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The significance of these vectors is that /bardblfrs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfrs/bardbl(respectively /bardblf∗
rs/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
rs/bardbl) will
turn out to be the projectoronto the 1-dimensionalsubspace Q0
rs(respectively ¯Q0
rs).41
Note that the fact that Qris Hermitian means
QT
r=Q∗
r (328)
(whereQ∗
ris the matrix whose elements are the complex conjugates of the cor re-
sponding elements of Qr). Consequently
/an}bracketle{t/an}bracketle{tt/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
(329)
for allr,s,t.
It is easily seen that /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
rs/an}bracketri}ht/an}bracketri}htare normalized. In fact, it follows from
Eqs. (116) and (120) that
/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht= (d+1)/an}bracketle{t/an}bracketle{ts/bardblQr/bardbls/an}bracketri}ht/an}bracketri}ht
=(d+1)2
dTrss−2(d+1)/an}bracketle{t/an}bracketle{ts/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht
=(d+1)2
d/parenleftbig
K2
rs−K4
rs/parenrightbig
= 1 (330)
for allr/ne}ationslash=s. In view of Eq. ( 329) we then have
/an}bracketle{t/an}bracketle{tf∗
rs/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=/parenleftig
/an}bracketle{t/an}bracketle{tfrs/bardblfrs/an}bracketri}ht/an}bracketri}ht/parenrightig∗
= 1 (331)
for allr/ne}ationslash=s. The fact that QrQT
r= 0 means we also have
/an}bracketle{t/an}bracketle{tfrs/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht= 0 (332)
for allr/ne}ationslash=s.
Note that, although we required that r/ne}ationslash=sin the definitions of /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht,
the definitions continue to make sense when r=s. However, the vectors are then
zero (as can be seen by setting r=sin Eq. (121)).
The vectors /bardblfrs/an}bracketri}ht/an}bracketri}ht,/bardblf∗
rs/an}bracketri}ht/an}bracketri}htsatisfy a number of identities, which it will be conve-
nient to collect in a lemma:
Lemma 18. For allr/ne}ationslash=s
/bardblfrs/an}bracketri}ht/an}bracketri}ht=−/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht+i/radicalbigg
2
d/parenleftig
/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
(333)
/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=−/bardblfsr/an}bracketri}ht/an}bracketri}ht−i/radicalbigg
2
d/parenleftig
/bardbles/an}bracketri}ht/an}bracketri}ht−/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
(334)
(where/bardbler/an}bracketri}ht/an}bracketri}htis the vector defined by Eq. ( 116))
Qr/bardblfrs/an}bracketri}ht/an}bracketri}ht=/bardblfrs/an}bracketri}ht/an}bracketri}ht QT
r/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht (335)
QT
r/bardblfrs/an}bracketri}ht/an}bracketri}ht= 0 Qr/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht= 0 (336)
Qs/bardblfrs/an}bracketri}ht/an}bracketri}ht=−1
d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht QT
s/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=−1
d+1/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht(337)
QT
s/bardblfrs/an}bracketri}ht/an}bracketri}ht=−d
d+1/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht Qs/bardblf∗
rs/an}bracketri}ht/an}bracketri}ht=−d
d+1/bardblfsr/an}bracketri}ht/an}bracketri}ht(338)
/an}bracketle{t/an}bracketle{tfrs/bardblfsr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
rs/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht=−1
d+1(339)42
/an}bracketle{t/an}bracketle{tfrs/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht=/an}bracketle{t/an}bracketle{tf∗
rs/bardblfsr/an}bracketri}ht/an}bracketri}ht=−d
d+1(340)
Proof.It follows from Eqs. ( 116) and (120) that
/an}bracketle{t/an}bracketle{tt/bardblfrs/an}bracketri}ht/an}bracketri}ht+/an}bracketle{t/an}bracketle{tt/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht=i√
d+1/parenleftbig
Qrts−Qsrt/parenrightbig
=i√
d+1/parenleftbiggd+1
d/parenleftbig
Trts−Tsrt/parenrightbig
−2/parenleftbig
/an}bracketle{t/an}bracketle{tt/bardbler/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{ter/bardbls/an}bracketri}ht/an}bracketri}ht−/an}bracketle{t/an}bracketle{tr/bardbles/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tes/bardblt/an}bracketri}ht/an}bracketri}ht/parenrightbigg