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d2/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht
−1
2/parenleftbigg2(d+1)
d/parenrightbigg3
2/parenleftig
K2
ra/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
rb/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+2(d+1)
dK2
raK2
rb/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht (358)
/an}bracketle{t/an}bracketle{ta/bardblQrTT
sQr/bardblb/an}bracketri}ht/an}bracketri}ht=(d+1)2
d2/an}bracketle{t/an}bracketle{ta/bardblTrTT
sTr/bardblb/an}bracketri}ht/an}bracketri}ht
−1
2/parenleftbigg2(d+1)
d/parenrightbigg3
2/parenleftig
K2
ra/an}bracketle{t/an}bracketle{ter/bardblTT
sTr/bardblb/an}bracketri}ht/an}bracketri}ht+K2
rb/an}bracketle{t/an}bracketle{ta/bardblTrTT
s/bardbler/an}bracketri}ht/an}bracketri}ht/parenrightig
+2(d+1)
dK2
raK2
rb/an}bracketle{t/an}bracketle{ter/bardblTT
s/bardbler/an}bracketri}ht/an}bracketri}ht (359)44
Using the definitions of Tr,/bardbler/an}bracketri}ht/an}bracketri}htand Eq. ( 23) (the 2-design property) we find, after
some algebra,
/an}bracketle{t/an}bracketle{ta/bardblTrTsTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
(d+1)2/parenleftig
K2
raTrsb+K2
rbTras+K2
rsTrab+K2
raK2
rb/parenrightig
(360)
/an}bracketle{t/an}bracketle{ter/bardblTsTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
2(d+1)/parenrightbigg3
2/parenleftig
2K2
rsK2
rb+K2
rb+Trsb/parenrightig
(361)
/an}bracketle{t/an}bracketle{ta/bardblTrTs/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
2(d+1)/parenrightbigg3
2/parenleftig
2K2
rsK2
ra+K2
ra+Tras/parenrightig
(362)
/an}bracketle{t/an}bracketle{ter/bardblTs/bardbler/an}bracketri}ht/an}bracketri}ht=d
2(d+1)/parenleftbig
3K2
rs+1/parenrightbig
(363)
and
/an}bracketle{t/an}bracketle{ta/bardblTrTT
sTr/bardblb/an}bracketri}ht/an}bracketri}ht=d2
(d+1)2/parenleftig
GraGasGsbGbr
+K2
raTrsb+K2
rbTras+K2
raK2
rb/parenrightig
=d2
(d+1)2/parenleftig
(d+1)TrasTrsb
+K2
raTrsb+K2
rbTras+K2
raK2
rb/parenrightig
(364)
/an}bracketle{t/an}bracketle{ter/bardblTT
sTr/bardblb/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
2(d+1)/parenrightbigg3
2/parenleftig
K2
rsK2
rb+K2
rb+2Trsb/parenrightig
(365)
/an}bracketle{t/an}bracketle{ta/bardblTrTT
s/bardbler/an}bracketri}ht/an}bracketri}ht= 2/parenleftbiggd
2(d+1)/parenrightbigg3
2/parenleftig
K2
rsK2
ra+K2
ra+2Tras/parenrightig
(366)
/an}bracketle{t/an}bracketle{ter/bardblTT