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Causality is rare: some topological properties of causal quantum channels

Robin Simmons

Abstract

Sorkin's impossible operations demonstrate that causality of a quantum channel in QFT is an additional constraint on quantum operations above and beyond the locality of the channel. What has not been shown in the literature so far is how much of a constraint it is. Here we answer this question in perhaps the strongest possible terms: the set of causal channels is nowhere dense in the set of local channels. We connect this result to quantum information, showing that the set of causal unitaries has Haar measure $0$ in the set of all unitaries acting on a lattice. Finally, we close with discussion on the implications and connections to recent QFT measurement models.

Causality is rare: some topological properties of causal quantum channels

Abstract

Sorkin's impossible operations demonstrate that causality of a quantum channel in QFT is an additional constraint on quantum operations above and beyond the locality of the channel. What has not been shown in the literature so far is how much of a constraint it is. Here we answer this question in perhaps the strongest possible terms: the set of causal channels is nowhere dense in the set of local channels. We connect this result to quantum information, showing that the set of causal unitaries has Haar measure in the set of all unitaries acting on a lattice. Finally, we close with discussion on the implications and connections to recent QFT measurement models.

Paper Structure

This paper contains 14 sections, 16 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Banach and operator spaces
  5. Banach spaces
  6. Operator spaces
  7. Quantum channels and topology
  8. Topologies on algebras and maps
  9. Quantum channels
  10. Closure and compactness
  11. QFTs and causality
  12. Causality is rare in QFT
  13. Unitary channels
  14. Conclusions

Key Result

Proposition 1

Let $n,m>1$ and define $\operatorname{U}(nm)\supseteq\operatorname{U}(n,m)\cong (\operatorname{U}(n)\times \operatorname{U}(m))/\operatorname{U}(1)$, then $\mu(\operatorname{U}(n,m))=0$ where $\mu$ is the Haar measure for $\operatorname{U}(nm)$.

Figures (2)

  • Figure 1: A spacetime scenario with $N=3$ spacelike separated systems. In this case, the first system locally prepares using $\Phi$, and the last two measure a joint observable $O$. We request that the effect of the preparation should be invisible to the expectation value of $U^\dagger OU$ if $U$ is causal.
  • Figure 2: The spacetime setup of a Sorkin scenario is shown, where $\mathbf{S}$ is before $\mathbf{K}$, which is before $\mathbf{R}$, yet $\mathbf{S}$ and $\mathbf{R}$ are spacelike separated. The lightcones are indicated by dashed lines.

Theorems & Definitions (30)

  • Proposition 1
  • proof
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • proof
  • Corollary 2
  • Proposition 3
  • proof
  • ...and 20 more