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Routing Quantum Control of Causal Order

T0 review · 0 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that every quantum circuit with quantum control of causal order can be obtained by fleshing out one generic routed graph per number of parties, giving all QC-QCs a routed circuit decomposition.

desk verdict This paper delivers a real structural result—every N-slot QC-QC has a routed circuit decomposition from one generic graph—and the proof is detailed enough to check; the main caveat is inherited reliance on the routed-circuit framework's central theorem. read the letter →

arxiv 2507.08781 v1 pith:QFNFV6OH submitted 2025-07-11 quant-ph

Maarten Grothus , Alastair A. Abbott , Augustin Vanrietvelde , Cyril Branciard This is my paper
classification quant-ph
keywords indefinitecausalorderquantumcontrolofroutedcircuitsskeletalsupermapbranchgraphswitchprocessmatrix
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

Quantum processes with indefinite causal order have been studied through two complementary lenses: quantum circuits with quantum control of causal order (QC-QCs), built bottom-up from physically motivated components, and routed quantum circuits, which decompose a process into a graph whose routes track where information can flow. Until now, only a handful of specific processes had been shown to admit a routed circuit decomposition. This paper proves that for any number $N$ of parties, a single routed graph, denoted $G_{\mathrm{QC-QC}}(N)$, can be fleshed out to produce any $N$-slot QC-QC, pure or mixed. If correct, the result makes routed circuits a general representation language for the entire QC-QC family, and gives a diagrammatic picture of the information flow inside these processes.

What carries the argument

The load-bearing object is the generic routed graph $G_{\mathrm{QC-QC}}(N)$ of Definition 4.2: a directed graph whose nodes are the $N$ agent slots $A_k$ and the $N+1$ internal-operation slots $V_{n+1}$, with arrows labelled by index values $X^k_n$ that are either a subset $K_n$ of agents containing $k$ or the null value. A route at each node is a relation constraining which index combinations may pass through that node, and the global index constraints require that a consistent assignment encodes one full ordering of the agents. The argument then shows the graph is valid, meaning bi-univocal with no non-weak loops in its branch graph, so the routed-circuit framework's main theorem certifies that its skeletal supermap is a routed superunitary. Fleshing out means filling each slot with a compatible routed operation; the paper's specific fleshing out uses isomorphisms $J^{\mathrm{in/out}}$ to translate between the graph's sectorised wires and the QC-QC's systems, and then plugs in the internal isometries and one-slot agent supermaps.

What would settle it

Take a specific $N$-slot QC-QC specified by its internal isometries, compute the Choi vector obtained by the paper's fleshing out of $G_{\mathrm{QC-QC}}(N)$ with those isometries, and compare it with the QC-QC process vector computed directly from the definition; a single mismatch would disprove Theorem 4.1. A more pointed test is the appendix's Zurich process for $N=4$, which the paper shows cannot be represented by the compact internal-node-free graph, so checking whether the full generic graph does represent it exercises exactly the part of the claim that goes beyond earlier examples.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 4.1: any $N$-slot quantum circuit with quantum control of causal order (QC-QC) can be implemented by fleshing out the skeletal supermap associated with the generic routed graph $G_{\mathrm{QC-QC}}(N)$. The graph has two kinds of nodes, agent slots $A_k$ and internal isometry slots $V_{n+1}$, and its arrows carry index values $X^k_n$ that record which subset of agents has already acted and which agent acts at time $n$. The routes at the nodes enforce that each agent acts exactly once and exactly one agent acts per time slot. The paper shows this graph is bi-univocal and its branch graph has no loops, so the routed-circuit framework's main theorem applies and the skeletal supermap is a routed superunitary. The authors then give explicit isomorphisms that insert the QC-QC's internal isometries and agent operations into the slots and compose the resulting Choi vectors with the link product, recovering exactly the QC-QC process vector; tracing out a final ancillary system extends the argument to mixed QC-QCs.

Load-bearing premise

The load-bearing premise is the routed-circuit framework's guarantee that a graph satisfying its validity conditions, namely bi-univocality and only harmless loops, yields a valid quantum process; the paper relies on that guarantee, and on the standard decomposition of a QC-QC into internal isometries, without reproving either.

Editorial extensions

If this is right

  • Every pure $N$-slot QC-QC, not just previously studied examples, acquires a routed circuit decomposition based on the single graph $G_{\mathrm{QC-QC}}(N)$.
  • Mixed QC-QCs are covered too: tracing out a final ancillary subsystem of the global future space turns the pure construction into the mixed process matrix.
  • The validity of $G_{\mathrm{QC-QC}}(N)$ independently certifies, through the routed-circuit framework, that QC-QCs are legitimate quantum processes.
  • The alternative graphs obtained by splitting and merging nodes recover the earlier routed decompositions of the quantum switch and the Grenoble process as special cases of the generic construction.
  • For more than three parties, the fully compact representation without internal V-nodes is generally impossible, so the generic graph is not just a convenience but a necessary level of description.

Reading between the lines

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

  • If the theorem holds, the subclass structure of QC-QCs, such as superpositions of fixed orders, classical control, and non-influenceable causal orders, could be read off as additional index constraints on the same routed graph, a programme the paper sketches but does not complete.
  • Combined with the belief that QC-QCs exhaust the processes realisable in a fixed spacetime, the result lends support to the conjecture that every such process has a routed circuit decomposition from a valid routed graph.
  • The index bookkeeping in $G_{\mathrm{QC-QC}}(N)$ suggests a design test for proposed causal-control processes: a structure is routable if its causal bookkeeping can be encoded as index constraints on a valid routed graph, which may help in searching for new indefinite-causal-order processes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The manuscript connects the routed quantum circuit framework of Vanrietvelde et al. with the QC-QC framework of Wechs et al. It proposes, for each N, a generic routed graph G_QC-QC(N) and proves that every N-slot QC-QC, pure or mixed, can be obtained by fleshing out the associated skeletal supermap. Sections 2 and 3 review both frameworks; Section 4 gives the construction, proves bi-univocality, checks that the branch graph is loop-free, and computes the Choi vector of the fleshed-out supermap, showing that it equals the QC-QC process vector. Section 5 discusses alternative routed graph descriptions, including recovery of earlier decompositions of the quantum switch and the Grenoble process. Appendices A-D contain the detailed calculations behind the main construction.

Significance. If the central result holds, this is a substantial step: it upgrades the routed circuit framework from a handful of examples to the entire QC-QC class, and it provides a concrete template for studying subclasses and open problems. The proof is constructive and largely self-contained, with explicit derivations of the choice function (Appendix A), branch graph acyclicity (Section 4.2), and the equivalence of the fleshed-out circuit to |w_QC-QC> (Appendix B). No fitted parameters or circular assumptions are involved. The main residual risk is the reliance on the external validity theorem (Theorem 2.23 of Ref. [39]) for the routed-graph-to-routed-superunitary step; the direct equality in Eq. (4.34) provides independent evidence for the concrete construction. Overall, this is a strong and useful contribution to the literature on indefinite causal order.

minor comments (6)
  1. [Section 2.6, Theorem 2.23] The paper relies on Theorem 2.23 of Ref. [39] to certify that a valid routed graph yields a routed superunitary, but the framework presented here is a simplified variant of the original one. Please add an explicit remark stating that Theorem 2.23 is quoted from Ref. [39] and confirming that the simplified definitions in Section 2 satisfy the hypotheses of the original theorem, or provide a short proof sketch of the transfer. This would remove a residual unclarity about the central validity certificate.
  2. [Section 4.1.4] Bi-univocality of the adjoint graph is justified by a compact "self-adjoint up to relabelling" argument. Since univocality of the adjoint is load-bearing for bi-univocality, please give the explicit involution on nodes, arrows, and index values that maps G_QC-QC(N) to its adjoint, and indicate why the associated choice relations are isomorphic. The current presentation is plausible but too terse for a formal proof.
  3. [Section 4.3.3, Eq. (4.30)] The notation (H_{\bar\alpha}\otimes H_{\bar C(k)})_{\mathrm{prac}} and the "dropping" of one-dimensional tensor factors in the practical subspaces deserve a clearer formal statement. Please specify explicitly that the identification is via the canonical isomorphism between a Hilbert space and its tensor product with one-dimensional spaces, and that this identification is compatible with the route relations at the corresponding nodes.
  4. [Section 5.1] When direct arrows V_n \to V_{n+1} are added to obtain G^\alpha_{\mathrm{QC-QC}}(N), the routes at the affected V-nodes are not explicitly updated. Please specify how the added index value \alpha_n enters the domain and codomain of \lambda_{V_n} and \lambda_{V_{n+1}}, and why the resulting routed graph remains valid under the stated conventions.
  5. [Section 5.3, Proposition 5.6] The proof of the forward direction of Proposition 5.6, especially the G_{\mathrm{merge}\downarrow} case, is sketched rather than fully argued. In particular, the claim that the restriction to a routed unitary at V is "without loss of generality" needs a more explicit construction showing how an arbitrary routed isometry at V is recovered from a routed unitary and a state preparation at the merged node, without violating the no-ancilla convention of Definition 2.26.
  6. [Section 6.3 and general presentation] There are several minor typographical and formatting issues, including "admissable" for "admissible" in Section 6.3 and various LaTeX rendering artifacts in equations. A careful copyedit would improve readability, especially in the appendices and figure captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generic routed graph is defined independently, and the proof of Theorem 4.1 is an explicit composition check that recovers the QC-QC process vector.

full rationale

I examined the derivation chain from Definition 4.2 through Theorem 4.1. The routed graph G_QC-QC(N) is defined combinatorially, with A-nodes, V-nodes, arrows labelled by subsets K_n, and route constraints such as requiring exactly one non-null index and imposing X_ell^{n+1} = X_k^n union ell; these conditions do not reference a particular QC-QC process vector or assume the target supermap. Bi-univocality and the absence of loops in the branch graph are checked directly in Sections 4.1.4 and 4.2.3. The fleshing out in Sections 4.3.2 through 4.3.3 inserts the QC-QC's own internal isometries and agent operations behind explicitly introduced isomorphisms J^{in/out}, and the calculation in Eq. (4.34), with the detailed derivation in Appendix B, links these ingredients to obtain exactly the QC-QC process vector |w_QC-QC> of Eq. (3.7). This is an existence proof by explicit construction, not a prediction that reduces to its input. The one load-bearing imported ingredient is Theorem 2.23 of Ref. [39], restated in Section 2.6, which certifies that a valid routed graph yields a routed superunitary; that theorem is stated and proved in the cited prior work, does not assume the present conclusion, and therefore counts as independent mathematical support rather than circularity. The self-citations to Refs. [34] and [39] reflect the authors' prior roles in developing both frameworks, but the paper does not use those citations to assume Theorem 4.1. Residual concerns, such as the fact that the simplified restatement of the routed-circuit framework in Section 2 deviates slightly from Ref. [39] without re-proving Theorem 2.23 under the new definitions, are correctness or robustness risks rather than circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The construction relies on the routed quantum circuit framework as an external formal system (Theorem 2.23 of [39]) and on the QC-QC definition from [34]. No physical entities are postulated and no data-fitting parameters are used; the dimension assignment is a construction choice, not a fitted quantity.

assumptions (3)
  • domain assumption Valid routed graphs yield routed superunitaries (Theorem 2.23 of [39])
    Used as the core tool to certify that the constructed skeletal supermap is a valid quantum process; stated in Section 2.6 and invoked in Section 4.2.3.
  • domain assumption QC-QC process vectors have the form of Eq. (3.7) in terms of internal isometries \tilde V_{n+1}
    The target set of processes is defined by this construction from [34]; the proof uses these isometries to flesh out the V-nodes.
  • standard math Finite-dimensional Hilbert spaces and standard linear algebra
    Throughout the paper, all Hilbert spaces are finite-dimensional; standard properties of Choi vectors and link products are used.

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Pith. "Pith review of Routing Quantum Control of Causal Order." pith.science (2026). https://pith.science/paper/QFNFV6OH

@misc{pith2026250708781,
  author       = {Pith},
  title        = {Pith review of: Routing Quantum Control of Causal Order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFNFV6OH}},
  note         = {Machine review of arXiv:2507.08781}
}
abstract

In recent years, various frameworks have been proposed for the study of quantum processes with indefinite causal order. In particular, quantum circuits with quantum control of causal order (QC-QCs) form a broad class of physical supermaps obtained from a bottom-up construction and are believed to represent all quantum processes physically realisable in a fixed spacetime. Complementarily, the formalism of routed quantum circuits introduces quantum operations constrained by "routes" to represent processes in terms of a more fine-grained routed circuit decomposition. This decomposition, formalised using a so-called routed graph, represents the information flow within the respective process. However, the existence of routed circuit decompositions has only been established for a small set of processes so far, including both certain specific QC-QCs and more exotic processes as examples. In this work, we remedy this fact by connecting these two frameworks. We prove that for any given $N$, one can use a single routed graph to systematically obtain a routed circuit decomposition for any QC-QC with $N$ parties. We detail this construction explicitly and contrast it with other routed circuit decompositions of QC-QCs, which we obtain from alternative routed graphs. We conclude by pointing out how this connection can be useful to tackle various open problems in the field of indefinite causal order, particularly establishing circuit representations of subclasses of QC-QCs.

Figures

Figures reproduced from arXiv: 2507.08781 by the authors.

Figure 2.1
Figure 2.1. Two relations λ : K → L1 ×L2 and µ : L2 ×L3 → M can be composed through L2, resulting in a relation λ ∗ µ : K × L3 → L1 × M that can be written as λ ∗ µ = TrL2 (λ ⊗ µ), as in Eq. (2.4). A (binary) relation λ over some sets K and L will be denoted λ : K → L. It can be represented as a Boolean matrix (λ l k ) l∈L k∈K, where the lower and upper indices represent the inputs and outputs of the relation, respectively, and… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figures from the paper (38 more)
Figure 2.2
Figure 2.2. Figure 2.2: Example of a branched relation λ : K → L, with its domain and codomain partitioned into branches (reproducing the example of [PITH_FULL_IMAGE:figures/full_fig_p007_2_2.png]
Figure 2.3
Figure 2.3. Figure 2.3: Augmenting the branched relation λ of [PITH_FULL_IMAGE:figures/full_fig_p007_2_3.png]
Figure 2
Figure 2. Figure 2: a [PITH_FULL_IMAGE:figures/full_fig_p008_2.png]
Figure 2.4
Figure 2.4. Figure 2.4: (a) Example of a routed graph (reproducing that of [PITH_FULL_IMAGE:figures/full_fig_p009_2_4.png]
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 2.5
Figure 2.5. Figure 2.5: Depiction of the skeletal supermap (cf. Definition 2.21) associated to the routed graph shown in [PITH_FULL_IMAGE:figures/full_fig_p016_2_5.png]
Figure 3.1
Figure 3.1. Figure 3.1: Generic depiction of a QC-QC. Figure reproduced from Ref. [34]. QC-QCs fit into the framework of [39] in terms of a suitable routed circuit decomposition, which will then allow us to see how their properties translate into diagrammatic features. 3 Quantum Circuits wi…
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 4.1
Figure 4.1. Figure 4.1: Neighbourhood of each type of node in the generic routed graph GQC-QC(N) of a QC-QC: Ak, V1, Vn+1 (for n = 1, . . . , N − 1), and VN+1. Hence, the practical input and output sets of each node Ak consist of (the same) lists of index values ( ∅, . . . , ∅, Kn, ∅, . . .…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 4.2
Figure 4.2. Figure 4.2: The generic routed graph GQC-QC(N) for N = 2 (left) and N = 3 (right). Here, A1, A2, A3 are denoted as A, B, C. 4.1.2 Identifying the branches A-nodes. The lists of index values ( ∅, . . . , ∅, Kn, ∅, . . . , ∅) in the practical input and output sets of each node Ak …
Figure 4.3
Figure 4.3. Figure 4.3: Branches of the V-nodes. For each node Vn+1, for n = 0, . . . , N, λVn+1 : Indin Vn+1 → Indout Vn+1 features N n  branches, as well as values outside its practical input and output sets (not shown). Each branch V Kn n+1 is associated with n input values (for n ≥ 1),…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 4.4
Figure 4.4. Figure 4.4: Augmented relations for the A- and V-nodes of the routed graph GQC-QC(N). We do not show any auxiliary input for the nodes Ak and VN+1, as there are no bifurcation choices to be made. Notice also that the auxiliary output of V1, as well as both outputs of VN+1, are c…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Generic branch graphs for GQC-QC(N) with N = 2 (left) and with N = 3 (right). For simplicity of notation, {A, B} and similar in the superscripts are written as AB. In the graph on the right, the unconnected green dashed arrows leaving from V∅ 1 are understood as bein…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]
Figure 4.6
Figure 4.6. Figure 4.6: Fleshing out for the V-nodes (with a channel). For each slot Vn+1, isomorphisms J in/out Vn+1 translate between the skeletal supermap and its associated spaces in the neighbourhood of the slot, and the QC-QC systems A˜ I/O n , αn+1 and C (′) n+1 within the slot. In-b…
Figure 4.7
Figure 4.7. Figure 4.7: Fleshing out for the A-nodes (with a 1-slot comb). After the fleshing out, an empty (un￾sectorised) slot A I k → A O k remains, associated with the free operation of an agent Ak, while in parallel, system C¯(k) maintains the control system of the QC-QC. Two isomorphi…
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p034_4.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p035_4.png]
Figure 4
Figure 4. Figure 4: a [PITH_FULL_IMAGE:figures/full_fig_p036_4.png]
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p037_3.png]
Figure 5
Figure 5. Figure 5: b [PITH_FULL_IMAGE:figures/full_fig_p039_5.png]
Figure 5.1
Figure 5.1. Figure 5.1: (a,b) The split-node generic routed graph G split QC-QC(N) (for N = 2, 3) according to Defini￾tion 5.1, splitting the isometries Vn+1 featured in the generic routed graph GQC-QC(N) shown in [PITH_FULL_IMAGE:figures/full_fig_p040_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: The branch graph for the split-node generic routed graph G split QC-QC(2) according to Defini￾tion 5.1, featuring additional branches V¯ Kn n+1 in addition to those present in the branch graph of GQC-QC(2) shown in [PITH_FULL_IMAGE:figures/full_fig_p041_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: A general example for a node-splitting procedure, obtaining the routed graph shown in (b) from the routed graph shown in (a). While for the latter, we have Bran(λN) = {Ni=1 , Nj=1 , Nk=1}, each of these branches is represented by an individual node Nˆ β with Bran(λNˆ…
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p042_5.png]
Figure 5.4
Figure 5.4. Figure 5.4: Routed and branch graph architectures which allow for node merging according to Proposi￾tion 5.6. For the branch graph excerpts, we only show the most relevant arrows for merging, which will no longer be featured in the branch graph obtained afterwards. In the associ…
Figure 5
Figure 5. Figure 5: a [PITH_FULL_IMAGE:figures/full_fig_p043_5.png]
Figure 5
Figure 5. Figure 5: d [PITH_FULL_IMAGE:figures/full_fig_p044_5.png]
Figure 5
Figure 5. Figure 5: b [PITH_FULL_IMAGE:figures/full_fig_p045_5.png]
Figure 4
Figure 4. Figure 4: a [PITH_FULL_IMAGE:figures/full_fig_p047_4.png]
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p053_4.png]
Figure 5
Figure 5. Figure 5: a [PITH_FULL_IMAGE:figures/full_fig_p057_5.png]
Figure 5
Figure 5. Figure 5: d [PITH_FULL_IMAGE:figures/full_fig_p058_5.png]

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Works this paper leans on

64 extracted references · 44 canonical work pages

  1. [39]

    Consistent circuits for indefinite causal order

    Augustin Vanrietvelde, Nick Ormrod, Hlér Kristjánsson, and Jonathan Barrett. “Consistent circuits for indefinite causal order” (2023). arXiv:2206.10042

    arXiv 2023

  2. [1]

    The direction of time

    Hans Reichenbach. “The direction of time”. University of California Press. Berkeley (1956)

    work page 1956

  3. [2]

    Causation, prediction, and search

    Peter Spirtes, Clark Glymour, and Richard Scheines. “Causation, prediction, and search”. Springer New York. NY (1993)

    work page 1993

  4. [3]

    Causality: Models, reasoning and inference

    Judea Pearl. “Causality: Models, reasoning and inference”. Cambridge University Press. (2009). 2nd edition. 51However, this alone does not imply the ability to interpret each (valid) skeletal supermap as a blueprint for a non-routed process! 49

    work page 2009

  5. [4]

    The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning

    Christopher J. Wood and Robert W. Spekkens. “The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning”. New J. Phys.17, 033002 (2015)

    work page 2015

  6. [5]

    Quantum common causes and quantum causal models

    John-Mark A. Allen, Jonathan Barrett, Dominic C. Horsman, Ciarán M. Lee, and Robert W. Spekkens. “Quantum common causes and quantum causal models”. Phys. Rev. X 7, 031021 (2017)

    work page 2017

  7. [6]

    Quantum causal models

    Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. “Quantum causal models” (2019). arXiv:1906.10726

    arXiv 2019

  8. [7]

    Quantum gravity computers: On the theory of computation with indefinite causal structure

    Lucien Hardy. “Quantum gravity computers: On the theory of computation with indefinite causal structure”. Pages 379–401. Springer Netherlands. (2009). arXiv:quant-ph/0701019

    work page Pith review arXiv 2009

Show all 64 references
  1. [8]

    Bounding quantum correlations with indefinite causal order

    Časlav Brukner. “Bounding quantum correlations with indefinite causal order”. New J. Phys. 17, 083034 (2015)

    2015

  2. [9]

    Quantum causal modelling

    Fabio Costa and Sally Shrapnel. “Quantum causal modelling”. New J. Phys.18, 063032 (2016)

    2016

  3. [10]

    Cyclic quantum causal models

    Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. “Cyclic quantum causal models”. Nature Commun.12 (2021)

    2021

  4. [11]

    Embedding cyclic information-theoretic structures in acyclic space-times: No-go results for indefinite causality

    V. Vilasini and Renato Renner. “Embedding cyclic information-theoretic structures in acyclic space-times: No-go results for indefinite causality”. Phys. Rev. A 110, 022227 (2024). arXiv:2203.11245

    2024 arXiv

  5. [12]

    Is there causation in fundamental physics? New insights from process matrices and quantum causal modelling

    Emily Adlam. “Is there causation in fundamental physics? New insights from process matrices and quantum causal modelling”. Synthese201, 152 (2023)

    2023

  6. [13]

    Fundamental limits for realizing quantum processes in space- time

    V. Vilasini and Renato Renner. “Fundamental limits for realizing quantum processes in space- time”. Phys. Rev. Lett.133, 080201 (2024). arXiv:2408.13387

    2024 arXiv

  7. [14]

    Transforming quantum operations: Quantum supermaps

    Giulio Chiribella, Giacomo Mauro D’Ariano, and Paulo Perinotti. “Transforming quantum operations: Quantum supermaps”. Europhys. Lett.83, 30004 (2008). arXiv:0804.0180

    2008 arXiv

  8. [15]

    Quantum correlations with no causal order

    Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. “Quantum correlations with no causal order”. Nature Commun.3, 1092 (2012). arXiv:1105.4464

    2012 arXiv

  9. [16]

    Bell’stheoremfortemporal order

    MagdalenaZych, FabioCosta, IgorPikovski, andČaslavBrukner. “Bell’stheoremfortemporal order”. Nature Commun.10, 3772 (2019)

    2019

  10. [17]

    Quantum switch in the gravity of earth

    Natália S. Móller, Bruna Sahdo, and Nelson Yokomizo. “Quantum switch in the gravity of earth”. Phys. Rev. A104, 042414 (2021). arXiv:2012.03989

    2021 arXiv

  11. [18]

    Noncausal Page-Wootters circuits

    Veronika Baumann, Marius Krumm, Philippe Allard Guérin, and Časlav Brukner. “Noncausal Page-Wootters circuits”. Phys. Rev. Res.4, 013180 (2022)

    2022

  12. [19]

    Quantum diffeomorphisms cannot make indefinite causal order definite

    Anne-Catherine de la Hamette, Viktoria Kabel, Marios Christodoulou, and Časlav Brukner. “Quantum diffeomorphisms cannot make indefinite causal order definite” (2024). arXiv:2211.15685

    2024

  13. [20]

    Indefinite order in the interface of quantum mechanics and gravity

    Bruna Sahdo. “Indefinite order in the interface of quantum mechanics and gravity”. Master’s thesis. Federal University of Minas Gerais. (2023). arXiv:2310.02290

    2023 arXiv

  14. [21]

    Quan- tum computations without definite causal structure

    Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. “Quan- tum computations without definite causal structure”. Phys. Rev. A 88, 022318 (2013). arXiv:0912.0195

    2013 arXiv

  15. [22]

    A purification postulate for quantum mechanics with indefinite causal order

    Mateus Araújo, Adrien Feix, Miguel Navascués, and Časlav Brukner. “A purification postulate for quantum mechanics with indefinite causal order”. Quantum1, 10 (2017)

    2017

  16. [23]

    Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios

    Ämin Baumeler, Adrien Feix, and Stefan Wolf. “Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios”. Phys. Rev. A90, 042106 (2014). arXiv:1403.7333

    2014 arXiv

  17. [24]

    The space of logically consistent classical processes without causal order

    Ämin Baumeler and Stefan Wolf. “The space of logically consistent classical processes without causal order”. New J. Phys.18, 013036 (2016)

    2016

  18. [25]

    Perfect discrimination of no-signalling channels via quantum superposition of causal structures

    Giulio Chiribella. “Perfect discrimination of no-signalling channels via quantum superposition of causal structures”. Phys. Rev. A86, 040301 (2012). arXiv:1109.5154

    2012 arXiv

  19. [26]

    Quantum superposition of the order of parties as a communication resource

    Adrien Feix, Mateus Araújo, and Časlav Brukner. “Quantum superposition of the order of parties as a communication resource”. Phys. Rev. A92, 052326 (2015). arXiv:1508.07840

    2015 arXiv

  20. [27]

    Exponential communication complexity advantage from quantum superposition of the direction of com- munication

    Philippe Allard Guérin, Adrien Feix, Mateus Araújo, and Časlav Brukner. “Exponential communication complexity advantage from quantum superposition of the direction of com- munication”. Phys. Rev. Lett.117, 100502 (2016). arXiv:1605.07372. 50

    2016 arXiv

  21. [28]

    Quan- tum computation with programmable connections between gates

    Timoteo Colnaghi, Giacomo Mauro D’Ariano, Stefano Facchini, and Paolo Perinotti. “Quan- tum computation with programmable connections between gates”. Phys. Lett. A376, 2940– 2943 (2012). arXiv:1109.5987

    2012 arXiv

  22. [29]

    Quantum circuits for the unitary permutation problem

    Stefano Facchini and Simon Perdrix. “Quantum circuits for the unitary permutation problem”. In Theory and Applications of Models of Computation. Pages 324–331. Springer International Publishing (2015). arXiv:1405.5205

    2015 arXiv

  23. [30]

    Computational advantage from quantum- controlled ordering of gates

    Mateus Araújo, Fabio Costa, and Časlav Brukner. “Computational advantage from quantum- controlled ordering of gates”. Phys. Rev. Lett.113, 250402 (2014). arXiv:1401.8127

    2014 arXiv

  24. [31]

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

    Márcio M. Taddei, Jaime Cariñe, Daniel Martínez, Tania García, Nayda Guerrero, Alastair A. Abbott, Mateus Araújo, Cyril Branciard, Esteban S. Gómez, Stephen P. Walborn, Leandro Aolita, and Gustavo Lima. “Computational advantage from the quantum superposition of multiple tempor...

    2021

  25. [32]

    Classical and quantum query com- plexity of Boolean functions under indefinite causal order

    Alastair A. Abbott, Mehdi Mhalla, and Pierre Pocreau. “Classical and quantum query com- plexity of Boolean functions under indefinite causal order”. In Proc. 22nd Int. Conf. Quantum Phys. Log. (QPL 2025). to appear in EPTCS (2025). arXiv:2506.05187

    2025 arXiv

  26. [33]

    Causal boxes: Quantum information-processing systems closed under composition

    Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Björn Tackmann. “Causal boxes: Quantum information-processing systems closed under composition”. IEEE Trans. Inf. Theory63, 3277–3305 (2017). arXiv:1512.02240

    2017 arXiv

  27. [34]

    Quantum cir- cuits with classical versus quantum control of causal order

    Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard. “Quantum cir- cuits with classical versus quantum control of causal order”. PRX Quantum2, 030335 (2021)

    2021

  28. [35]

    PBS-calculus: A graphical language for coher- ent control of quantum computations

    Alexandre Clément and Simon Perdrix. “PBS-calculus: A graphical language for coher- ent control of quantum computations”. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Volume 170 of Leibniz International Pro- ceedings in Informati...

    2020 arXiv

  29. [36]

    Coherentcontroland distinguishability of quantum channels via PBS-diagrams

    CyrilBranciard, AlexandreClément, MehdiMhalla, andSimonPerdrix. “Coherentcontroland distinguishability of quantum channels via PBS-diagrams”. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Pro- ceedings in Inf...

    2021 arXiv

  30. [37]

    Quantum theory cannot violate a causal inequality

    Tom Purves and Anthony J. Short. “Quantum theory cannot violate a causal inequality”. Phys. Rev. Lett.127, 110402 (2021). arXiv:2101.09107

    2021 arXiv

  31. [38]

    Ad- dressable quantum gates

    Pablo Arrighi, Christopher Cedzich, Marin Costes, Ulysse Rémond, and Benoît Valiron. “Ad- dressable quantum gates”. ACM Trans. Quantum Comput. (2023). arXiv:2109.08050

    2023

  32. [40]

    The Many-Worlds Calculus

    Kostia Chardonnet, Marc de Visme, Benoît Valiron, and Renaud Vilmart. “The Many-Worlds Calculus”. Log. Methods Comput. Sci21, 2 (2025)

    2025

  33. [41]

    Connecting indefinite causal order processes to composable quantum pro- tocols in a spacetime

    Matthias Salzger. “Connecting indefinite causal order processes to composable quantum pro- tocols in a spacetime”. Master’s thesis. ETH Zurich. (2022). arXiv:2304.06735

    2022 arXiv

  34. [42]

    Mapping indefinite causal order processes to composable quantum protocols in a spacetime

    Matthias Salzger and V Vilasini. “Mapping indefinite causal order processes to composable quantum protocols in a spacetime”. New J. Phys.27, 023002 (2025)

    2025

  35. [43]

    Routed quantum circuits

    Augustin Vanrietvelde, Hlér Kristjánsson, and Jonathan Barrett. “Routed quantum circuits”. Quantum 5, 503 (2021)

    2021

  36. [44]

    Quantum Shannon theory with superpositions of trajectories

    Giulio Chiribella and Hlér Kristjánsson. “Quantum Shannon theory with superpositions of trajectories”. Proc. Roy. Soc. A475, 20180903 (2019)

    2019

  37. [45]

    Quantum circuit architecture

    G. Chiribella, G. M. D’Ariano, and P. Perinotti. “Quantum circuit architecture”. Phys. Rev. Lett. 101, 060401 (2008). arXiv:0712.1325

    2008 arXiv

  38. [46]

    Theoretical framework for quantum networks

    Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. “Theoretical framework for quantum networks”. Phys. Rev. A80, 022339 (2009). arXiv:0904.4483

    2009 arXiv

  39. [47]

    Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning

    Bob Coecke and Aleks Kissinger. “Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning”. Cambridge University Press. (2017)

    2017

  40. [48]

    Unilateral determination of causal order in a cyclic process

    Ilyass Mejdoub and Augustin Vanrietvelde. “Unilateral determination of causal order in a cyclic process” (2025). arXiv:2506.18540

    2025 arXiv

  41. [49]

    Completely positive linear maps on complex matrices

    Man-Duen Choi. “Completely positive linear maps on complex matrices”. Linear Algebra Appl. 10, 285–290 (1975). 51

    1975

  42. [50]

    Scalable quantum control of causal order with single use of operations

    Fabio Costa and Lee Rozema. “Scalable quantum control of causal order with single use of operations” (2025). In preparation; presented at Relativistic Quantum Information - North 2024

    2025

  43. [51]

    Causal structure in the presence of sectorial constraints, with application to the quantum switch

    Nick Ormrod, Augustin Vanrietvelde, and Jonathan Barrett. “Causal structure in the presence of sectorial constraints, with application to the quantum switch”. Quantum7, 1028 (2023)

    2023

  44. [52]

    Operational causality in the Page-Wootters framework

    Lukas Schmitt. “Operational causality in the Page-Wootters framework”. Master’s thesis. ETH Zurich. (2023)

    2023

  45. [53]

    What an event is not: unravelling the identity of events in quantum theory and gravity

    Anne-Catherine de la Hamette, Viktoria Kabel, and Časlav Brukner. “What an event is not: unravelling the identity of events in quantum theory and gravity” (2024). arXiv:2404.00159

    2024 arXiv

  46. [54]

    Optimal strategies of quantum metrology with a strict hierarchy

    Qiushi Liu, Zihao Hu, Haidong Yuan, and Yuxiang Yang. “Optimal strategies of quantum metrology with a strict hierarchy”. Phys. Rev. Lett.130, 070803 (2023). arXiv:2203.09758

    2023 arXiv

  47. [55]

    Reassessing the advantage of indefinite causal orders for quantum metrology

    Raphaël Mothe, Cyril Branciard, and Alastair A. Abbott. “Reassessing the advantage of indefinite causal orders for quantum metrology”. Phys. Rev. A 109, 062435 (2024). arXiv:2312.12172

    2024 arXiv

  48. [56]

    Correlations and quantum circuits with dynamical causal order

    Raphaël Mothe, Alastair A. Abbott, and Cyril Branciard. “Correlations and quantum circuits with dynamical causal order” (2025). arXiv:2507.07992

    2025

  49. [57]

    A decompositional framework for process theories in spacetime

    Matthias Salzger and John H. Selby. “A decompositional framework for process theories in spacetime” (2025). arXiv:2411.08266

    2025

  50. [58]

    A fine-grained perspective on indefinite causal order: Modelling agents in spacetime

    Emilien de Bank, V. Vilasini, and Cyril Branciard. “A fine-grained perspective on indefinite causal order: Modelling agents in spacetime” (2025). In preparation

    2025

  51. [59]

    Causal and compositional structure of unitary trans- formations

    Robin Lorenz and Jonathan Barrett. “Causal and compositional structure of unitary trans- formations”. Quantum5, 511 (2021)

    2021

  52. [60]

    Causal decompositions of 1D quantum cellular automata

    Augustin Vanrietvelde, Octave Mestoudjian, and Pablo Arrighi. “Causal decompositions of 1D quantum cellular automata” (2025). arXiv:2506.22219

    2025 arXiv

  53. [61]

    A complete logic for causal consistency

    Will Simmons and Aleks Kissinger. “A complete logic for causal consistency” (2024). arXiv:2403.09297

    2024 arXiv

  54. [62]

    A categorical semantics for causal structure

    Aleks Kissinger and Sander Uijlen. “A categorical semantics for causal structure”. Log. Meth- ods Comput. Sci.15, 3 (2019)

    2019

  55. [63]

    Admissible causal structures and correla- tions

    Eleftherios-Ermis Tselentis and Ämin Baumeler. “Admissible causal structures and correla- tions”. PRX Quantum4, 040307 (2023)

    2023

  56. [64]

    No-signalling constrains quantum computation with indefinite causal structure

    Luca Apadula, Alessandro Bisio, and Paolo Perinotti. “No-signalling constrains quantum computation with indefinite causal structure”. Quantum8, 1241 (2024). 52 Appendix A Derivation of the choice function forGQC−QC(N) Here we will evaluate explicitly the choice relationΛGQC-QC...

    2024

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