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X|a/an}bracketri}ht=|a+1/an}bracketri}ht (414)
Z|a/an}bracketri}ht=ωa|a/an}bracketri}ht (415)
whereω=e2πi
dand the addition of indices in the first equation is modd. We then
define the Weyl-Heisenberg displacement operators by (adopting t he convention
used in, for example, ref. [ 16])
Dp=τp1p2Xp1Zp2(416)
wherepis the vector ( p1,p2) (p1,p2being integers) and τ=e(d+1)πi
d. Generally
speaking the decision to insert the phase τp1p2is a matter of convention, and many
authors define it differently, or else omit altogether. However, for the purposes of
this section it is essential, as a different choice of phase at this stage would lead to a
different gauge in the class of POVMs to be defined below, and the Gra m projector
would then typically not have the P-PTproperty.
Note thatτ2=τd2=ωin every dimension. If the dimension is odd we can write
τ=ωd+1
2. Soτis adthroot of unity. However, if the dimension is even τd=−1.
This has the consequence that
Dp+du= (−1)u1p2+u2p1Dp (417)
Soin even dimension p=q(modd) does notnecessarilyimply Dp=Dq(although
the operators are, of course, equal if p=q(mod 2d))
In every dimension (even or odd) we have
D†
p=D−p (418)
for allp
(Dp)n=Dnp (419)
for allp,nand
DpDq=τ/angbracketleftp,q/angbracketrightDp+q (420)
for allp,q. In the last expression /an}bracketle{tp,q/an}bracketri}htis the symplectic form
/an}bracketle{tp,q/an}bracketri}ht=p2q1−p1q2 (421)
Now let|ψ/an}bracketri}htbe any normalized vector (not necessarily a SIC-fiducial vector), and
define
|ψp/an}bracketri}ht=Dp|ψ/an}bracketri}ht (422)
Let
L=/summationdisplay
p∈Z2
d|ψp/an}bracketri}ht/an}bracketle{tψp| (423)
It is easily seen that/bracketleftbig
Dp,L/bracketrightbig
= 0 (424)
for allp.51
We now appeal to the fact that there is no non-trivial subspace of Hdwhich
the displacement operators leave invariant. To see this assume the contrary. Then
there would exist non-zero vectors |φ/an}bracketri}ht,|χ/an}bracketri}htsuch that
/an}bracketle{tφ|Dp|χ/an}bracketri}ht= 0 (425)
for allp. Writing the left-hand side out in full this gives
d−1/summationdisplay
a=0ωp2a/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (426)
for allp1,p2. Taking the discrete Fourier transform with respect to p2, we have
/an}bracketle{tφ|a+p1/an}bracketri}ht/an}bracketle{ta|χ/an}bracketri}ht= 0 (427)
for alla,p1, implying that either |φ/an}bracketri}ht= 0 or|χ/an}bracketri}ht= 0—contrary to assumption. We
can therefore use Schur’s lemma [ 55] to deduce that
L=kI (428)
for some constant k. Taking the trace on both sides of this equation we infer
thatk=d. We conclude that1
d|ψp/an}bracketri}ht/an}bracketle{tψp|is a POVM. We refer to POVMs of this
general class as Weyl-Heisenberg covariant POVMs. We refer to th e vector |ψ/an}bracketri}ht
which generates the POVM as the fiducial vector (with no implication t hat it is
necessarily a SIC-fiducial).
Now consider the Gram projector
P=/summationdisplay
p,q∈Z2
dPp,q/bardblp/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardbl (429)
where
Pp,q=1
d/an}bracketle{tψp|ψq/an}bracketri}ht (430)
and where we label the matrix elements of Pand the standard basis kets with the
vectorsp,qrather than with the single integer indices r,sas in the rest of this
paper. We know from Theorem 1thatPis a rankdprojector.
In view of Eqs. ( 418) and (420) we have
/an}bracketle{t/an}bracketle{tp/bardblP/bardblq/an}bracketri}ht/an}bracketri}ht=Pp,q
=1
dτ−/angbracketleftp,q/angbracketright/an}bracketle{tψ|Dq−p|ψ/an}bracketri}ht
=1
dd−1/summationdisplay
a=0τp1p2+q1q2ωaq2−(q1+a)p2/an}bracketle{tψ|a+q1−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht(431)
Hence
/an}bracketle{t/an}bracketle{tp/bardblPPT/bardblq/an}bracketri}ht/an}bracketri}ht=/summationdisplay
u∈Zd/an}bracketle{t/an}bracketle{tp/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tq/bardblP/bardblu/an}bracketri}ht/an}bracketri}ht
=1
d2d−1/summationdisplay
a,b,u1,u2=0τp1p2+q1q2ωu2(u1+a+b)−(u1+a)p2−(u1+b)q2
×/an}bracketle{tψ|a+u1−p1/an}bracketri}ht/an}bracketle{tψ|b+u1−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht52
=1
dd−1/summationdisplay
a,b=0τp1p2+q1q2ωp2b+q2a/an}bracketle{tψ|−b−p1/an}bracketri}ht/an}bracketle{tb|ψ/an}bracketri}ht/an}bracketle{tψ|−a−q1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht
=/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardblq/an}bracketri}ht/an}bracketri}ht (432)
where/bardblh/an}bracketri}ht/an}bracketri}htis the vector with components
/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=1√
dd−1/summationdisplay
a=0τp1p2ωp2a/an}bracketle{tψ|−a−p1/an}bracketri}ht/an}bracketle{ta|ψ/an}bracketri}ht (433)
It is easily verified that /bardblh/an}bracketri}ht/an}bracketri}htis normalized, and that /an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}htis real.
Finally, suppose that the dimension is odd. Then the Wigner function o f the
state|ψ/an}bracketri}htis [56,57]
W(p) =1
d/an}bracketle{tψ|DpUPD†
p|ψ/an}bracketri}ht=1
d/an}bracketle{tψ|D2pUP|ψ/an}bracketri}ht (434)
whereUPistheparityoperator,whoseactiononthestandardbasisis UP|a/an}bracketri}ht=|−a/an}bracketri}ht.
It is straightforward to show