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Indefinite Quantum Causality

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Indefinite causal order between quantum operations can function as a resource for information processing.

desk verdict This is a review paper summarizing the process matrix formalism for indefinite causal order, with no new results or derivations. read the letter →

arxiv 2606.19438 v1 pith:DFLHW34J submitted 2026-06-17 quant-ph

Fabio Costa , Giulia Rubino , Cyril Branciard , Časlav Brukner , Marco Túlio Quintino This is my paper · ORCID
classification quant-ph
keywords indefinitecausalorderprocessmatrixformalismquantuminformationprocessingfoundationscausalityhigher-orderoperations
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper surveys the process matrix formalism as a way to describe quantum processes whose causal order is not fixed in advance. It collects the main theoretical results, experimental demonstrations, and applications that follow once the usual assumption of a definite sequence of operations is relaxed. A sympathetic reader would care because this relaxation is presented as both a tool for new quantum protocols and a bridge between quantum theory and questions about spacetime causality.

What carries the argument

The process matrix formalism, which represents quantum processes as matrices that encode possible causal relations without requiring a fixed order between operations.

What would settle it

An experiment or consistency proof demonstrating that every valid quantum process must possess a definite causal order, or a derivation showing that the process matrix approach violates the no-signaling principle or leads to negative probabilities.

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Extended reading notes

Core claim

Indefinite causal order is a feasible resource for quantum information processing, as developed within the process matrix formalism.

Load-bearing premise

The process matrix formalism correctly captures the possibility of quantum indefiniteness in causal order without introducing inconsistencies with standard quantum theory or relativity.

Editorial extensions

If this is right

  • Higher-order quantum operations become available for computation beyond standard circuit models.
  • New experimental protocols can test or exploit causal indefiniteness using photonic or other quantum systems.
  • The formalism supplies a language for studying the interface between quantum mechanics and general relativity.
  • Resource theories of causal order can be developed to quantify advantages in communication or computation tasks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the formalism holds, it may suggest that causal structure itself can be treated as a quantum degree of freedom in future theories of quantum gravity.
  • Practical implementations could extend to quantum algorithms that use causal superposition to reduce query complexity in certain tasks.
  • The approach invites direct comparison with other frameworks that relax temporal order, such as those based on causal sets or quantum reference frames.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript is a review article surveying indefinite quantum causality. It describes the process matrix formalism as the central methodology for modeling quantum processes with indefinite causal order, summarizes key theoretical results establishing this as a resource for quantum information processing, reviews experimental implementations, and discusses recent advances motivated by quantum foundations, quantum gravity interfaces, and higher-order quantum computation.

Significance. If the synthesis of the literature is accurate, the review consolidates an emerging area that relaxes the definite causal order assumption in quantum theory. It provides a useful entry point for researchers working at the intersection of quantum information, foundations, and potential gravitational effects, explicitly crediting the process matrix formalism for enabling new protocols beyond standard quantum circuits. No novel derivations or data are presented; the value lies in the overview of existing results.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a growing literature is emerging' would benefit from a specific citation to the foundational process matrix paper (e.g., Oreshkov et al.) to immediately anchor the review for readers.
  2. [Experimental section] The discussion of experimental implementations would be strengthened by a table summarizing the physical platforms, achieved process matrices, and measured advantages over definite-order protocols.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. The report correctly identifies the manuscript as a review of the process matrix formalism and its applications. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: review paper with no internal derivations

full rationale

This is a review article summarizing the process matrix formalism, prior theoretical results, and experiments on indefinite causal order. The abstract and structure indicate synthesis of existing literature rather than any novel derivation chain, predictions, or first-principles results. No equations or claims are presented that reduce by construction to fitted inputs, self-definitions, or self-citation chains within the paper itself. The central claim follows from cited prior work, which is treated as external. This matches the default expectation of no circularity for review/synthesis papers.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

As a review paper, no new free parameters, axioms, or invented entities are introduced by this work; the ledger reflects the underlying literature it summarizes.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Indefinite Quantum Causality." pith.science (2026). https://pith.science/paper/DFLHW34J

@misc{pith2026260619438,
  author       = {Pith},
  title        = {Pith review of: Indefinite Quantum Causality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFLHW34J}},
  note         = {Machine review of arXiv:2606.19438}
}
read the original abstract

In recent years, operational approaches to quantum foundations have been developed as a means of understanding the core principles and distinctive features of quantum theory. Such approaches typically view physical processes as sequences of operations, with earlier operations serving as causes of later effects. However, a growing literature is emerging on the possibility of relaxing this assumption and allowing for quantum indefiniteness in the causal order. This development stems from a variety of motivations, both fundamental and applied, including exploring the role of causality in quantum theory, the interplay between quantum theory and general relativity, and higher-order quantum computing. A prominent offshoot of this development is the emergence of indefinite causal order as a feasible resource for quantum information processing. This review provides an overview of the current state of the art in the field, covering the methodology underlying indefinite quantum causality within the so-called "process matrix formalism", outlining key results and experimental implementations, and discussing recent advances.

Figures

Figures reproduced from arXiv: 2606.19438 by the authors.

Figure 1
Figure 1. Scheme of a bipartite quantum switch. Two parties, Al [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Pictorial representation of the operation-state duality: A [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Quantum circuit and Choi representation of quantum [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (29 more)
Figure 4
Figure 4. Figure 4: Circuit illustration of a bipartite ordered quantum process, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Pictorial representation of a bipartite process matrix, where [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Pictorial representation of the most general measurement [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Mapping a process to a state. The state (density matrix) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Pictorial representation of the correspondence between [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Depiction of the convex set of causally separable pro [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: a) Pictorial representation of a one-slot process which transforms a quantum channel from AI to AO into a quantum chan￾nel mapping P (past) to F (future). b) Pictorial representation of a two-slot process which transforms a pair of quantum channels, one from AI to AO,…
Figure 11
Figure 11. Figure 11: A classical process is defined by transition probabilities [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: Qualitative depiction of polytopes of classical processes. [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]
Figure 13
Figure 13. Figure 13: a) Illustrative example of a mixture involving two non￾valid process matrices: a circular identity channel combined with a circular bit-flip channel. b) Process that sends the output of each party to the next in a loop, with the direction determined by whether the maj…
Figure 14
Figure 14. Figure 14: The quantum switch. The connection between two quan [PITH_FULL_IMAGE:figures/full_fig_p032_14.png]
Figure 15
Figure 15. Figure 15: Quantum switch through multiple black-box calls to the [PITH_FULL_IMAGE:figures/full_fig_p035_15.png]
Figure 16
Figure 16. Figure 16: Quantum switch through multiple calls to quantum [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]
Figure 17
Figure 17. Figure 17: Pictorial representation of the scenario considered in [PITH_FULL_IMAGE:figures/full_fig_p042_17.png]
Figure 18
Figure 18. Figure 18: Schematic of a communication complexity setup in a causally ordered scenario and using the quantum switch. a) Based on her [PITH_FULL_IMAGE:figures/full_fig_p048_18.png]
Figure 19
Figure 19. Figure 19: Schematic of a two-party quantum switch with target sys [PITH_FULL_IMAGE:figures/full_fig_p051_19.png]
Figure 20
Figure 20. Figure 20: Illustration of the entangled quantum switch configura [PITH_FULL_IMAGE:figures/full_fig_p052_20.png]
Figure 21
Figure 21. Figure 21: Schematic of a 4-partite quantum switch with target sys [PITH_FULL_IMAGE:figures/full_fig_p053_21.png]
Figure 22
Figure 22. Figure 22: Experimental setup from (Cao et al., 2022) demonstrating a refrigeration cycle based on indefinite causality. A beam splitter (BS1) creates a superposition of two spatial modes, encoding the control qubit. In one mode, the polarization qubit undergoes the causal order…
Figure 24
Figure 24. Figure 24: Schematic of a quantum-optical switch in a Sagnac con [PITH_FULL_IMAGE:figures/full_fig_p055_24.png]
Figure 23
Figure 23. Figure 23: Schematic of a quantum-optical switch implementing [PITH_FULL_IMAGE:figures/full_fig_p055_23.png]
Figure 25
Figure 25. Figure 25: Schematic of a two-party quantum switch with the tar [PITH_FULL_IMAGE:figures/full_fig_p056_25.png]
Figure 26
Figure 26. Figure 26: Schematic of a two-party switch where the target qubit [PITH_FULL_IMAGE:figures/full_fig_p057_26.png]
Figure 28
Figure 28. Figure 28: Schematic of the fiber-based quantum switch from ( [PITH_FULL_IMAGE:figures/full_fig_p058_28.png]
Figure 29
Figure 29. Figure 29: Schematic of a two-party quantum switch with control [PITH_FULL_IMAGE:figures/full_fig_p059_29.png]
Figure 30
Figure 30. Figure 30: a) Quantum circuit reproducing a quantum switch of ther [PITH_FULL_IMAGE:figures/full_fig_p060_30.png]
Figure 31
Figure 31. Figure 31: Quantum control of causal order due to gravitational time [PITH_FULL_IMAGE:figures/full_fig_p061_31.png]
Figure 32
Figure 32. Figure 32: A Directed Acyclic Graph (DAG) comprises nodes con￾nected by directed edges, with no directed loops allowed. In a causal model, nodes represent observable variables and edges direct cause￾effect relations. dence P(B, C|A) = P(B|A)P(C|A). This condition is not compatib…
Figure 33
Figure 33. Figure 33: Operation with quantum-controlled timing. The time at [PITH_FULL_IMAGE:figures/full_fig_p089_33.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Self-testing Quantum Supermaps

    quant-ph 2026-06 unverdicted novelty 8.0 of 10

    Quantum supermaps are shown to be self-testable device-independently, achieving identification up to local embedding combs or extracting/injecting maps, with demonstrations on identity, error-correcting, Grover, and q...

  2. Causality from the spectrum: Emergence of causal order from process-matrix mereology

    quant-ph 2026-07 conditional novelty 7.0 of 10

    A process matrix can be unitarily transformed to a fixed-causal-order process exactly when its eigenvalue multiplicities are divisible by the final output dimension; generic high-dimensional spectra are close to such spectra.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages · cited by 2 Pith papers

  1. [1]

    SIAM Journal on Computing , volume =

    causal models,” Nature Communications12, arXiv:2002.12157 [quant-ph]. S. Bartlett, T. Rudolph, and R. Spekkens (2007), “Reference frames, superselection rules, and quantum information,” Rev. Mod. Phys. 79, 555–609, arXiv:quant-ph/0610030 [quant-ph]. V . Baumann, Ä. Baumeler, and E.-E. Tselentis (2025), “No quantum advantage for violating fixed-order inequ...

    work page doi:10.1137/s0097539796300921 2002

  2. [2]

    Measurement-device-independent entanglement witnesses for all entangled quantum states,

    Leibniz Inter- national Proceedings in Informatics (LIPIcs),202, 22:1–22:20, arXiv:2103.02073 [quant-ph]. C. Branciard, D. Rosset, Y .-C. Liang, and N. Gisin (2013), “Measurement-device-independent entanglement witnesses for all entangled quantum states,” Phys. Rev. Lett.110, 060405, arXiv:1210.8037 [quant-ph]. S. L. Braunstein, and C. M. Caves (1988), “I...

    work page arXiv 2013

  3. [3]

    Bounding quantum correlations with indefinite causal order

    WE- Heraeus-Seminar conference: Quantum Contextuality, Non-Locality, and the Foundations of Quantum Mechanics. ˇC. Brukner (2015), “Bounding quantum correlations with indef- inite causal order,” New Journal of Physics17(8), 083034, arXiv:1404.0721 [quant-ph]. N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner (2014), “Bell nonlocality,” Rev...

    work page Pith review arXiv 2015

  4. [4]

    Probabilistic Microcausality in a Thermal Bath of Gravitons

    G. Cintia, F. Piazza, and S. Ramos (2026), “Probabilistic micro- causality in a thermal bath of gravitons,” arXiv:2606.02729 [hep- th]. B. S. Cirel’son (1980), “Quantum generalizations of bell’s inequal- ity,” Letters in Mathematical Physics4(2), 93–100. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt (1969), “Pro- posed experiment to test local hi...

    work page Pith review arXiv 2026

  5. [5]

    Optimal design for uni- versal multiport interferometers,

    Leibniz International Proceedings in Informatics (LIPIcs),170, 24:1–24:14, arXiv:2002.09387 [quant- ph]. W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Koltham- mer, and I. A. Walmsley (2016), “Optimal design for uni- versal multiport interferometers,” Optica3(12), 1460–1465, arXiv:1603.08788 [physics.optics]. D. Collins, N. Gisin, N. Linden, S. Ma...

    work page arXiv 2002

  6. [6]

    Rapid solution of problems by quantum computation,

    D. Deutsch, and R. Jozsa (1992), “Rapid solution of problems by quantum computation,” Proceedings of the Royal Society A439, 553–558. P. R. Dieguez, V . F. Lisboa, and R. M. Serra (2023), “Thermal de- vices powered by generalized measurements with indefinite causal order,” Phys. Rev. A107, 012423, arXiv:2205.14406 [quant-ph]. A. Dimi´c, M. Milivojevi´c, D...

    work page arXiv 1992

  7. [7]

    Quantum network probing with indefinite routing,

    M. Frey (2021), “Quantum network probing with indefinite routing,” Quantum Inf. Process.20(1),

    2021

  8. [8]

    Implementing quantum control for unknown subroutines

    N. Friis, V . Dunjko, W. Dür, and H. J. Briegel (2014), “Implementing Quantum Control for Unknown Subroutines,” Phys. Rev. A89, 030303, arXiv:1401.8128 [quant-ph]. M. Frisch (2022),The Stanford Encyclopedia of Philosophy, Spring 2022 ed., edited by E. N. Zalta (Metaphysics Research Lab, Stan- ford University). T. Fujita, and M. Sasaki (2026), “Quantum mat...

    work page Pith review arXiv doi:10.1103/phys- 2014

Show all 15 references
  1. [9]

    Optimal strategies of quantum metrology with a strict hierarchy,

    Q. Liu, Z. Hu, H. Yuan, and Y . Yang (2023a), “Optimal strategies of quantum metrology with a strict hierarchy,” Phys. Rev. Lett.130, 070803, arXiv:2203.09758 [quant-ph]. W.-Q. Liu, Z. Meng, B.-W. Song, J. Li, Q.-Y . Wu, X.-X. Chen, J.-Y . Hong, A.-N. Zhang, and Z. qi Yin (202...

    2022

  2. [10]

    Reconstructing quantum theory from diagrammatic postulates,

    M. O. Scully, and M. S. Zubairy (1997),Quantum Optics(Cam- bridge University Press). J. H. Selby, C. M. Scandolo, and B. Coecke (2021), “Reconstructing quantum theory from diagrammatic postulates,” Quantum5, 445, arXiv:1802.00367 [quant-ph]. K. Sengupta (2024), “Achieving maxi...

    1997

  3. [11]

    Equivalence determi- nation of unitary operations,

    A. Shimbo, A. Soeda, and M. Murao (2018), “Equivalence determi- nation of unitary operations,” arXiv:1803.11414 [quant-ph]. S. Shrapnel, and F. Costa (2018), “Causation does not explain con- textuality,” Quantum2, 63, arXiv:1708.00137 [quant-ph]. S. Shrapnel, F. Costa, and G. ...

    2018 arXiv

  4. [12]

    Work extraction from coherently activated maps via quantum switch,

    K. Simonov, G. Francica, G. Guarnieri, and M. Paternostro (2022), “Work extraction from coherently activated maps via quantum switch,” Phys. Rev. A105, 032217, arXiv:2009.11265 [quant-ph]. P. Skrzypczyk, and D. Cavalcanti (2023),Semidefinite Programming in Quantum Information ...

    2022

  5. [13]

    Simulating noncausality with quantum control of causal orders,

    A. Steffinlongo, and H. Dourdent (2026), “Simulating noncausality with quantum control of causal orders,” Phys. Rev. Res.8, 013127, arXiv:2502.15579 [quant-ph]. W. F. Stinespring (1955), “Positive functions on c*-algebras,” Pro- ceedings of the American Mathematical Society6(2...

    2026

  6. [14]

    Computational advantage from the quantum superposition of multiple temporal orders of photonic gates,

    M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima (2021), “Computational advantage from the quantum superposition of multiple temporal orders of photonic gates,” PRX Quantum2,...

    2021

  7. [15]

    controlled-time

    Operation with quantum-controlled timing. The time at which the operationMis applied on a target system depends on the state of the control: timet 1 for|0⟩and timet 2 for|1⟩. In such a situation, it is possible to identify a subsystem decomposition such thatMacts on asinglesub...

    2019

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