Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group

Matthias Christandl, Graeme Mitchison

Abstract

Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples.

The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group

Abstract

Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples.

Paper Structure

This paper contains 7 sections, 7 theorems, 32 equations.

Table of Contents

  1. I. INTRODUCTION
  2. II. YOUNG DIAGRAMS AND THE SPECTRUM OF A DENSITY OPERATOR
  3. III. THE CONTENT EXPANSION
  4. IV. MAIN RESULT
  5. V. CONCLUSIONS
  6. ACKNOWLEDGEMENTS
  7. NOTE

Key Result

Theorem 1

Let $\rho$ be a density operator with spectrum $r=\text{spec}(\rho)$, and let $P_\lambda$ be the projection onto ${\cal U}_\lambda \otimes {\cal V}_\lambda$. Then with $D(p||q)=\sum_i p_i (\log p_i -\log q_i)$ the Kullback-Leibler distance of two normalised probability distributions $p$ and $q$. Note that $D(p||q)=0$ if and only if $p=q$.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 2 more