text
stringlengths 0
44.4k
|
|---|
r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (267) |
Tr=d |
d+1Qr+2d |
d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (268) |
(see Section 3). As we noted previously, it is possible to define everything in terms |
of the adjoint representation matrices Jrand the rank-1 projectors /bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl: |
Qr=1 |
2Jr(Jr+I) (269) |
¯Rr=J2 |
r (270) |
Rr=J2 |
r+4/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (271) |
Tr=d |
2(d+1)Jr(Jr+I)+2d |
d+1/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbl (272) |
In that sense the structure constants of the Lie algebra, supple mented with the |
vectors/bardbler/an}bracketri}ht/an}bracketri}ht, determine everything else. |
In the next section we will show that there are some interesting geo metrical |
relationships between the hyperplanes onto which Qr,QT |
rand¯Rrproject. In this |
section, as a preliminary to that investigation, we prove a number of identities |
satisfied by the Q,Jand¯Rmatries. We start by computing their Hilbert-Schmidt |
inner products: |
Theorem 13. For allr,s |
Tr/parenleftbig |
QrQs/parenrightbig |
=d3δrs+d2−d−1 |
(d+1)2(273) |
Tr/parenleftbig |
QrQT |
s/parenrightbig |
=d2(1−δrs) |
(d+1)2(274) |
Tr/parenleftbig |
JrJs/parenrightbig |
=2(d2δrs−1) |
d+1(275) |
Tr/parenleftbig¯Rr¯Rs/parenrightbig |
=2(d−1)(d2δrs+2d+1) |
(d+1)2(276) |
Tr/parenleftbig |
Jr¯Rs/parenrightbig |
= 0 (277) |
Proof.Let us first calculate some auxiliary quantities. It follows from the de finition |
ofTr, andthe factthat the matrix P=1 |
dGdefined byEq.( 63) isarankdprojector, |
that |
Tr(TrTs) =d2/summationdisplay |
u,v=1TruvTsvu34 |
=d2/summationdisplay |
u,v=1K2 |
uvGruGusGsvGvr |
=d |
d+1d2/summationdisplay |
u=1K2 |
ruK2 |
su+d4 |
d+1d2/summationdisplay |
u,v=1PruPusPsvPvr |
=d2(dδrs+d+2) |
(d+1)3+d4 |
d+1/vextendsingle/vextendsinglePrs/vextendsingle/vextendsingle2 |
=d2(dδrs+d+2) |
(d+1)3+d2 |
d+1K2 |
rs |
=d2/parenleftbig |
d(d+2)δrs+2d+3/parenrightbig |
(d+1)3(278) |
Also |
Tr/parenleftbig |
TrTT |
s/parenrightbig |
=d2/summationdisplay |
u,v=1TruvTsuv |
=d2/summationdisplay |
u=1GruGsu |
d2/summationdisplay |
v=1GuvGuvGvrGvs |
|
=2d |
d+1d2/summationdisplay |
u=1GruGsuGurGus |
=2d2 |
(d+1)2/parenleftbig |
1+K2 |
rs/parenrightbig |
=2d2(dδrs+d+2) |
(d+1)3(279) |
where we made two applications of Eq. ( 23) (i.e.the fact that every SIC-POVM is |
a 2-design). Finally, it is a straightforward consequence of the defi nitions ofTr,TT |
r |
and/bardbler/an}bracketri}ht/an}bracketri}htthat |
/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{ter/bardblTT |
s/bardbler/an}bracketri}ht/an}bracketri}ht |
=d+1 |
2dd2/summationdisplay |
u,v=1TsuvK2 |
ruK2 |
rv |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.