alx-ai/arxiv_papers · Datasets at Hugging Face

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challenges . PhD thesis, LMU M¨ unchen: Faculty of Physics, 2009.
http://edoc.ub.uni-muenchen.de/10474/.
[51] L. Bedulli and L. Vezzoni, The Ricci tensor of SU(3)-manifolds ,J. Geom. Phys. 57(2007)
1125–1146, [ math/0606786 ].
[52] M. J. Duﬀ, B. E. W. Nilsson, and C. N. Pope, Spontaneous supersymmetry breaking by the
squashed seven-sphere ,Phys. Rev. Lett. 50(1983) 2043.
[53] F. M¨ uller-Hoissen and R. St¨ uckl, Coset spaces and ten-dimensional uniﬁed theories ,Class.
Quant. Grav. 5(1988) 27.
[54] P. van Nieuwenhuizen, General theory of coset manifolds and antisymmetric tensor s applied
to Kaluza-Klein supergravity , inSupersymmetry and supergravity ’84 , World Scientiﬁc,
Singapore, 1985.
[55] T. W. Grimm and J. Louis, The eﬀective action of N= 1Calabi-Yau orientifolds ,Nucl.
Phys.B699(2004) 387–426, [ hep-th/0403067 ].
[56] T. W. Grimm and J. Louis, The eﬀective action of type IIA Calabi-Yau orientifolds ,Nucl.
Phys.B718(2005) 153–202, [ hep-th/0412277 ].
[57] I. Benmachiche and T. W. Grimm, Generalized N= 1orientifold compactiﬁcations and the
Hitchin functionals ,Nucl. Phys. B748(2006) 200–252, [ hep-th/0602241 ].
[58] P. Koerber and L. Martucci, From ten to four and back again: how to generalize the
geometry ,JHEP08(2007) 059, [ arXiv:0707.1038 ].
– 20 –[59] D. Cassani, Reducing democratic type II supergravity on SU(3) ×SU(3) structures ,JHEP06
(2008) 027, [ arXiv:0804.0595 ].
[60] J. Louis and A. Micu, Type II theories compactiﬁed on Calabi-Yau threefolds in th e presence
of background ﬂuxes ,Nucl. Phys. B635(2002) 395–431, [ hep-th/0202168 ].
[61] S. Gurrieri, J. Louis, A. Micu, and D. Waldram, Mirror symmetry in generalized Calabi-Yau
compactiﬁcations ,Nucl. Phys. B654(2003) 61–113, [ hep-th/0211102 ].
[62] U. H. Danielsson, P. Koerber, and T. Van Riet, Universal de Sitter solutions at tree-level ,
arXiv:1003.3590 .
[63] N. Hitchin, The geometry of three-forms in six and seven dimensions ,math/0010054 .
[64] S. Chiossi and S. Salamon, The intrinsic torsion of su(3) and g 2structures ,Ann. Mat. Pura
e Appl.282(1980) 35–58, [ math/0202282 ].
[65] E. Bergshoeﬀ, R. Kallosh, T. Ort´ ın, D. Roest, and A. Van Proe yen,New formulations of
D= 10supersymmetry and D8-O8 domain walls ,Class. Quant. Grav. 18(2001) 3359–3382,
[hep-th/0103233 ].
– 21 –
arXiv:1001.0004v1 [quant-ph] 31 Dec 2009The Lie Algebraic Signiﬁcance of
Symmetric Informationally Complete Measurements
D.M. Appleby, Steven T. Flammia and Christopher A. Fuchs
Perimeter Institute for Theoretical Physics
Waterloo, Ontario N2L 2Y5, Canada
December 30, 2009
Abstract
Examplesofsymmetric informationallycomplete positiveoperatorva lued mea-
sures (SIC-POVMs) have been constructed in every dimension ≤67. However,
it remains an open question whether they exist in all ﬁnite dimensions. A SIC-
POVM is usually thought of as a highly symmetric structure in quantum state
space. However, its elements can equally well be regarded as a basis for the Lie
algebra gl(d,C). In this paper we examine the resulting structure constants,
which are calculated from the traces of the triple products of the S IC-POVM
elements and which, it turns out, characterize the SIC-POVM up to unitary
equivalence. We show that the structure constants have numero us remarkable
properties. In particular we show that the existence of a SIC-POV M in di-
mensiondis equivalent to the existence of a certain structure in the adjoint
representation of gl( d,C). We hope that transforming the problem in this way,
from a question about quantum state space to a question about Lie algebras,
may help to make the existence problem tractable.
Contents
1. Introduction 1
2. The Angle Tensors 7
3. Spectral Decompositions 14
4. TheQ-QTProperty 18
5. Lie Algebraic Formulation of the Existence Problem 21
6. The Algebra sl( d,C) 31
7. Further Identities 33
8. Geometrical Considerations 36
9. TheP-PTProperty 49
10. Conclusion 52
11. Acknowledgements 53
References 531
1.Introduction
Symmetric informationally complete positive operator-valued measu res (SIC-
POVMs) present us with what is, simultaneously, one of the most inte resting, and
one of the most diﬃcult and tantalizing problems in quantum informatio n [1–46].
SIC-POVMs are important practically, with applications to quantum t omography
and cryptography [ 4,8,12,15,20,29], and to classical signal processing [ 24,36].
However, without in any way wishing to impugn the signiﬁcance of the a pplications
which have so far been proposed, it appears to us that the interes t of SIC-POVMs
stems less from these particular proposed uses than from rather broader, more gen-
eral considerations: the sense one gets that SICs are telling us so mething deep,
and hitherto unsuspected about the structure of quantum stat e space. In spite of
its being the central object about which the rest of quantum mech anics rotates,
and notwithstanding the eﬀorts of numerous investigators [ 47], the geometry of
quantum state space continues to be surprisingly ill-understood. T he hope which
inspires our eﬀorts is that a solution to the SIC problem will prove to b e the key,
not just to SIC-POVMs narrowly conceived, but to the geometry o f state space in
general. Such things are, by nature, unpredictable. However, it is not unreasonable
to speculate that a better theoretical understanding of the geo metry of quantum
state space might have important practical consequences: not o nly the applica-
tions listed above, but perhaps other applications which have yet to be conceived.
On a more foundational level one may hope that it will lead to a much imp roved
understanding of the conceptual message of quantum mechanics [7,43,45,48].
Having said why we describe the problem as interesting, let us now exp lain why
we describe it as tantalizing. The trouble is that, although there is an abundance of
reasons for suspecting that SIC-POVMs exist in every ﬁnite dimens ion (exact and
high-precision numerical examples [ 1,2,5,11,16,19,28,39,46] having now been
constructed in every dimension up to 67), and in spite of the intense eﬀorts of many
people [1–46] extending over a period of more than ten years, a general existe nce
proof continues to elude us. In their seminal paper on the subject , published in
2004, Renes et al[5] say “A rigorous proof of existence of SIC-POVMs in all ﬁnite
dimensions seems tantalizingly close, yet remains somehow distant.” T hey could
have said the same if they were writing today.

challenges . PhD thesis, LMU M¨ unchen: Faculty of Physics, 2009.

http://edoc.ub.uni-muenchen.de/10474/.

[51] L. Bedulli and L. Vezzoni, The Ricci tensor of SU(3)-manifolds ,J. Geom. Phys. 57(2007)

1125–1146, [ math/0606786 ].

[52] M. J. Duﬀ, B. E. W. Nilsson, and C. N. Pope, Spontaneous supersymmetry breaking by the

squashed seven-sphere ,Phys. Rev. Lett. 50(1983) 2043.

[53] F. M¨ uller-Hoissen and R. St¨ uckl, Coset spaces and ten-dimensional uniﬁed theories ,Class.

Quant. Grav. 5(1988) 27.

[54] P. van Nieuwenhuizen, General theory of coset manifolds and antisymmetric tensor s applied

to Kaluza-Klein supergravity , inSupersymmetry and supergravity ’84 , World Scientiﬁc,

Singapore, 1985.

[55] T. W. Grimm and J. Louis, The eﬀective action of N= 1Calabi-Yau orientifolds ,Nucl.

Phys.B699(2004) 387–426, [ hep-th/0403067 ].

[56] T. W. Grimm and J. Louis, The eﬀective action of type IIA Calabi-Yau orientifolds ,Nucl.

Phys.B718(2005) 153–202, [ hep-th/0412277 ].

[57] I. Benmachiche and T. W. Grimm, Generalized N= 1orientifold compactiﬁcations and the

Hitchin functionals ,Nucl. Phys. B748(2006) 200–252, [ hep-th/0602241 ].

[58] P. Koerber and L. Martucci, From ten to four and back again: how to generalize the

geometry ,JHEP08(2007) 059, [ arXiv:0707.1038 ].

– 20 –[59] D. Cassani, Reducing democratic type II supergravity on SU(3) ×SU(3) structures ,JHEP06

(2008) 027, [ arXiv:0804.0595 ].

[60] J. Louis and A. Micu, Type II theories compactiﬁed on Calabi-Yau threefolds in th e presence

of background ﬂuxes ,Nucl. Phys. B635(2002) 395–431, [ hep-th/0202168 ].

[61] S. Gurrieri, J. Louis, A. Micu, and D. Waldram, Mirror symmetry in generalized Calabi-Yau

compactiﬁcations ,Nucl. Phys. B654(2003) 61–113, [ hep-th/0211102 ].

[62] U. H. Danielsson, P. Koerber, and T. Van Riet, Universal de Sitter solutions at tree-level ,

arXiv:1003.3590 .

[63] N. Hitchin, The geometry of three-forms in six and seven dimensions ,math/0010054 .

[64] S. Chiossi and S. Salamon, The intrinsic torsion of su(3) and g 2structures ,Ann. Mat. Pura

e Appl.282(1980) 35–58, [ math/0202282 ].

[65] E. Bergshoeﬀ, R. Kallosh, T. Ort´ ın, D. Roest, and A. Van Proe yen,New formulations of

D= 10supersymmetry and D8-O8 domain walls ,Class. Quant. Grav. 18(2001) 3359–3382,

[hep-th/0103233 ].

– 21 –

arXiv:1001.0004v1 [quant-ph] 31 Dec 2009The Lie Algebraic Signiﬁcance of

Symmetric Informationally Complete Measurements

D.M. Appleby, Steven T. Flammia and Christopher A. Fuchs

Perimeter Institute for Theoretical Physics

Waterloo, Ontario N2L 2Y5, Canada

December 30, 2009

Abstract

Examplesofsymmetric informationallycomplete positiveoperatorva lued mea-

sures (SIC-POVMs) have been constructed in every dimension ≤67. However,

it remains an open question whether they exist in all ﬁnite dimensions. A SIC-

POVM is usually thought of as a highly symmetric structure in quantum state

space. However, its elements can equally well be regarded as a basis for the Lie

algebra gl(d,C). In this paper we examine the resulting structure constants,

which are calculated from the traces of the triple products of the S IC-POVM

elements and which, it turns out, characterize the SIC-POVM up to unitary

equivalence. We show that the structure constants have numero us remarkable

properties. In particular we show that the existence of a SIC-POV M in di-

mensiondis equivalent to the existence of a certain structure in the adjoint

representation of gl( d,C). We hope that transforming the problem in this way,

from a question about quantum state space to a question about Lie algebras,

may help to make the existence problem tractable.

Contents

1. Introduction 1

2. The Angle Tensors 7

3. Spectral Decompositions 14

4. TheQ-QTProperty 18

5. Lie Algebraic Formulation of the Existence Problem 21

6. The Algebra sl( d,C) 31

7. Further Identities 33

8. Geometrical Considerations 36

9. TheP-PTProperty 49

10. Conclusion 52

11. Acknowledgements 53

References 531

1.Introduction

Symmetric informationally complete positive operator-valued measu res (SIC-

POVMs) present us with what is, simultaneously, one of the most inte resting, and

one of the most diﬃcult and tantalizing problems in quantum informatio n [1–46].

SIC-POVMs are important practically, with applications to quantum t omography

and cryptography [ 4,8,12,15,20,29], and to classical signal processing [ 24,36].

However, without in any way wishing to impugn the signiﬁcance of the a pplications

which have so far been proposed, it appears to us that the interes t of SIC-POVMs

stems less from these particular proposed uses than from rather broader, more gen-

eral considerations: the sense one gets that SICs are telling us so mething deep,

and hitherto unsuspected about the structure of quantum stat e space. In spite of

its being the central object about which the rest of quantum mech anics rotates,

and notwithstanding the eﬀorts of numerous investigators [ 47], the geometry of

quantum state space continues to be surprisingly ill-understood. T he hope which

inspires our eﬀorts is that a solution to the SIC problem will prove to b e the key,

not just to SIC-POVMs narrowly conceived, but to the geometry o f state space in

general. Such things are, by nature, unpredictable. However, it is not unreasonable

to speculate that a better theoretical understanding of the geo metry of quantum

state space might have important practical consequences: not o nly the applica-

tions listed above, but perhaps other applications which have yet to be conceived.

On a more foundational level one may hope that it will lead to a much imp roved

understanding of the conceptual message of quantum mechanics [7,43,45,48].

Having said why we describe the problem as interesting, let us now exp lain why

we describe it as tantalizing. The trouble is that, although there is an abundance of

reasons for suspecting that SIC-POVMs exist in every ﬁnite dimens ion (exact and

high-precision numerical examples [ 1,2,5,11,16,19,28,39,46] having now been

constructed in every dimension up to 67), and in spite of the intense eﬀorts of many

people [1–46] extending over a period of more than ten years, a general existe nce

proof continues to elude us. In their seminal paper on the subject , published in

2004, Renes et al[5] say “A rigorous proof of existence of SIC-POVMs in all ﬁnite

dimensions seems tantalizingly close, yet remains somehow distant.” T hey could

have said the same if they were writing today.