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Invariance under quantum permutations rules out parastatistics

Manuel Mekonnen, Thomas D. Galley, Markus P. Mueller

Abstract

Quantum systems invariant under particle exchange are either Bosons or Fermions, even though quantum theory in principle admits more general behavior under permutations. But why do we not observe such "paraparticles" in nature? The analysis of this question was previously limited primarily to specific quantum field theory models. Here we give two independent arguments that rule out parastatistics universally, originating in quantum information theory and recent research on internal quantum reference frames. First, we introduce a notion of complete invariance: quantum systems should not only preserve their local state under permutations, but also the quantum information they carry about other systems, in analogy to the notion of complete positivity in quantum information theory. Second, we demand that quantum systems are invariant under quantum permutations, i.e. permutations conditioned on values of permutation-invariant observables. For both, we show that the respective principle is fulfilled if and only if the particle is a Boson or Fermion. Our results show how quantum reference frames can shed light on a longstanding problem of quantum physics, they underline the crucial role played by the compositional structure of quantum information, and demonstrate the explanatory power but also subtle limitations of recently proposed quantum covariance principles.

Invariance under quantum permutations rules out parastatistics

Abstract

Quantum systems invariant under particle exchange are either Bosons or Fermions, even though quantum theory in principle admits more general behavior under permutations. But why do we not observe such "paraparticles" in nature? The analysis of this question was previously limited primarily to specific quantum field theory models. Here we give two independent arguments that rule out parastatistics universally, originating in quantum information theory and recent research on internal quantum reference frames. First, we introduce a notion of complete invariance: quantum systems should not only preserve their local state under permutations, but also the quantum information they carry about other systems, in analogy to the notion of complete positivity in quantum information theory. Second, we demand that quantum systems are invariant under quantum permutations, i.e. permutations conditioned on values of permutation-invariant observables. For both, we show that the respective principle is fulfilled if and only if the particle is a Boson or Fermion. Our results show how quantum reference frames can shed light on a longstanding problem of quantum physics, they underline the crucial role played by the compositional structure of quantum information, and demonstrate the explanatory power but also subtle limitations of recently proposed quantum covariance principles.

Paper Structure

This paper contains 26 sections, 14 theorems, 119 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Standard permutation invariance and parastatistics
  3. Complete permutation invariance
  4. Quantum permutation invariance
  5. Consequences and outlook
  6. Methods
  7. Representation theory and permutation-invariance
  8. Formalism of complete invariance
  9. Formalism of quantum permutations
  10. Quantum reference frame descriptions
  11. Consequences of Schur's Lemma
  12. Descriptions of parastatistics
  13. Examples and classification
  14. Parastatistics in second quantization
  15. Complete invariance for compact groups
  16. ...and 11 more sections

Key Result

Theorem 1

A quantum state $\rho_S$ is completely invariant under permutations if and only if it is fully supported on either the Bosonic or the Fermionic subspace.

Figures (2)

  • Figure 1: Complete invariance. All physical predictions are invariant under permutations $\pi$ of the system of particles $S$. This includes the statistics and correlations of measurements of $S$ and any ancillary system $A$, even if $SA$ is potentially in a pure entangled state.
  • Figure 2: Invariance under quantum permutations. The system of particles $S$ is permuted "branchwise", i.e. conditioned on the value of a permutation-invariant observable. This must be possible in a way that preserves the statistics of all measurements on the system. We also consider what kinds of correlations with ancillary systems $A$ are allowed by global invariance of $SA$.

Theorems & Definitions (33)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • ...and 23 more