Causality is rare: some topological properties of causal quantum channels

TL;DR

This paper investigates the rarity of causal quantum channels, showing they form a negligible subset among local channels and unitaries, with implications for quantum field theory and quantum information.

Contribution

It demonstrates that causal quantum channels are extremely rare within the set of local channels, highlighting topological and measure-theoretic properties.

Findings

Causal channels are nowhere dense among local channels.

The set of causal unitaries has Haar measure zero among all unitaries.

Implications for quantum field theory measurement models.

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.