alx-ai/arxiv_papers · Datasets at Hugging Face

text stringlengths 0 44.4k
/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=√
dW(−2−1p) (435)
where 2−1= (d+1)/2 is the multiplicative inverse of 2 considered as an element of
Zd:i.e.the unique integer 0 ≤m<dsuch that 2 m= 1 (modd).
10.Conclusion
A curious fact about SIC-POVMs is that, although they are charac terized by
their being highly symmetric structures, they do not wear this prop erty on their
sleeve (so to speak). If one casually inspects the components of a SIC-ﬁducial,
without knowing in advance that that is what they are, there does n ot seem to be
anything special about them at all. Indeed, so far from there being any obvious
pattern to the components, they seem, to a casual inspection, lik e a completely
random collection of numbers. Moreover, this is just as true of an e xact ﬁducial as
it is of a numerical one (see, for instance, the tabulations in Scott a nd Grassl [ 46]).
It is only when one looks at them through the right pair of spectacles , and takes the
trouble to calculate the overlaps Tr(Π rΠs), that the symmetry becomes apparent.
The situation is a little reminiscent of a hologram, which only takes on th e aspect of
a meaningful image when it is viewed in the right way. If one wanted to s ummarize
the content of this paper in a nutshell it could be said that we have ex hibited some
other pairs of spectacles—other ways of looking at a SIC—which cau se its inner
secrets (or at any rate some of its inner secrets) to become manif est.
Rather than focusing on the SIC-vectors \|ψr/an}bracketri}ht, as is usually done, we havefocused
on the angle tensors θrsandθrst, and on the T,JandRmatrices deﬁned in
terms of them. This is an important change of emphasis because, ra ther than
being tied to any particular SIC, these quantities characterize ent ire families of
unitarily equivalent SICs. Like the components of a SIC-ﬁducial, the angle tensors
appear, to a casual inspection, like a random collection of numbers. However, if
one examines the spectra of the T,JandRmatrices one realizes that, underlying
the appearance of randomness, there is a high degree of order. I f one then goes on
to examine the geometrical relationships between the subspaces o nto which the Q,
QTand¯Rmatrices project, as we did in Section 8, one ﬁnds yet more instances
of structure and order. To our minds what is particularly interestin g about all of53
this is that none of it is obviously suggested by the deﬁning property of a SIC, that
Tr(ΠrΠs) = 1/(d+1) forr/ne}ationslash=s.
In the course of this paper we have several times expressed the h ope that the
Lie algebraic perspective on a SIC will lead to a solution to the existenc e problem.
Of course, that is only a hope, and it may not be fulﬁlled. However, we feel on
rather safer ground when we suggest that the solution is likely to co me, if not from
this investigation, then from one which is like it to the extent that it fo cuses on a
feature of a SIC which is not immediately apparent to the eye.
Specializing to the case of a Weyl-Heisenberg covariant SIC, a ﬁducia l vector\|ψ/an}bracketri}ht
is a solution to the equations
/vextendsingle/vextendsingle/an}bracketle{tψ\|Dp\|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=dδp,0+1
d+1(436)
Allowing for the arbitrariness of the overall phase of \|ψ/an}bracketri}ht, and taking \|ψ/an}bracketri}htto be nor-
malized, this gives us d2−1 conditions on only 2 d−2 independent real parameters.
The equations are thus over-determined, and very highly over-de termined when d
is large. Nevertheless, they have turned out to be soluble in every c ase which has
been investigated to date. It seems likely that progress will depend on ﬁnding the
structural feature which is responsible for this remarkable fact. The motivation for
this paper is the belief that it may be structural features of the Lie algebra gl(d,C)
which are responsible. That suggestion may or may not be correct. But if it turns
out to be incorrect, the amount of eﬀort which has been expended on this problem
over a period of more than ten years, so far without fruit, sugges ts to us that the
solution will depend on ﬁnding some other structural feature of a S IC, which is not
obvious, and which has hitherto escaped attention.
11.Acknowledgements
The authors thank I. Bengtsson for discussions. Two authors, D MA and CAF,
were supported in part by the U. S. Oﬃce of Naval Research (Gran t No. N00014-
09-1-0247). Research at Perimeter Institute is supported by th e Government of
Canada through Industry Canada and by the Province of Ontario t hrough the
Ministry of Research & Innovation.
References
[1] S.G. Hoggar, Geom. Dedic. 69, 287 (1998).
[2] G. Zauner, “Quantum designs—foundations of a non-commutat ive theory of
designs” (in German), Ph.D. thesis, University of Vienna, 1999. Ava ilable on-
line athttp://www.mat.univie.ac.at/˜neum/papers/physpapers.html .
[3] C.M. Caves, “Symmetric Informationally Complete POVMs,” UNM
Information Physics Group internal report. Available online at
http://info.phys.unm.edu/caves/reports/reports.html (1999).
[4] C.A. Fuchs and M. Sasaki, Quant. Inf. Comp. 3, 277 (2003). Also available as
quant-ph/030292.
[5] J.M. Renes, R. Blume-Kohout, A.J. Scott and C.M. Caves, J. Math. Phys. 45,
2171 (2004). Also available as quant-ph/0310075.
[6] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum and Semicl. Optics
6, L19 (2005). Also available as math-ph/0403057.54
[7] C.A. Fuchs, Quantum Information and Computation 4, 467 (2004). Also avail-
able as quant-ph/0404122.
[8] J.ˇReh´ aˇ cek, B.-G. Englert and D. Kaszlikowski, Phys. Rev. A 70, 052321
(2004). Also available as quant-ph/0405084.
[9] W.K. Wootters, Found. Phys. 36, 112 (2006). Also available as quant-
ph/0406032.
[10] I. Bengtsson, quant-ph/0406174.
[11] M. Grassl, in Proceedings ERATO Conference on Quantum Information Sci-
ence 2004 (Tokyo, 2004). Also available as quant-ph/0406175.
[12] J.M. Renes, Quant. Inf. Comp. 5, 80 (2005). Also available as quant-
ph/0409043.
[13] I. Bengtsson and ˚A. Ericsson, Open Sys. and Information Dyn. 12, 187 (2005).
Also available as quant-ph/0410120.
[14] R. K¨ onig and R. Renner, J. Math. Phys. 46, 122108 (2005) Also available as
quant-ph/0410229.
[15] M. Ziman and V. Buˇ zek, Phys. Rev. A 72, 022343 (2005). Also available as
quant-ph/0411135.
[16] D.M. Appleby, J. Math. Phys. 46, 052107 (2005). Also available as quant-
ph/0412001.
[17] A. Klappenecker and M. R¨ otteler, Proc. 2005 IEEE International Sympo-
sium on Information Theory, Adelaide , 1740 (2005). Also available as quant-
ph/0502031.
[18] A. Klappenecker, M. R¨ otteler, I. Shparlinski and A. Winterho f,J. Math. Phys.
46, 082104 (2005). Also available as quant-ph/0503239.
[19] M. Grassl, Electronic Notes in Discrete Mathematics 20, 151 (2005).

/an}bracketle{t/an}bracketle{tp/bardblh/an}bracketri}ht/an}bracketri}ht=√

dW(−2−1p) (435)

where 2−1= (d+1)/2 is the multiplicative inverse of 2 considered as an element of

Zd:i.e.the unique integer 0 ≤m<dsuch that 2 m= 1 (modd).

10.Conclusion

A curious fact about SIC-POVMs is that, although they are charac terized by

their being highly symmetric structures, they do not wear this prop erty on their

sleeve (so to speak). If one casually inspects the components of a SIC-ﬁducial,

without knowing in advance that that is what they are, there does n ot seem to be

anything special about them at all. Indeed, so far from there being any obvious

pattern to the components, they seem, to a casual inspection, lik e a completely

random collection of numbers. Moreover, this is just as true of an e xact ﬁducial as

it is of a numerical one (see, for instance, the tabulations in Scott a nd Grassl [ 46]).

It is only when one looks at them through the right pair of spectacles , and takes the

trouble to calculate the overlaps Tr(Π rΠs), that the symmetry becomes apparent.

The situation is a little reminiscent of a hologram, which only takes on th e aspect of

a meaningful image when it is viewed in the right way. If one wanted to s ummarize

the content of this paper in a nutshell it could be said that we have ex hibited some

other pairs of spectacles—other ways of looking at a SIC—which cau se its inner

secrets (or at any rate some of its inner secrets) to become manif est.

Rather than focusing on the SIC-vectors |ψr/an}bracketri}ht, as is usually done, we havefocused

on the angle tensors θrsandθrst, and on the T,JandRmatrices deﬁned in

terms of them. This is an important change of emphasis because, ra ther than

being tied to any particular SIC, these quantities characterize ent ire families of

unitarily equivalent SICs. Like the components of a SIC-ﬁducial, the angle tensors

appear, to a casual inspection, like a random collection of numbers. However, if

one examines the spectra of the T,JandRmatrices one realizes that, underlying

the appearance of randomness, there is a high degree of order. I f one then goes on

to examine the geometrical relationships between the subspaces o nto which the Q,

QTand¯Rmatrices project, as we did in Section 8, one ﬁnds yet more instances

of structure and order. To our minds what is particularly interestin g about all of53

this is that none of it is obviously suggested by the deﬁning property of a SIC, that

Tr(ΠrΠs) = 1/(d+1) forr/ne}ationslash=s.

In the course of this paper we have several times expressed the h ope that the

Lie algebraic perspective on a SIC will lead to a solution to the existenc e problem.

Of course, that is only a hope, and it may not be fulﬁlled. However, we feel on

rather safer ground when we suggest that the solution is likely to co me, if not from

this investigation, then from one which is like it to the extent that it fo cuses on a

feature of a SIC which is not immediately apparent to the eye.

Specializing to the case of a Weyl-Heisenberg covariant SIC, a ﬁducia l vector|ψ/an}bracketri}ht

is a solution to the equations

/vextendsingle/vextendsingle/an}bracketle{tψ|Dp|ψ/an}bracketri}ht/vextendsingle/vextendsingle2=dδp,0+1

d+1(436)

Allowing for the arbitrariness of the overall phase of |ψ/an}bracketri}ht, and taking |ψ/an}bracketri}htto be nor-

malized, this gives us d2−1 conditions on only 2 d−2 independent real parameters.

The equations are thus over-determined, and very highly over-de termined when d

is large. Nevertheless, they have turned out to be soluble in every c ase which has

been investigated to date. It seems likely that progress will depend on ﬁnding the

structural feature which is responsible for this remarkable fact. The motivation for

this paper is the belief that it may be structural features of the Lie algebra gl(d,C)

which are responsible. That suggestion may or may not be correct. But if it turns

out to be incorrect, the amount of eﬀort which has been expended on this problem

over a period of more than ten years, so far without fruit, sugges ts to us that the

solution will depend on ﬁnding some other structural feature of a S IC, which is not

obvious, and which has hitherto escaped attention.

11.Acknowledgements

The authors thank I. Bengtsson for discussions. Two authors, D MA and CAF,

were supported in part by the U. S. Oﬃce of Naval Research (Gran t No. N00014-

09-1-0247). Research at Perimeter Institute is supported by th e Government of

Canada through Industry Canada and by the Province of Ontario t hrough the

Ministry of Research & Innovation.

References

[1] S.G. Hoggar, Geom. Dedic. 69, 287 (1998).

[2] G. Zauner, “Quantum designs—foundations of a non-commutat ive theory of

designs” (in German), Ph.D. thesis, University of Vienna, 1999. Ava ilable on-

line athttp://www.mat.univie.ac.at/˜neum/papers/physpapers.html .

[3] C.M. Caves, “Symmetric Informationally Complete POVMs,” UNM

Information Physics Group internal report. Available online at

http://info.phys.unm.edu/caves/reports/reports.html (1999).

[4] C.A. Fuchs and M. Sasaki, Quant. Inf. Comp. 3, 277 (2003). Also available as

quant-ph/030292.

[5] J.M. Renes, R. Blume-Kohout, A.J. Scott and C.M. Caves, J. Math. Phys. 45,

2171 (2004). Also available as quant-ph/0310075.

[6] M. Saniga, M. Planat and H. Rosu, J. Opt. B: Quantum and Semicl. Optics

6, L19 (2005). Also available as math-ph/0403057.54

[7] C.A. Fuchs, Quantum Information and Computation 4, 467 (2004). Also avail-

able as quant-ph/0404122.

[8] J.ˇReh´ aˇ cek, B.-G. Englert and D. Kaszlikowski, Phys. Rev. A 70, 052321

(2004). Also available as quant-ph/0405084.

[9] W.K. Wootters, Found. Phys. 36, 112 (2006). Also available as quant-

ph/0406032.

[10] I. Bengtsson, quant-ph/0406174.

[11] M. Grassl, in Proceedings ERATO Conference on Quantum Information Sci-

ence 2004 (Tokyo, 2004). Also available as quant-ph/0406175.

[12] J.M. Renes, Quant. Inf. Comp. 5, 80 (2005). Also available as quant-

ph/0409043.

[13] I. Bengtsson and ˚A. Ericsson, Open Sys. and Information Dyn. 12, 187 (2005).

Also available as quant-ph/0410120.

[14] R. K¨ onig and R. Renner, J. Math. Phys. 46, 122108 (2005) Also available as

quant-ph/0410229.

[15] M. Ziman and V. Buˇ zek, Phys. Rev. A 72, 022343 (2005). Also available as

quant-ph/0411135.

[16] D.M. Appleby, J. Math. Phys. 46, 052107 (2005). Also available as quant-

ph/0412001.

[17] A. Klappenecker and M. R¨ otteler, Proc. 2005 IEEE International Sympo-

sium on Information Theory, Adelaide , 1740 (2005). Also available as quant-

ph/0502031.

[18] A. Klappenecker, M. R¨ otteler, I. Shparlinski and A. Winterho f,J. Math. Phys.

46, 082104 (2005). Also available as quant-ph/0503239.

[19] M. Grassl, Electronic Notes in Discrete Mathematics 20, 151 (2005).