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Parity erasure: a foundational principle for indefinite causal order

Zixuan Liu, Ognyan Oreshkov

Abstract

Processes with indefinite causal order can arise when quantum theory is locally valid. Here, we identify an information-theoretic principle, termed parity erasure, that completely characterizes such processes. Our characterization does not rely on the formalism of quantum theory itself, but instead is derived from a set of axioms for general operational probabilistic theories, and thus holds also for a large class of theories beyond quantum theory. This informational approach reveals a fundamental property of information exchange in scenarios with indefinite causal structure.

Parity erasure: a foundational principle for indefinite causal order

Abstract

Processes with indefinite causal order can arise when quantum theory is locally valid. Here, we identify an information-theoretic principle, termed parity erasure, that completely characterizes such processes. Our characterization does not rely on the formalism of quantum theory itself, but instead is derived from a set of axioms for general operational probabilistic theories, and thus holds also for a large class of theories beyond quantum theory. This informational approach reveals a fundamental property of information exchange in scenarios with indefinite causal structure.

Paper Structure

This paper contains 2 sections, 4 theorems, 10 equations, 2 figures.

Table of Contents

  1. Proof of Theorem \ref{['theo:weak']}
  2. Proof of Lemma \ref{['theo:induction']}

Key Result

Theorem 1

For any local input-output relation of a supermap in a causal OPT, and any nonempty subset $I$ of $\{1, \cdots, N\}$, the joint outcome $(a_i)_{i \in I}$ is independent of the parity $\bigoplus_{i \in I} x_i$.

Figures (2)

  • Figure 1: Two-party supermaps. (a) Supermaps are transformations of channels; (b) A local input-output relation of a supermap is the statistics of an experiment in which the experimenter in each laboratory measures their incoming system and prepares an encoding of a random bit on the outgoing systems.
  • Figure 2: One-way-signaling decomposability. Under this assumption, every channel satisfying the no-influence condition $X_1 \not\to A_1$ can be implemented by a circuit with a slot from $A_1$ to $X_1$.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1: Weak principle of parity erasure
  • Definition 2
  • Corollary 1: Strong principle of parity erasure
  • Lemma 1
  • Theorem 2