Tracing the Loop: Non-Causal Computation, Partial Traces, & Postselected Entanglement
Abstract
This paper gives a categorical interpretation of Baumeler \& Wolf's logically consistent non-causal circuits, connecting them to postselected quantum teleportation. Looped feedback is represented by a trace in the category of non-negative matrices, and it is shown that the traced process is stochastic precisely when the induced loop transition matrix has trace $1$ for every external input, a condition shown to be equivalent to a unique fixed point for the loop for each input to a deterministic circuit. A classical non-causal circuit is represented by a measure-and-prepare quantum channel with an internal register utilising a maximally entangled Bell-state with postselection. The main result is that classical logical consistency is equivalent to the postselected Bell outcome having, for loop dimension d, probability exactly 1/d2 for each classical input distribution. The subsequent normalised conditional output then agrees exactly with the classical categorical trace. This work identifies a class of quantum Bell-postselection constructions whose conditional evolution maintains linear dependence on classical input distributions.
Related
- What Moved the Alarm Score? Causal Attribution with Exact Computation and Scarce-Incident Estimation
- Partial Invariance for Classification under Structural Causal Models
- Optimal Experiments for Partial Causal Effect Identification
- Unlocking Exponential and Unbounded Robust Gains in Shannon Capacity of Classical Multiple Access Channels with Causal CSIT via Quantum Entanglement Assistance
- Causal effect estimation for ordinal outcomes: a performance comparison of G-computation, IPW, and TMLE-based methods.
BibTeX
@misc{carney2026tracing,
title = {{Tracing the Loop: Non-Causal Computation, Partial Traces, & Postselected Entanglement}},
author = {Mark Carney},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08716},
eprint = {2609.08716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08716},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tracing the Loop: Non-Causal Computation, Partial Traces, & Postselected Entanglement
AU - Mark Carney
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.08716
UR - https://arxiv.org/abs/2609.08716
AB - This paper gives a categorical interpretation of Baumeler \& Wolf's logically consistent non-causal circuits, connecting them to postselected quantum teleportation. Looped feedback is represented by a trace in the category of non-negative matrices, and it is shown that the traced process is stochastic precisely when the induced loop transition matrix has trace $1$ for every external input, a condition shown to be equivalent to a unique fixed point for the loop for each input to a deterministic circuit. A classical non-causal circuit is represented by a measure-and-prepare quantum channel with an internal register utilising a maximally entangled Bell-state with postselection. The main result is that classical logical consistency is equivalent to the postselected Bell outcome having, for loop dimension d, probability exactly 1/d^2 for each classical input distribution. The subsequent normalised conditional output then agrees exactly with the classical categorical trace. This work identifies a class of quantum Bell-postselection constructions whose conditional evolution maintains linear dependence on classical input distributions.
ER -