alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
srK2
sb=d
(d+1)/parenleftigg
d+2
d+1/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg
+d
(d+1)3δab+d+2
(d+1)3/parenrightigg
(406)
wherewe usedEq. ( 23) to derivethe ﬁrstexpression. Substituting these expressions
into Eq. ( 402) and using
/an}bracketle{t/an}bracketle{ta/bardblQT
r/bardblb/an}bracketri}ht/an}bracketri}ht=d+1
d/parenleftigg
Trba−/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg/parenleftigg/radicalbigg
d−1
d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1
d/parenrightigg/parenrightigg
(407)
we deduce Eq. ( 394). Taking complex conjugates on both sides we obtain Eq. ( 395).
Eq. (396) is an immediate consequence of Eq. ( 392) and the fact that /bardblgsr/an}bracketri}ht/an}bracketri}ht=
−/bardblgrs/an}bracketri}ht/an}bracketri}htfor allr,s.49
To prove Eq. ( 397) observe that it follows from Eqs. ( 394)–(396) that
d2/summationdisplay
s=1
(s/negationslash=r)/parenleftig
/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
sr/bardbl+/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl/parenrightig
=d2/summationdisplay
s=1
(s/negationslash=r)/parenleftig
2/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl−/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
sr/bardbl/parenrightig
= 2/parenleftig
¯Rr−(d−1)/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl
−1
d+1/parenleftig
I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig/parenrightbigg
(408)
Hence
2
d−3d2/summationdisplay
s=1
(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=1
d−3d2/summationdisplay
s=1
(s/negationslash=r)/parenleftig
/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl+/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
sr/bardbl
−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
sr/bardbl−/bardblf∗
sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗
sr/bardbl/parenrightig
=¯Rr+4(d−1)
d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4
(d+1)(d−3)/parenleftig
I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig
(409)
/square
9.TheP-PTProperty
In the preceding sections the Q-QTproperty has played a prominent role. In
this section we show that in the particular case ofa Weyl-Heisenberg covariantSIC-
POVM, and with the appropriate choice of gauge, the Gram project or (deﬁned in
Eq. (63)) has an analogous property, which we call the P-PTproperty. Speciﬁcally
one has
PPT=PTP=/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (410)
where/bardblh/an}bracketri}ht/an}bracketri}htis a normalized vector whose components in the standard basis are a ll
real. In odd dimensions the components of /bardblh/an}bracketri}ht/an}bracketri}htin the standard basis can be simply
expressed in terms of the Wigner function of the ﬁducial vector. I t could be said
thattheprojectors PandPTarealmostorthogonal(bycontrastwiththeprojectors
QrandQT
rwhich are completely orthogonal). More precisely Phas the spectral
decomposition
P=¯P+/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (411)
where¯Pis a rank (d−1) projector with the property
¯P¯PT= 0 (412)
This means that the matrix
JP=P−PT(413)
is a pure imaginary Hermitian matrix with the property that J2
Pis a real rank
2d−2 projector ( c.f.the discussion in Section 4).
Although we are mainly interested in the P-PTproperty as it applies to SIC-
POVMs, itshould benoted that itactuallyholdsforanyWeyl-Heisenbe rgcovariant
POVM (with the appropriate choice of gauge). So we will prove the ab ove propo-
sitions for this more general case.50
Let us begin by ﬁxing notation. Let \|0/an}bracketri}ht,...,\|d−1/an}bracketri}htbe an orthonormal basis for
d-dimensional Hilbert space and let XandZbe the operators whose action on the
\|r/an}bracketri}htis

srK2

sb=d

(d+1)/parenleftigg

d+2

d+1/parenleftigg/radicalbigg

d−1

d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1

d/parenrightigg/parenleftigg/radicalbigg

d−1

d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1

d/parenrightigg

+d

(d+1)3δab+d+2

(d+1)3/parenrightigg

(406)

wherewe usedEq. ( 23) to derivethe ﬁrstexpression. Substituting these expressions

into Eq. ( 402) and using

/an}bracketle{t/an}bracketle{ta/bardblQT

r/bardblb/an}bracketri}ht/an}bracketri}ht=d+1

d/parenleftigg

Trba−/parenleftigg/radicalbigg

d−1

d+1/an}bracketle{t/an}bracketle{ta/bardbl¯er/an}bracketri}ht/an}bracketri}ht+1

d/parenrightigg/parenleftigg/radicalbigg

d−1

d+1/an}bracketle{t/an}bracketle{t¯er/bardblb/an}bracketri}ht/an}bracketri}ht+1

d/parenrightigg/parenrightigg

(407)

we deduce Eq. ( 394). Taking complex conjugates on both sides we obtain Eq. ( 395).

Eq. (396) is an immediate consequence of Eq. ( 392) and the fact that /bardblgsr/an}bracketri}ht/an}bracketri}ht=

−/bardblgrs/an}bracketri}ht/an}bracketri}htfor allr,s.49

To prove Eq. ( 397) observe that it follows from Eqs. ( 394)–(396) that

d2/summationdisplay

s=1

(s/negationslash=r)/parenleftig

/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗

sr/bardbl+/bardblf∗

sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl/parenrightig

=d2/summationdisplay

s=1

(s/negationslash=r)/parenleftig

2/bardblgsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tgsr/bardbl−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl−/bardblf∗

sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗

sr/bardbl/parenrightig

= 2/parenleftig

¯Rr−(d−1)/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl

−1

d+1/parenleftig

I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig/parenrightbigg

(408)

Hence

2

d−3d2/summationdisplay

s=1

(s/negationslash=r)/bardbl¯gsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯gsr/bardbl=1

d−3d2/summationdisplay

s=1

(s/negationslash=r)/parenleftig

/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tfsr/bardbl+/bardblf∗

sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗

sr/bardbl

−/bardblfsr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗

sr/bardbl−/bardblf∗

sr/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tf∗

sr/bardbl/parenrightig

=¯Rr+4(d−1)

d−3/bardbl¯er/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{t¯er/bardbl+4

(d+1)(d−3)/parenleftig

I−/bardblv0/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{tv0/bardbl/parenrightig

(409)

/square

9.TheP-PTProperty

In the preceding sections the Q-QTproperty has played a prominent role. In

this section we show that in the particular case ofa Weyl-Heisenberg covariantSIC-

POVM, and with the appropriate choice of gauge, the Gram project or (deﬁned in

Eq. (63)) has an analogous property, which we call the P-PTproperty. Speciﬁcally

one has

PPT=PTP=/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (410)

where/bardblh/an}bracketri}ht/an}bracketri}htis a normalized vector whose components in the standard basis are a ll

real. In odd dimensions the components of /bardblh/an}bracketri}ht/an}bracketri}htin the standard basis can be simply

expressed in terms of the Wigner function of the ﬁducial vector. I t could be said

thattheprojectors PandPTarealmostorthogonal(bycontrastwiththeprojectors

QrandQT

rwhich are completely orthogonal). More precisely Phas the spectral

decomposition

P=¯P+/bardblh/an}bracketri}ht/an}bracketri}ht/an}bracketle{t/an}bracketle{th/bardbl (411)

where¯Pis a rank (d−1) projector with the property

¯P¯PT= 0 (412)

This means that the matrix

JP=P−PT(413)

is a pure imaginary Hermitian matrix with the property that J2

Pis a real rank

2d−2 projector ( c.f.the discussion in Section 4).

Although we are mainly interested in the P-PTproperty as it applies to SIC-

POVMs, itshould benoted that itactuallyholdsforanyWeyl-Heisenbe rgcovariant

POVM (with the appropriate choice of gauge). So we will prove the ab ove propo-

sitions for this more general case.50

Let us begin by ﬁxing notation. Let |0/an}bracketri}ht,...,|d−1/an}bracketri}htbe an orthonormal basis for

d-dimensional Hilbert space and let XandZbe the operators whose action on the

|r/an}bracketri}htis