# 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 $\mathrm{A}$,
- $\mathsf{Transf}\left(\mathrm{A},\mathrm{B}\right)$ 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

- ${F}_{\mathsf{I}}:=\left\{\rho \in {\mathsf{St}}_{1}\left(\mathrm{A}\right):\left({a}_{\mathsf{I}}|\rho \right)=1\right\}$;
- ${F}_{\mathsf{I}}^{\perp}:=\left\{\rho \in {\mathsf{St}}_{1}\left(\mathrm{A}\right):\left({a}_{\mathsf{I}}|\rho \right)=0\right\}$,

**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.**

- $0\le {F}_{\u2020}\left(\rho ,\sigma \right)\le 1$;
- ${F}_{\u2020}\left(\rho ,\sigma \right)=0$ if and only if ρ and σ are perfectly distinguishable;
- ${F}_{\u2020}\left(\rho ,\sigma \right)=1$ if and only if $\rho =\sigma $;
- ${F}_{\u2020}\left(\mathcal{U}\rho ,\mathcal{U}\sigma \right)={F}_{\u2020}\left(\rho ,\sigma \right)$, for every reversible channel $\mathcal{U}$.

**Proof.**

- Recall that $\u2329\rho ,\sigma \u232a=\left({\rho}^{\u2020}|\sigma \right)\ge 0$, whence ${F}_{\u2020}\left(\rho ,\sigma \right)\ge 0$. Moreover, by Schwarz inequality, $\u2329\rho ,\sigma \u232a\le {\u2225\rho \u2225}_{\u2020}{\u2225\sigma \u2225}_{\u2020}$, so ${F}_{\u2020}\left(\rho ,\sigma \right)\le 1$.
- Suppose $\rho $ and $\sigma $ are perfectly distinguishable, then by Lemma A1 $\u2329\rho ,\sigma \u232a=0$, implying ${F}_{\u2020}\left(\rho ,\sigma \right)=0$. Now suppose ${F}_{\u2020}\left(\rho ,\sigma \right)=0$; then $\u2329\rho ,\sigma \u232a=0$. 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 ${\sum}_{i=1}^{r}{p}_{i}\left({\alpha}_{i}^{\u2020}|\sigma \right)=0$, which means that $\left({\alpha}_{i}^{\u2020}|\sigma \right)=0$ for $i=1,\dots ,r$. This means that we can build an observation-test that distinguishes $\rho $ and $\sigma $ perfectly by taking $\left\{a,u-a\right\}$, where $a={\sum}_{i=1}^{r}{\alpha}_{i}^{\u2020}$.
- Clearly, if $\rho =\sigma $, $\u2329\rho ,\sigma \u232a={\u2225\rho \u2225}_{\u2020}^{2}$, whence ${F}_{\u2020}\left(\rho ,\sigma \right)=1$. Conversely, suppose ${F}_{\u2020}\left(\rho ,\sigma \right)=1$. This means that $\u2329\rho ,\sigma \u232a={\u2225\rho \u2225}_{\u2020}{\u2225\sigma \u2225}_{\u2020}$. 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.**

## References

- Feynman, R.P.; Leighton, R.; Sands, M. The Feynman Lectures on Physics. The Definitive and Extended Edition; Addison Wesley: Boston, MA, USA, 2005. [Google Scholar]
- Sorkin, R.D. Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A
**1994**, 9, 3119–3127. [Google Scholar] [CrossRef] - Sorkin, R.D. Quantum Classical Correspondence: The 4th Drexel Symposium on Quantum Nonintegrability; Chapter Quantum Measure Theory and Its Interpretation; International Press: Boston, MA, USA, 1997; pp. 229–251. [Google Scholar]
- Barnum, H.; Müller, M.P.; Ududec, C. Higher-order interference and single-system postulates characterizing quantum theory. New J. Phys.
**2014**, 16, 123029. [Google Scholar] [CrossRef] - Bolotin, A. On the ongoing experiments looking for higher-order interference: What are they really testing? arXiv
**2016**. [Google Scholar] - Dakić, B.; Paterek, T.; Brukner, Č. Density cubes and higher-order interference theories. New J. Phys.
**2014**, 16, 023028. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H. Deriving grover’s lower bound from simple physical principles. New J. Phys.
**2016**, 18, 093047. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H. Generalised phase kick-back: The structure of computational algorithms from physical principles. New J. Phys.
**2016**, 18, 033023. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H. Higher-order interference in extensions of quantum theory. Found. Phys.
**2017**, 47, 89–112. [Google Scholar] [CrossRef] - Niestegge, G. Three-slit experiments and quantum nonlocality. Found. Phys.
**2013**, 43, 805–812. [Google Scholar] [CrossRef] - Ududec, C. Perspectives on the Formalism of Quantum Theory. Ph.D. Thesis, University of Waterloo, Waterloo, ON, Canada, 2012. [Google Scholar]
- Ududec, C.; Barnum, H.; Emerson, J. Probabilistic Interference in Operational Models. 2009; in preparation. [Google Scholar]
- Ududec, C.; Barnum, H.; Emerson, J. Three slit experiments and the structure of quantum theory. Found. Phys.
**2011**, 41, 396–405. [Google Scholar] [CrossRef] - Lee, C.M.; Selby, J.H. A no-go theorem for theories that decohere to quantum mechanics. arXiv
**2017**. [Google Scholar] - Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A. Generalized no-broadcasting theorem. Phys. Rev. Lett.
**2007**, 99, 240501. [Google Scholar] [CrossRef] [PubMed] - Barnum, H.; Wilce, A. Information processing in convex operational theories. Electron. Notes Theor. Comput. Sci.
**2011**, 270, 3–15. [Google Scholar] [CrossRef] - Barrett, J. Information processing in generalized probabilistic theories. Phys. Rev. A
**2007**, 75, 032304. [Google Scholar] [CrossRef] - Barrett, J.; de Beaudrap, N.; Hoban, M.J.; Lee, C.M. The computational landscape of general physical theories. arXiv
**2017**. [Google Scholar] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Probabilistic theories with purification. Phys. Rev. A
**2010**, 81, 062348. [Google Scholar] [CrossRef] - Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Informational derivation of quantum theory. Phys. Rev. A
**2011**, 84, 012311. [Google Scholar] [CrossRef] - Chiribella, G.; Spekkens, R.W. (Eds.) Quantum Theory: Informational Foundations and Foils; Fundamental Theories of Physics; Springer: Dordrecht, The Netherlands, 2016; Volume 181. [Google Scholar]
- Dakić, B.; Brukner, Č. Quantum Theory and Beyond: Is Entanglement Special; Cambridge University Press: Cambridge, UK, 2011; pp. 365–392. [Google Scholar]
- Hardy, L. Quantum Theory From Five Reasonable Axioms. arXiv
**2001**. [Google Scholar] - Hardy, L. Foliable Operational Structures for General Probabilistic Theories; Cambridge University Press: Cambridge, UK, 2011; pp. 409–442. [Google Scholar]
- Lee, C.M.; Barrett, J. Computation in generalised probabilistic theories. New J. Phys.
**2015**, 17, 083001. [Google Scholar] [CrossRef] - Lee, C.M.; Hoban, M.J. Bounds on the power of proofs and advice in general physical theories. Proc. R. Soc. A
**2016**, 472, 20160076. [Google Scholar] [CrossRef] [PubMed] - Lee, C.M.; Hoban, M.J. The information content of systems in general physical theories. In Proceedings of the 7th International Workshop on Physics and Computation, Manchester, UK, 14 July 2016; Volume 214, pp. 22–28. [Google Scholar]
- Masanes, L.; Müller, M.P. A derivation of quantum theory from physical requirements. New J. Phys.
**2011**, 13, 063001. [Google Scholar] [CrossRef] - Hardy, L. Reformulating and reconstructing quantum theory. arXiv
**2011**. [Google Scholar] - Chiribella, G.; Scandolo, C.M. Entanglement as an axiomatic foundation for statistical mechanics. arXiv
**2016**. [Google Scholar] - Krumm, M.; Barnum, H.; Barrett, J.; Müller, M.P. Thermodynamics and the structure of quantum theory. New J. Phys.
**2017**, 19, 043025. [Google Scholar] [CrossRef] - Jin, F.; Liu, Y.; Geng, J.; Huang, P.; Ma, W.; Shi, M.; Duan, C.; Shi, F.; Rong, X.; Du, J. Experimental test of born’s rule by inspecting third-order quantum interference on a single spin in solids. Phys. Rev. A
**2017**, 95, 012107. [Google Scholar] [CrossRef] - Kauten, T.; Keil, R.; Kaufmann, T.; Pressl, B.; Brukner, Č.; Weihs, G. Obtaining tight bounds on higher-order interferences with a 5-path interferometer. New J. Phys.
**2017**, 19, 033017. [Google Scholar] [CrossRef] - Park, D.K.; Moussa, O.; Laflamme, R. Three path interference using nuclear magnetic resonance: A test of the consistency of born’s rule. New J. Phys.
**2012**, 14, 113025. [Google Scholar] [CrossRef] - Sinha, A.; Vijay, A.H.; Sinha, U. On the superposition principle in interference experiments. Sci. Rep.
**2015**, 5, 10304. [Google Scholar] [CrossRef] [PubMed] - Sinha, U.; Couteau, C.; Jennewein, T.; Laflamme, R.; Weihs, G. Ruling out multi-order interference in quantum mechanics. Science
**2010**, 329, 418–421. [Google Scholar] [CrossRef] [PubMed] - Barnum, H.; Graydon, M.; Wilce, A. Composites and categories of Euclidean Jordan algebras. arXiv
**2016**. [Google Scholar] - Chiribella, G. Dilation of states and processes in operational-probabilistic theories. In Proceedings of the 11th workshop on Quantum Physics and Logic, Kyoto, Japan, 4–6 June 2014; Volume 172, pp. 1–14. [Google Scholar]
- Chiribella, G.; D’Ariano, G.M.; Perinotti, P. Quantum Theory: Informational Foundations and Foils; Chapter Quantum from Principles; Springer: Dordrecht, The Netherlands, 2016; pp. 171–221. [Google Scholar]
- Hardy, L. Quantum Theory: Informational Foundations and Foils; Chapter Reconstructing Quantum Theory; Springer: Dordrecht, The Netherlands, 2016; pp. 223–248. [Google Scholar]
- Abramsky, S.; Coecke, B. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, Turku, Finland, 13–17 July 2004; pp. 415–425. [Google Scholar]
- Coecke, B. Kindergarten quantum mechanics: Lecture notes. AIP Conf. Proc.
**2006**, 810, 81–98. [Google Scholar] - Coecke, B. Quantum picturalism. Contemp. Phys.
**2010**, 51, 59. [Google Scholar] [CrossRef] - Coecke, B.; Duncan, R.; Kissinger, A.; Wang, Q. Quantum Theory: Informational Foundations and Foils; Chapter Generalised Compositional Theories and Diagrammatic Reasoning; Springer: Dordrecht, The Netherlands, 2016; pp. 309–366. [Google Scholar]
- Coecke, B.; Kissinger, A. Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Selinger, P. A survey of graphical languages for monoidal categories. In New Structures for Physics; Coecke, B., Ed.; Springer: Berlin, Germany, 2011; pp. 289–356. [Google Scholar]
- Wootters, W.K. Local accessibility of quantum states. In Complexity, Entropy and the Physics of Information; Zurek, W.H., Ed.; Westview Press: Boulder, CO, USA, 1990; pp. 39–46. [Google Scholar]
- Chiribella, G.; Scandolo, C.M. Entanglement and thermodynamics in general probabilistic theories. New J. Phys.
**2015**, 17, 103027. [Google Scholar] [CrossRef] - Chiribella, G.; Scandolo, C.M. Purity in microcanonical thermodynamics: A tale of three resource theories. arXiv
**2016**. [Google Scholar] - Gour, G.; Müller, M.P.; Narasimhachar, V.; Spekkens, R.W.; Yunger Halpern, N. The resource theory of informational nonequilibrium in thermodynamics. Phys. Rep.
**2015**, 583, 1–58. [Google Scholar] [CrossRef] - Horodecki, M.; Horodecki, P.; Oppenheim, J. Reversible transformations from pure to mixed states and the unique measure of information. Phys. Rev. A
**2003**, 67, 062104. [Google Scholar] [CrossRef] - Selby, J.H.; Coecke, B. Leaks: Quantum, classical, intermediate, and more. Entropy
**2017**, 19, 174. [Google Scholar] [CrossRef] - Coecke, B. Terminality implies non-signalling. In Proceedings of the 11th workshop on Quantum Physics and Logic, Kyoto, Japan, 4–6 June 2014; Volume 172, pp. 27–35. [Google Scholar]
- Chiribella, G.; Scandolo, C.M. Operational axioms for diagonalizing states. In Proceedings of the 12th International Workshop on Quantum Physics and Logic, Oxford, UK, 15–17 July 2015; Volume 195, pp. 96–115. [Google Scholar]
- Chiribella, G.; Scandolo, C.M. Conservation of information and the foundations of quantum mechanics. EPJ Web Conf.
**2015**, 95, 03003. [Google Scholar] [CrossRef] - Disilvestro, L.; Markham, D. Quantum protocols within Spekkens’ toy model. Phys. Rev. A
**2017**, 95, 052324. [Google Scholar] [CrossRef] - D’Ariano, G.M.; Manessi, F.; Perinotti, P.; Tosini, A. Fermionic computation is non-local tomographic and violates monogamy of entanglement. Europhys. Lett.
**2014**, 107, 20009. [Google Scholar] [CrossRef] - D’Ariano, G.M.; Manessi, F.; Perinotti, P.; Tosini, A. The Feynman problem and fermionic entanglement: Fermionic theory versus qubit theory. Int. J. Mod. Phys. A
**2014**, 29, 1430025. [Google Scholar] [CrossRef] - Chiribella, G.; Yuan, X. Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality. Inf. Comput.
**2016**, 250, 15–49. [Google Scholar] [CrossRef] - Pfister, C.; Wehner, S. An information-theoretic principle implies that any discrete physical theory is classical. Nat. Commun.
**2013**, 4, 1851. [Google Scholar] [CrossRef] [PubMed] - Alfsen, E.M.; Shultz, F.W. Geometry of State Spaces of Operator Algebras; Mathematics Theory & Applications; Birkhäuser: Basel, Switzerland, 2003. [Google Scholar]
- Barnum, H.; Barrett, J.; Krumm, M.; Müller, M.P. Entropy, majorization and thermodynamics in general probabilistic theories. In Proceedings of the 12th International Workshop on Quantum Physics and Logic, Oxford, UK, 15–17 July 2015; Volume 195, pp. 43–58. [Google Scholar]
- Chiribella, G.; Yuan, X. Measurement sharpness cuts nonlocality and contextuality in every physical theory. arXiv
**2014**. [Google Scholar] - Alfsen, E.M.; Shultz, F.W. State spaces of Jordan algebras. Acta Math.
**1978**, 140, 155–190. [Google Scholar] [CrossRef] - Iochum, B. Cônes Autopolaires et Algèbres de Jordan; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1358; Volume 1049, (In French). [Google Scholar] [CrossRef]
- Coecke, B.; Selby, J.; Tull, S. Two roads to classicality. arXiv
**2017**. [Google Scholar] - Barnum, H. Spectrality as a Tool for Quantum Reconstruction: Higher-Order Interference, Jordan State Space Characterizations, Aug. 2009. Talk Given at the Conference “Reconstructing Quantum Theory”, August 9–11, Perimeter Institute for Theoretical Physics. Available online: http://pirsa.org/09080016/ (accessed on 26 May 2017).
- Niestegge, G. Conditional probability, three-slit experiments, and the jordan algebra structure of quantum mechanics. Adv. Math. Phys.
**2012**, 2012, 156573. [Google Scholar] [CrossRef] - Selby, J.H.; Coecke, B. Process-theoretic characterisation of the hermitian adjoint. arXiv
**2016**. [Google Scholar] - Müller, M.P.; Oppenheim, J.; Dahlsten, O.C.O. The black hole information problem beyond quantum theory. J. High Energy Phys.
**2012**, 2012, 9. [Google Scholar] [CrossRef] - Müller, M.P.; Ududec, C. Structure of reversible computation determines the self-duality of quantum theory. Phys. Rev. Lett.
**2012**, 108, 130401. [Google Scholar] [CrossRef] [PubMed] - Marshall, A.W.; Olkin, I.; Arnold, B.C. Inequalities: Theory of Majorization and Its Applications; Springer Series in Statistics; Springer: New York, NY, USA, 2011. [Google Scholar]
- Müller, M.P.; Dahlsten, O.C.O.; Vedral, V. Unifying typical entanglement and coin tossing: On randomization in probabilistic theories. Commun. Math. Phys.
**2012**, 316, 441–487. [Google Scholar] [CrossRef] - Wilde, M.M. Quantum Information Theory, 2nd ed.; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Selinger, P. Dagger compact closed categories and completely positive maps. Electron. Notes Theor. Comput. Sci.
**2007**, 170, 139–163. [Google Scholar] [CrossRef]

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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