# Causal and compositional structure of unitary transformations

**Authors:** Robin Lorenz, Jonathan Barrett  
**arXiv:** [2001.07774](https://arxiv.org/abs/2001.07774)  
**Date:** 2020-01-21  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2001.07774v1/

## Key points

- Causal structure is crucial in determining influence between input and output subsystems in unitary transformations.
- In a quantum circuit, lack of path from input system A to output system B means A cannot influence B.
- Certain unitaries may not have a circuit decomposition revealing all causal constraints.
- 'Extended circuit diagrams' formalism introduced to address limitations, incorporating direct sum structures alongside traditional compositions.
- Causally faithful extended circuit decomposition ensures clear path from input to output only if there is actual influence.
- Hypothesis suggests every finite-dimensional unitary transformation can have causally faithful extended circuit decomposition.
- Understanding causal and compositional structure through extended circuit diagrams provides insights into relationships within quantum systems.

## Summary

In the study of unitary transformations, the causal structure plays a crucial role in determining the possible influence between input and output subsystems. This causal structure can be analyzed in terms of the compositional structure of the unitary transformation itself. It has been observed that in a quantum circuit, if there is no path from an input system A to an output system B, then system A cannot influence system B. Conversely, if a unitary U exhibits a no-influence relation from input A to output B, it implies that there exists a circuit decomposition of U with no path from A to B. However, it has been noted that certain unitaries do not have a circuit decomposition that simultaneously reveals all causal constraints. To address this limitation, a new formalism of 'extended circuit diagrams' has been introduced. This extended formalism goes beyond traditional quantum circuits by incorporating direct sum structures alongside sequential and tensor product compositions. A causally faithful extended circuit decomposition for a unitary U is one where there is a clear path from an input A to an output B only if there is actual influence from A to B in U. By deriving causally faithful extended circuit decompositions for a broad class of unitaries, it has been shown that the decomposition aligns with the respective causal structure of each unitary. Furthermore, there is a hypothesis put forward that every finite-dimensional unitary transformation can have a causally faithful extended circuit decomposition. This suggests that understanding the causal and compositional structure of unitary transformations through extended circuit diagrams can provide valuable insights into the intricate relationships within quantum systems. Additionally, extending this formalism to non-unitary channels and higher-order maps could offer further advancements in comprehending the causal structure within quantum models.