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Journal of Mathematical Physics / Volume 53 / Issue 1 / ARTICLES / Methods of Mathematical Physics
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J. Math. Phys. 53, 013501 (2012); http://dx.doi.org/10.1063/1.3672064 (22 pages)

Composite parameterization and Haar measure for all unitary and special unitary groups

Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr

Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

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(Received 14 June 2011; accepted 1 December 2011; published online 3 January 2012)

We adopt the concept of the composite parameterization of the unitary group U(d) to the special unitary group SU(d). Furthermore, we also consider the Haar measure in terms of the introduced parameters. We show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on U(d) and SU(d). With regard to possible applications of our results, we consider the computation of high-order integrals over unitary groups.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. COMPOSITE PARAMETERIZATION OF THE UNITARY GROUP U(d)
  3. COMPOSITE PARAMETERIZATION OF THE SPECIAL UNITARY GROUP SU(d)
    1. Remarks on the composite parameterization
  4. HAAR MEASURE ON THE UNITARY GROUP U(d)
  5. HAAR MEASURE ON THE SPECIAL UNITARY GROUP SU(d)
  6. REMARKS ON INTEGRALS OVER UNITARY GROUPS
    1. Example 1
    2. Example 2
    3. Example 3
  7. SUMMARY

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KEYWORDS and PACS

Keywords

Hilbert spaces, integration, SU(N) theory, unitary symmetry

PACS

  • 03.65.Fd

    Algebraic methods

  • 03.65.Ta

    Foundations of quantum mechanics; measurement theory

  • 02.30.Cj

    Measure and integration

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3672064

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

  1. These can be considered as generalizations of the Pauli matrix sigmay.
  2. The order of the product is [product]<sub>i = 1</sub><sup>N</sup>Ai=A1·A2[centered ellipsis]AN.
  3. Note that the indices m and n again run from 1 to d except that there is no lambdad,d.
  4. Intuitively: When changing from the orthogonal coordinates {uk} to the non-orthogonal coordinates {alphal} the infinitesimal volume element transforms as [product]<sub>k = 1</sub><sup>d[sup 2]</sup>duk=|det(([partial-derivative](u[sub 1],...,u[sub d[sup 2]]))/([partial-derivative](alpha[sub 1],...,alpha[sub d[sup 2]])))|[product]<sub>l = 1</sub><sup>d[sup 2]</sup>dalphal according to the Jacobian determinant.
  5. The independence of |detM1| on lambdam,n with min {m, n} >= 2 can also be confirmed by exploiting again the left and right invariance of the Haar measure.
  6. A basis for the vector space of operators [openface C]d×[openface C]d has d2 elements. The constraint detU=1 on special unitary operators implies that the trace of any derivative [partial-derivative]U/[partial-derivative]alphal with respect to an arbitrary parameter is always zero. Traceless operators form a d2 − 1 dimensional subspace of [openface C]d×[openface C]d. As we have d2 − 1 parameters {alphal} it is required to express the derivatives [partial-derivative]U/[partial-derivative]alphal in a basis of this subspace to obtain d2 − 1 linearly independent column vectors.
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Figures (click on thumbnails to view enlargements)

FIG.1
Schematic illustration of the Jacobian matrix (jk,l′) for the operator basis ( 34 ). All entries outside the gray-shaded blocks are zero.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
Schematic illustration of the matrix block math1. Only the gray-shaded entries are non-zero.

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
Schematic illustration of the Jacobian matrix (jk,l′) for the operator basis ( 85 ).

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
The average entanglement ( 122 ) of a bipartite qudit system measured by the (squared) concurrence C2(|ψ〉) for the dimensions d = 2, …, 12. The average 〈C2〉 increases with the size of the Hilbert space HA⊗HB = mathd⊗mathd.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint



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