RepLAB: A Computational/Numerical
Approach to Representation Theory
Denis Rosset, Felipe Montealegre-Mora, and Jean-Daniel Bancal
Abstract We present a MATLAB/Octave toolbox to decompose finite dimensional
representations of compact groups. Surprisingly, little information about the group
and the representation is needed to perform that task. We discuss applications to
semidefinite programming.
Keywords Representation theory · Compact groups
Early in the development of quantum formalism, some saw group theory as a mere
nuisance. The label Gruppenpest is attributed to Pauli’s talent for derision [1].
Today, the usefulness of group theory is no longer debated, usually a handful
of groups are employed: symmetric, cyclic, Pauli and Clifford groups, and the
(special) unitary group. It turns out that a larger variety of groups are present
in quantum information computations. Thousands of Bell inequalities [2] have
been discovered [3]: most with some form of symmetry [4]. And even a well
understood problem (Schur-Weyl duality) required recent extensive work [5–9]
to cater for partial transposition. In parallel, quantum information has seen the
rise of numerical methods, especially those based on semidefinite positive (SDP)
optimization. Density operators correspond naturally to SDP matrices, and so
do a variety of quantum objects [10–12]; see, for example, SDP relaxations to
test entanglement [13–16]. Moment-based (SDP) relaxations bound the set of
quantum correlations [17–21]. Such SDP relaxations inherit the symmetries of the
objects studied. We thus need an approach to decompose representations of a large
class of symmetry groups. This paper presents the first compact, self-contained
D. Rosset ()
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
F. Montealegre-Mora
Institute for Theoretical Physics, Universität zu Köln, Cologne, Germany
e-mail: fmonteal@thp.uni-koeln.de
J.-D. Bancal
Département de Physique Appliquée, Université de Genève, Genève, Switzerland
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_60
643
Approach to Representation Theory
Denis Rosset, Felipe Montealegre-Mora, and Jean-Daniel Bancal
Abstract We present a MATLAB/Octave toolbox to decompose finite dimensional
representations of compact groups. Surprisingly, little information about the group
and the representation is needed to perform that task. We discuss applications to
semidefinite programming.
Keywords Representation theory · Compact groups
Early in the development of quantum formalism, some saw group theory as a mere
nuisance. The label Gruppenpest is attributed to Pauli’s talent for derision [1].
Today, the usefulness of group theory is no longer debated, usually a handful
of groups are employed: symmetric, cyclic, Pauli and Clifford groups, and the
(special) unitary group. It turns out that a larger variety of groups are present
in quantum information computations. Thousands of Bell inequalities [2] have
been discovered [3]: most with some form of symmetry [4]. And even a well
understood problem (Schur-Weyl duality) required recent extensive work [5–9]
to cater for partial transposition. In parallel, quantum information has seen the
rise of numerical methods, especially those based on semidefinite positive (SDP)
optimization. Density operators correspond naturally to SDP matrices, and so
do a variety of quantum objects [10–12]; see, for example, SDP relaxations to
test entanglement [13–16]. Moment-based (SDP) relaxations bound the set of
quantum correlations [17–21]. Such SDP relaxations inherit the symmetries of the
objects studied. We thus need an approach to decompose representations of a large
class of symmetry groups. This paper presents the first compact, self-contained
D. Rosset ()
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada
F. Montealegre-Mora
Institute for Theoretical Physics, Universität zu Köln, Cologne, Germany
e-mail: fmonteal@thp.uni-koeln.de
J.-D. Bancal
Département de Physique Appliquée, Université de Genève, Genève, Switzerland
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_60
643