Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1202.3559

A remarkable representation of the Clifford group

Ingemar Bengtsson

Published Feb 16, 2012• 0 citations
arXivHTMLPDF

Abstract

The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group --- the automorphism group of the Heisenberg group --- is represented by monomial phase-permutation matrices. This has a beneficial influence on the amount of calculation that must be done to find Symmetric Informationally Complete POVMs. I make some comments on the equations obeyed by the absolute values of the components of the SIC vectors, and on the fact that the representation partly suggests a preferred tensor product structure.

DOI
10.48550/arxiv.1202.3559

Related papers

A remarkable representation of the Clifford group

Ingemar Bengtsson, Andrei Yu. Khrennikov, Gregg S. Jaeger +2 more

The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group—the automorphism group of the Heisenberg group—is represented by monomial phase‐permutation matrices. This has a beneficial influence on the amount of calculation that must be done to find Symmetric Informationally Complete POVMs. I make some comments on the equations obeyed by the absolute values of the components of the SIC vectors, and on the fact that the representation partly suggests a preferred tensor product structure.

AIP conference proceedings•Jan 1, 2011•1 citation

The monomial representations of the Clifford group

D. M. Appleby, Ingemar Bengtsson, STEPHEN D. BRIERLEY +3 more

We show that the Clifford group---the normaliser of the Weyl-Heisenberg group---can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.

arXiv:1102.1268•Quantum Information and Computation•May 1, 2012•19 citations

The monomanial representations of the Clifford group

Appleby, D Markus, Bengtsson, Ingemar, Brierley, Steve +3 more

We show that the Clifford group—the normaliser of the Weyl-Heisenberg group—can be represented by monomial phase-permutation matrices if and only if the dimension is a square number. This simplifies expressions for SIC vectors, and has other applications to SICs and to Mutually Unbiased Bases. Exact solutions for SICs in dimension 16 are presented for the first time.

•Jan 1, 2012•0 citations

Systems of imprimitivity for the Clifford group

Appleby, D.M., Bengtsson, I., Brierley, S. +3 more

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

•Mar 1, 2014•14 citations

Systems of Imprimitivity for the Clifford group

D. M. Appleby, Ingemar Bengtsson, STEPHEN D. BRIERLEY +3 more

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of $k$-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

arXiv:1210.1055•Quantum Information and Computation•Mar 1, 2014•10 citations

Structure of Clifford groups of composite finite quantum systems

Miroslav Korbelář, Jiří Tolar

In this paper the Clifford groups of general multipartite quantum systems with configuration space $\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}$, $k\geq 1$ are studied. It is known that the Clifford group is a natural semidirect product provided the dimension $N=n_1\cdots n_k$ of the corresponding Hilbert space is an odd number. For even $N$ special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group $\mathrm{Sp}_{[n_1,\dots,n_k]}$ we prove that for even $N=n_1\cdots n_k$ both Clifford group and the projective Clifford group are natural semidirect products if and only if $N$ is not divisible by four.

math-phCross-listedarXiv:2606.08215•Jun 9, 2026•0 citations
The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group --- the automorphism group of the Heisenberg group --- is represented by monomial phase-permutation matrices. This has a beneficial influence on the amount of calculation that must be done to find Symmetric Informationally Complete POVMs. I make some comments on the equations obeyed by the absolute values of the components of the SIC vectors, and on the fact that the representation partly suggests a preferred tensor product structure.
The finite Heisenberg group knows when the dimension of Hilbert space is a square number. Remarkably, it then admits a representation such that the entire Clifford group—the automorphism group of the Heisenberg group—is represented by monomial phase‐permutation matrices. This has a beneficial influence on the amount of calculation that must be done to find Symmetric Informationally Complete POVMs. I make some comments on the equations obeyed by the absolute values of the components of the SIC vectors, and on the fact that the representation partly suggests a preferred tensor product structure.
It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).
It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of kkk-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).
In this paper the Clifford groups of general multipartite quantum systems with configuration space Zn1⊕⋯⊕Znk\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}Zn1​​⊕⋯⊕Znk​​, k≥1k\geq 1k≥1 are studied. It is known that the Clifford group is a natural semidirect product provided the dimension N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ of the corresponding Hilbert space is an odd number. For even NNN special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group Sp[n1,…,nk]\mathrm{Sp}_{[n_1,\dots,n_k]}Sp[n1​,…,nk​]​ we prove that for even N=n1⋯nkN=n_1\cdots n_kN=n1​⋯nk​ both Clifford group and the projective Clifford group are natural semidirect products if and only if NNN is not divisible by four.