# Ruling out Higher-Order Interference from Purity Principles

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## Abstract

**:**

## 1. Introduction

## 2. Framework

#### 2.1. States, Transformations, and Effects

- $\mathsf{St}\left(\mathrm{A}\right)$ as the set of states of system $\mathrm{A}$,
- $\mathsf{Eff}\left(\mathrm{A}\right)$ as the set of effects on $\mathrm{A}$,
- $\mathsf{Transf}\left(\right)open="("\; close=")">\mathrm{A},\mathrm{B}$ as the set of transformations from $\mathrm{A}$ to $\mathrm{B}$, and $\mathsf{Transf}\left(\mathrm{A}\right)$ as the set of transformations from $\mathrm{A}$ to $\mathrm{A}$,
- $\mathcal{B}\circ \mathcal{A}$ (or $\mathcal{B}\mathcal{A}$, for short) as the sequential composition of two transformations $\mathcal{A}$ and $\mathcal{B}$, with the input of $\mathcal{B}$ matching the output of $\mathcal{A}$,
- $\mathcal{A}\otimes \mathcal{B}$ as the parallel composition (or tensor product) of the transformations $\mathcal{A}$ and $\mathcal{B}$.

**Definition**

**1.**

#### 2.2. Tests and Channels

**Definition**

**2.**

#### 2.3. Pure Transformations

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.4. Causality

**Axiom**

**1**

## 3. Higher-Order Interference

**Definition**

**6.**

**Definition**

**7.**

## 4. Sharp Theories with Purification

**Axiom**

**2**

**Axiom**

**3**

**Axiom**

**4**

#### Properties of Sharp Theories With Purifications

**Proposition**

**1.**

**Theorem**

**5.**

**Theorem**

**6**

**Proposition**

**2.**

**Proposition**

**3.**

## 5. Sharp Theories with Purification Have No Higher-Order Interference

#### 5.1. Self-Duality

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### 5.2. Existence of Pure Orthogonal Projectors

- ${F}_{\mathsf{I}}:=\left(\right)open="\{"\; close="\}">\rho \in {\mathsf{St}}_{1}\left(\mathrm{A}\right):\left(\right)open="("\; close=")">{a}_{\mathsf{I}}|\rho $;
- ${F}_{\mathsf{I}}^{\perp}:=\left(\right)open="\{"\; close="\}">\rho \in {\mathsf{St}}_{1}\left(\mathrm{A}\right):\left(\right)open="("\; close=")">{a}_{\mathsf{I}}|\rho $,

**Definition**

**8.**

- if $\rho \in {F}_{\mathsf{I}}$, then ${P}_{\mathsf{I}}\rho =\rho $;
- if $\rho \in {F}_{\mathsf{I}}^{\perp}$, then ${P}_{\mathsf{I}}\rho =0$.

**Proposition**

**6.**

**Proof.**

- ${\rho}^{\u2020}{P}_{\mathsf{I}}={\rho}^{\u2020}$ if $\rho \in {F}_{\mathsf{I}}$
- ${\rho}^{\u2020}{P}_{\mathsf{I}}=0$ if $\rho \in {F}_{\mathsf{I}}^{\perp}$

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

#### 5.3. Main Result

**Theorem**

**7.**

#### 5.4. Jordan-Algebraic Structure

**Theorem**

**8.**

**Proof.**

## 6. Discussion and Conclusions

#### Finding Higher Order Interference

- The transformations corresponding to blocking slits satisfy: ${T}_{\mathsf{I}}{T}_{\mathsf{J}}={T}_{\mathsf{I}\cap \mathsf{J}}$. By this we mean that they share several properties with the projectors ${P}_{\mathsf{I}}$ of Section 5: if we define the effects ${a}_{\mathsf{I}}=u{T}_{\mathsf{I}}$ and the faces ${F}_{\mathsf{I}}$ and ${F}_{\mathsf{I}}^{\perp}$ as in Section 5.2, i.e., as the 1-set and 0-set of ${a}_{\mathsf{I}}$, then the ${T}_{\mathsf{I}}$ are assumed to be orthogonal projectors in the sense of Definition 8, and to be both idempotent and “orthogonal” (${T}_{\mathsf{I}}{T}_{\mathsf{J}}=0$) if $\mathsf{I}$ and $\mathsf{J}$ are disjoint (as in Proposition 7).
- The ${T}_{\mathsf{I}}$’s map pure states to pure states
- The ${T}_{\mathsf{I}}$’s are self-adjoint.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Norms and Fidelity

#### Appendix A.1. Operational Norm and Dagger Norm

**Proposition**

**A1.**

**Proof.**

**Definition**

**A1.**

**Proposition**

**A2.**

**Proof.**

#### Appendix A.2. Dagger Fidelity

**Definition**

**A2.**

**Lemma**

**A1.**

**Proof.**

**Proposition**

**A3.**

- $0\le {F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $;
- ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $ if and only if ρ and σ are perfectly distinguishable;
- ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $ if and only if $\rho =\sigma $;
- ${F}_{\u2020}\left(\right)open="("\; close=")">\mathcal{U}\rho ,\mathcal{U}\sigma $, for every reversible channel $\mathcal{U}$.

**Proof.**

- Recall that $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma \ge 0$, whence ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $. Moreover, by Schwarz inequality, $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma $, so ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $.
- Suppose $\rho $ and $\sigma $ are perfectly distinguishable, then by Lemma A1 $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma $, implying ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $. Now suppose ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $; then $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma $. Let $\rho ={\sum}_{i=1}^{r}{p}_{i}{\alpha}_{i}$ be a diagonalisation of $\rho $, with ${p}_{i}>0$, for all $i=1,\dots ,r$, and $r\le d$. We have ${\sum}_{i=1}^{r}{p}_{i}\left(\right)open="("\; close=")">{\alpha}_{i}^{\u2020}|\sigma $, which means that $\left(\right)open="("\; close=")">{\alpha}_{i}^{\u2020}|\sigma $ for $i=1,\dots ,r$. This means that we can build an observation-test that distinguishes $\rho $ and $\sigma $ perfectly by taking $\left(\right)$, where $a={\sum}_{i=1}^{r}{\alpha}_{i}^{\u2020}$.
- Clearly, if $\rho =\sigma $, $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma $, whence ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $. Conversely, suppose ${F}_{\u2020}\left(\right)open="("\; close=")">\rho ,\sigma $. This means that $\left(\right)open="\langle "\; close="\rangle ">\rho ,\sigma $. By Schwarz inequality, this is true if and only if $\rho =\lambda \sigma $, for some $\lambda \in \mathbb{R}$. Since both states are normalised, $\lambda =1$, yielding $\rho =\sigma $.
- This property follows by Proposition 4, because the inner product and the dagger norm are invariant under reversible channels.

**Proposition**

**A4.**

**Lemma**

**A2.**

**Proof.**

**Proof**

**of**

**Proposition**

**A4**

## Appendix B. Dagger of All Transformations

**Definition**

**A3.**

**Proposition**

**A5.**

**Proof.**

**Proposition**

**A6.**

**Proof.**

**Proposition**

**A7.**

**Proof.**

**Proposition**

**A8.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Proposition**

**A9.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Proposition**

**A10.**

**Proof.**

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**MDPI and ACS Style**

Barnum, H.; Lee, C.M.; Scandolo, C.M.; Selby, J.H.
Ruling out Higher-Order Interference from Purity Principles. *Entropy* **2017**, *19*, 253.
https://doi.org/10.3390/e19060253

**AMA Style**

Barnum H, Lee CM, Scandolo CM, Selby JH.
Ruling out Higher-Order Interference from Purity Principles. *Entropy*. 2017; 19(6):253.
https://doi.org/10.3390/e19060253

**Chicago/Turabian Style**

Barnum, Howard, Ciarán M. Lee, Carlo Maria Scandolo, and John H. Selby.
2017. "Ruling out Higher-Order Interference from Purity Principles" *Entropy* 19, no. 6: 253.
https://doi.org/10.3390/e19060253