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1
0
...
0
,|2/an}bracketri}ht=
0
1
...
0
, ... |d/an}bracketri}ht=
0
0
...
1
(147)
In terms of these basis vectors we have
P=d/summationdisplay
r,s=1Prs|r/an}bracketri}ht/an}bracketle{ts| (148)
Now let |a1/an}bracketri}ht,...,|an/an}bracketri}htbe an orthonormal basis for the subspace onto which P
projects, and let |a∗
r/an}bracketri}htbe the column vector which is obtained from |ar/an}bracketri}htby tak-
ing the complex conjugate of each of its components. Taking comple x conjugates
on each side of the equation
P|ar/an}bracketri}ht=|ar/an}bracketri}ht (149)
gives
P∗|a∗
r/an}bracketri}ht=|a∗
r/an}bracketri}ht (150)
So|a∗
1/an}bracketri}ht,...,|a∗
n/an}bracketri}htis an orthonormal basis for the subspace onto which PT=P∗
projects. Since PTis orthogonal to Pwe conclude that
/an}bracketle{tar|a∗
s/an}bracketri}ht= 0 (151)
for allr,s.
Next define vectors |b1/an}bracketri}ht,...,|b2n/an}bracketri}htby
|b2r−1/an}bracketri}ht=1√
2/parenleftbig
|a∗
r/an}bracketri}ht−|ar/an}bracketri}ht/parenrightbig
(152)
|b2r/an}bracketri}ht=i√
2/parenleftbig
|a∗
r/an}bracketri}ht+|ar/an}bracketri}ht/parenrightbig
(153)
By construction these vectors are orthonormal and real. So we c an extend them
to an orthonormal basis for the full space by adding a further d−2nvectors
|b2n+1/an}bracketri}ht,...,|bd/an}bracketri}ht, which can also be chosen to be real. We have
P=n/summationdisplay
r=1|ar/an}bracketri}ht/an}bracketle{tar|
=1
2n/summationdisplay
r=1/parenleftig
|b2r−1/an}bracketri}ht/an}bracketle{tb2r−1|−i|b2r−1/an}bracketri}ht/an}bracketle{tb2r|+i|b2r/an}bracketri}ht/an}bracketle{tb2r−1|+|b2r/an}bracketri}ht/an}bracketle{tb2r|/parenrightig
(154)21
So if we define
S=d/summationdisplay
r=1|br/an}bracketri}ht/an}bracketle{tr| (155)
thenSis a real orthogonal matrix such that
P=SDST(156)
where
D=1
2n/summationdisplay
r=1/parenleftig
|2r−1/an}bracketri}ht/an}bracketle{t2r−1|−i|2r−1/an}bracketri}ht/an}bracketle{t2r|+i|2r/an}bracketri}ht/an}bracketle{t2r−1|+|2r/an}bracketri}ht/an}bracketle{t2r|/parenrightig
(157)
is the matrix defined by Eq. ( 145). /square
This result implies the following alternative characterization of the cla ss of ma-
trices to which the Jmatrices belong
Corollary 6. LetAbe a Hermitian matrix. Then the following statements are
equivalent:
(1)Ahas the spectral decomposition
A=P−PT(158)
wherePis a projector which is orthogonal to its own transpose.
(2)There exists a real orthogonal matrix Ssuch that
A=SDST(159)
whereDhas the block diagonal form
D=
σy...0 0...0
............
0... σ y0...0
0...0 0...0
............
0...0 0...0
(160)
σybeing the Pauli matrix
σy=/parenleftbigg0−i
i0/parenrightbigg
(161)
In other words Dhasncopies ofσyon the diagonal, where n=1
2rank(A),
and0everywhere else (note that a matrix of this type must have eve n rank).
Proof.Immediate consequence of Theorem 5. /square
5.Lie Algebraic Formulation of the Existence Problem
This section is the core of the paper. We show that the problem of pr oving the
existence of a SIC-POVM in dimension dis equivalent to the problem of proving
the existence of an Hermitian basis for gl( d,C) all of whose elements have the Q-QT
property. We hope that this new way of thinking will help make the SIC -existence
problem more amenable to solution.