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The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography

Matthias Christandl

TL;DR

This work develops a dual perspective on bipartite quantum states by uniting group representation theory with cryptographic entanglement measures. It proves an asymptotic equivalence between the spectral compatibility problem for bipartite states and the nonvanishing of Kronecker coefficients, leveraging Schur–Weyl duality and spectrum-estimation techniques. It further introduces squashed entanglement as a cryptographically motivated, additive, strongly subadditive entanglement measure, clarifying its role in quantum information tasks. Together, these results connect spectral constraints, representation-theoretic invariants, and entanglement quantification, with implications for quantum marginal problems and entropy inequalities.

Abstract

This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.

The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography

TL;DR

This work develops a dual perspective on bipartite quantum states by uniting group representation theory with cryptographic entanglement measures. It proves an asymptotic equivalence between the spectral compatibility problem for bipartite states and the nonvanishing of Kronecker coefficients, leveraging Schur–Weyl duality and spectrum-estimation techniques. It further introduces squashed entanglement as a cryptographically motivated, additive, strongly subadditive entanglement measure, clarifying its role in quantum information tasks. Together, these results connect spectral constraints, representation-theoretic invariants, and entanglement quantification, with implications for quantum marginal problems and entropy inequalities.

Abstract

This thesis presents a study of the structure of bipartite quantum states. In the first part, the representation theory of the unitary and symmetric groups is used to analyse the spectra of quantum states. In particular, it is shown how to derive a one-to-one relation between the spectra of a bipartite quantum state and its reduced states, and the Kronecker coefficients of the symmetric group. In the second part, the focus lies on the entanglement of bipartite quantum states. Drawing on an analogy between entanglement distillation and secret-key agreement in classical cryptography, a new entanglement measure, `squashed entanglement', is introduced.

Paper Structure

This paper contains 72 sections, 110 theorems, 507 equations, 16 figures, 5 tables.

Table of Contents

  1. Abstract
  2. Acknowledgements
  3. Foreword
  4. Preliminaries
  5. Insights from Group Theory
  6. The Symmetric and the Unitary Groups
  7. Schur-Weyl Duality I
  8. Minimal Projections and Irreducible Representations
  9. A Duality Theorem
  10. Schur-Weyl Duality
  11. The Irreducible Representations of the Symmetric Group
  12. The Young Symmetriser
  13. Combinatorics of Young Tableaux
  14. The Irreducible Representations of the Unitary Group
  15. Lie Groups and Lie Algebras
  16. ...and 57 more sections

Key Result

Lemma 4.2

Let $\{p_i, \rho_i\}$ and $\{q_i, \sigma_i\}$ be two ensembles of mixed states, then where $D(P||Q)=\sum_i p_i (\log p_i -\log q_i)$, the Kullback-Leibler distance of two probability distributions. Since $D(P||Q)=0$ for $P=Q$, this implies convexity of $S(\rho||\sigma)$.

Figures (16)

  • Figure 4.1: A state $\rho$ on system ${\cal H}$ is sent through a channel $\Lambda$ implemented by the unitary $U$. Ancilla systems are the lose ends of lines. The state on ${\cal H}\otimes {\cal K}$ is $|\psi\rangle$ and $\Lambda(\rho)$, the state on ${\cal H}'$, is the channel output.
  • Figure 5.2: Young frame $(3,2)$, top row: standard Young tableaux, bottom row: semistandard Young tableaux with numbering $\{1, 2\}$.
  • Figure 5.3: Hook of box $(1,2)$ in Young frame $(4,3,1)$
  • Figure 5.4: Clebsch-Gordan decomposition: $\hbox{spin} 2 \,\otimes \hbox{spin} 1= \hbox{spin} 3\, \oplus \hbox{spin} 2\, \oplus \hbox{spin} 1$
  • Figure 5.5: Building a semistandard Young tableau from three skew diagrams
  • ...and 11 more figures

Theorems & Definitions (125)

  • Definition 4.1
  • Lemma 4.2: Joint convexity of relative entropy
  • Lemma 4.3: Concavity of von Neumann entropy
  • Theorem 4.4: Holevo's bound Holevo73a
  • Lemma 4.5: OhyPet04Book
  • Lemma 4.6: Fannes' inequality Fannes73
  • Theorem 4.7
  • Lemma 4.8: Schur's lemma
  • Theorem 4.9
  • Theorem 4.10: Orthogonality Relations
  • ...and 115 more