
PT=P∗(137)
whereP∗is the matrix whose elements are the complex conjugates of the cor re-
sponding elements of P. So Eq. ( 136) implies that the components of Aare pure
imaginary. Since PPT= 0 it also implies that A2is a projector.
To show that (2) = ⇒(1) observe that the fact that A2is a projector means
that the eigenvalues of A=±1 or 0. So
A=P−P′(138)
whereP,P′are orthogonal projectors. Since Ais pure imaginary we must have
PT−(P′)T=AT=A∗=−A=P′−P (139)
PTand (P′)Tare also orthogonal projectors. So if PT\|ψ/an}bracketri}ht=\|ψ/an}bracketri}ht, and\|ψ/an}bracketri}htis nor-
malized, we must have
1 =/an}bracketle{tψ\|PT\|ψ/an}bracketri}ht
=/angbracketleftbig
ψ/vextendsingle/vextendsingle/parenleftbig
PT−(P′)T/parenrightbig/vextendsingle/vextendsingleψ/angbracketrightbig
=/an}bracketle{tψ\|P′\|ψ/an}bracketri}ht−/an}bracketle{tψ\|P\|ψ/an}bracketri}ht (140)
Since
0≤ /an}bracketle{tψ\|P′\|ψ/an}bracketri}ht ≤1 (141)
0≤ /an}bracketle{tψ\|P\|ψ/an}bracketri}ht ≤1 (142)
we must have /an}bracketle{tψ\|P′\|ψ/an}bracketri}ht= 1, implying P′\|ψ/an}bracketri}ht=\|ψ/an}bracketri}ht. Similarly P′\|ψ/an}bracketri}ht=\|ψ/an}bracketri}htimplies
PT\|ψ/an}bracketri}ht=\|ψ/an}bracketri}ht. So
P′=PT(143)
/square
We also have the following statement, inspired in part by Ref. [ 51],
Theorem 5. The necessary and suﬃcient condition for a matrix Pto be a projector
which is orthogonal to its own transpose is that
P=SDST(144)
whereSis an any real orthogonal matrix and Dhas the block-diagonal form
D=
σ ... 0 0...0
............
0... σ 0...0
0...0 0...0
............
0...0 0...0
(145)20
with
σ=1
2/parenleftbigg
1−i
i1/parenrightbigg
(146)
In other words Dhasncopies ofσon the diagonal, where n= rank(P), and0
everywhere else.
Proof.Suﬃciency is an immediate consequence of the fact that σis a rank 1 pro-
jector such that σσT= 0.
To prove necessity let dbe the dimension of the space and nthe rank of P. It
will be convenient to deﬁne
\|1/an}bracketri}ht=

r=4d2

(d+1)2/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (130)

and consequently

QrQT

r= 0 (131)

Eqs. (102) and (103) are immediate consequences of the results already proved

and the deﬁnitions of Jr,Rr.

We deﬁnedthe Jmatricestobe theadjointrepresentativesofthe SIC-projecto rs,

considered as a basis for the Lie algebra gl( d,C), and that is certainly a most

important fact about them. However, the results of this section s how that, along

with the vectors /bardbler/an}bracketri}ht/an}bracketri}ht, they actually determine the whole structure. Speciﬁcally,

we have

Qr=1

2/parenleftbig

Jr+J2

r/parenrightbig

(132)

Rr=J2

r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (133)

Tr=d

2(d+1)/parenleftig

Jr+J2

r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl/parenrightig

(134)

Moreover, if we know the Tmatrices then we know the order-3 angle tensor, which

in view of Theorem 3means we can reconstruct the SIC-projectors. Since the

vectors/bardbler/an}bracketri}ht/an}bracketri}htare given, once and for all, this means that the problem of proving th e

existenceofa SIC-POVMin dimension dis equivalent to the problem ofprovingthe

existence of a certain remarkable structure in the adjoint repres entation of gl( d,C)

(as we will see in more detail in Section 5).

In the Introduction webegan with the concept ofa SIC-POVM,and then deﬁned

theJmatrices in terms of it. However, one could, if one wished, go in the op posite

direction, and take the Lie algebraic structure to be primary, with t he SIC-POVM

being the secondary, derivative entity.

4.TheQ-QTProperty

The next ﬁve sections are devoted to a study of the Jmatrices which, as we will

see, have numerous interesting properties. We begin our investiga tion by trying to

get some additional insight into what we will call the Q-QTproperty: namely, the

fact that the Jmatrices have the spectral decomposition

Jr=Qr−QT

r (135)

whereQrisarankd−1projectorwhichisorthogonaltoits owntranspose. We wish

to characterize the general class of matrices which are of this typ e. The following

theorem provides one such characterization.

Theorem 4. LetAbe a Hermitian matrix. Then the following statements are

equivalent:19

(1)Ahas the spectral decomposition

A=P−PT(136)

wherePis a projector which is orthogonal to its own transpose.

(2)Ais pure imaginary and A2is a projector.

Proof.To show that (1) = ⇒(2) observe that the fact that Pis Hermitian means

PT=P∗(137)

whereP∗is the matrix whose elements are the complex conjugates of the cor re-

sponding elements of P. So Eq. ( 136) implies that the components of Aare pure

imaginary. Since PPT= 0 it also implies that A2is a projector.

To show that (2) = ⇒(1) observe that the fact that A2is a projector means

that the eigenvalues of A=±1 or 0. So

A=P−P′(138)

whereP,P′are orthogonal projectors. Since Ais pure imaginary we must have

PT−(P′)T=AT=A∗=−A=P′−P (139)

PTand (P′)Tare also orthogonal projectors. So if PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht, and|ψ/an}bracketri}htis nor-

malized, we must have

1 =/an}bracketle{tψ|PT|ψ/an}bracketri}ht

=/angbracketleftbig

ψ/vextendsingle/vextendsingle/parenleftbig

PT−(P′)T/parenrightbig/vextendsingle/vextendsingleψ/angbracketrightbig

=/an}bracketle{tψ|P′|ψ/an}bracketri}ht−/an}bracketle{tψ|P|ψ/an}bracketri}ht (140)

Since

0≤ /an}bracketle{tψ|P′|ψ/an}bracketri}ht ≤1 (141)

0≤ /an}bracketle{tψ|P|ψ/an}bracketri}ht ≤1 (142)

we must have /an}bracketle{tψ|P′|ψ/an}bracketri}ht= 1, implying P′|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. Similarly P′|ψ/an}bracketri}ht=|ψ/an}bracketri}htimplies

PT|ψ/an}bracketri}ht=|ψ/an}bracketri}ht. So

P′=PT(143)

/square

We also have the following statement, inspired in part by Ref. [ 51],

Theorem 5. The necessary and suﬃcient condition for a matrix Pto be a projector

which is orthogonal to its own transpose is that

P=SDST(144)

whereSis an any real orthogonal matrix and Dhas the block-diagonal form

D=

σ ... 0 0...0

............

0... σ 0...0

0...0 0...0

............

0...0 0...0

(145)20

with

σ=1

2/parenleftbigg

1−i

i1/parenrightbigg

(146)

In other words Dhasncopies ofσon the diagonal, where n= rank(P), and0

everywhere else.

Proof.Suﬃciency is an immediate consequence of the fact that σis a rank 1 pro-

jector such that σσT= 0.

To prove necessity let dbe the dimension of the space and nthe rank of P. It

will be convenient to deﬁne

|1/an}bracketri}ht=