# Leaks: Quantum, Classical, Intermediate and More

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### Related Work

## 2. Process Theories with Discarding...

- the boxes that appear in the diagram and
- how these boxes are wired together, including the overall ordering of inputs/outputs.

**Remark**

**1.**

**Example**

**1**

**Example**

**2**

**Example**

**3**

**Example**

**4**

## 3. ... and Leaks

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Remark**

**2.**

## 4. Quality of a Leak

**Example**

**5.**

## 5. A Representation for All Classical-Quantum Leaks

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 6. The Leak-Construction

**Theorem**

**1.**

- systems stay the same;
- one restricts processes to those of the form:

**Proof.**

**Example**

**6**

**Example**

**7**

**Example**

**8.**

**Theorem**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

## 7. Process-Purity from Leaks

**Proposition**

**6.**

**Proof.**

**Definition**

**2.**

**Example**

**9.**

**Example**

**10**

**Example**

**11**

**Proof.**

**Example**

**12.**

**Proof.**

**Example**

**13**

**Proof.**

**Proposition**

**7.**

**Proof.**

## 8. Conclusions

- that quantum theory is a leak-free theory, whilst classical theory is maximally leaking, giving a clear separation between the theories for which quantum theory is optimal.
- how to construct sub-theories via a “leak construction”, which can be thought of as the sub-theories that can be obtained from a dynamical decoherence mechanism. For quantum theory, we can obtain classical theory, composite quantum classical theory and, generally, finite dimensional C*-algebras from this construction [9].
- a characterisation of the leaks and pure processes for quantum, classical and composite systems; in particular, we demonstrate that there is no pure way to transform quantum systems into classical systems or vice versa.
- that leaks are essential to defining purity of processes; we therefore introduce a novel definition of purity of processes, which makes sense both for quantum theory and for classical theory.

#### Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Mathematical Tools for Proofs

**Definition**

**A1.**

**Definition**

**A2.**

**Definition**

**A3.**

## Appendix B. Dilations of Classical Processes

## Appendix C. Pure Quantum-Classical Composite Processes

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**MDPI and ACS Style**

Selby, J.; Coecke, B.
Leaks: Quantum, Classical, Intermediate and More. *Entropy* **2017**, *19*, 174.
https://doi.org/10.3390/e19040174

**AMA Style**

Selby J, Coecke B.
Leaks: Quantum, Classical, Intermediate and More. *Entropy*. 2017; 19(4):174.
https://doi.org/10.3390/e19040174

**Chicago/Turabian Style**

Selby, John, and Bob Coecke.
2017. "Leaks: Quantum, Classical, Intermediate and More" *Entropy* 19, no. 4: 174.
https://doi.org/10.3390/e19040174