pith. sign in

arxiv: 2605.22807 · v1 · pith:CXMFS4BQnew · submitted 2026-05-21 · 🪐 quant-ph

How many systems can be dephased before the quantum switch becomes causally definite?

Yassine Benhaj , Kuntal Sengupta , Cyril Branciard This is my paper

Pith reviewed 2026-05-22 05:07 UTC · model grok-4.3

classification 🪐 quant-ph
keywords causal nonseparabilityindefinite causal orderdephasingquantum processesquantum circuits with quantum controlbipartite processesquantum switch
0
0 comments X

The pith

For bipartite processes, dephasing all systems or only the future one makes them causally separable, while nonseparability persists if any non-future system stays undephased; the same holds for multipartite QC-QCs.

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

The paper examines how dephasing affects causal nonseparability in quantum processes that lack a definite causal order. It proves that in bipartite cases with open past and future, full dephasing or dephasing everything except the future system forces the process to become causally separable. Yet processes can keep their causal nonseparability when even one non-future system remains coherent. The identical threshold appears for multipartite quantum circuits with quantum control. This identifies the minimal coherence needed to maintain indefinite causal order against decoherence.

Core claim

For bipartite processes with open past and future, dephasing all systems or only the future system renders the process causally separable. If any single system other than the future remains undephased, there exist processes that retain causal nonseparability. The same separation threshold holds for multipartite QC-QCs: dephasing all systems or only the future one yields causal separability, while leaving any non-future system undephased can preserve causal nonseparability.

What carries the argument

Selective application of a completely positive dephasing map to individual systems in the process matrix or QC-QC, which removes off-diagonal coherence terms while leaving the overall process structure intact.

If this is right

  • Causal nonseparability in these processes requires coherence in at least one non-future system.
  • Full dephasing or future-only coherence eliminates any advantage that indefinite causal order might provide.
  • The quantum switch and similar constructions lose causal nonseparability under the same dephasing conditions.
  • The result supplies a concrete criterion for when noise destroys causal indefiniteness in open quantum processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experiments testing indefinite causal order may tolerate decoherence on the future system more readily than on earlier ones.
  • The threshold could guide error-correction strategies that protect only the minimal set of systems needed for nonseparability.
  • Similar dephasing analysis might apply to other classes of causally nonseparable processes beyond QC-QCs.

Load-bearing premise

The multipartite conclusions assume the processes belong to the class of quantum circuits with quantum control and that dephasing acts as an independent map on each system.

What would settle it

Finding a specific bipartite process matrix or multipartite QC-QC that remains causally nonseparable after dephasing every system, or after dephasing all systems except one non-future system, would disprove the stated thresholds.

Figures

Figures reproduced from arXiv: 2605.22807 by Cyril Branciard, Kuntal Sengupta, Yassine Benhaj.

Figure 1
Figure 1. Figure 1: FIG. 1. Any bipartite process matrix in which the global [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Three explicit constructions derived from a standard quantum switch showing that causal indefiniteness can [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
read the original abstract

Quantum processes with indefinite causal order -- so-called causally nonseparable processes -- can exhibit various advantages over quantum circuits with a fixed or a well-defined causal structure. A natural question is how much nonclassicality is required for a process to display causal nonseparability. Here we address this by investigating how many systems can be dephased (or decohered) before this property vanishes. First, for bipartite processes with open past and future we show that if all systems are dephased, or if only the future system is kept undephased, then the process becomes causally separable. However, if any single system other than the future system remains undephased, then there exist processes that retain causal nonseparability. Next, we demonstrate a similar behaviour in the multipartite case, when restricted to the physically motivated class of quantum circuits with quantum control (QC-QCs). Namely, dephasing all systems or keeping only the future system undephased renders any QC-QC causally separable; while causal nonseparability can persist if any non-future system is left undephased.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates how dephasing affects causal nonseparability in quantum processes. For bipartite processes with open past and future, it shows that dephasing all systems or keeping only the future system undephased renders the process causally separable, while nonseparability can persist if any single non-future system remains undephased. Similar results are established for multipartite processes restricted to the class of quantum circuits with quantum control (QC-QCs).

Significance. If the derivations hold, the work quantifies the minimal coherence needed to sustain indefinite causal order, distinguishing the role of the future system from others. This provides concrete bounds on decoherence effects in causally nonseparable processes and strengthens the connection between standard quantum channels and causal structure, with potential relevance for resource theories in quantum information.

minor comments (3)
  1. The title refers to the quantum switch becoming 'causally definite,' yet the abstract and body address general bipartite processes and QC-QCs. Clarify whether the quantum switch is the primary example or if the claims are intended more broadly.
  2. The modeling of dephasing as a specific completely positive map acting independently on each system is central to the claims; ensure the explicit form of this map (e.g., Kraus operators) is stated in the main text with a dedicated equation or definition.
  3. In the multipartite section, the restriction to QC-QCs is noted in the abstract but could be reiterated briefly in the introduction or conclusion to prevent misreading as applying to all causally nonseparable processes.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly captures our main results on the effects of dephasing on causal nonseparability for bipartite processes and for QC-QCs in the multipartite case. We appreciate the recognition that this work provides concrete bounds on decoherence and distinguishes the role of the future system, with potential relevance to resource theories. Since the report lists no specific major comments, we have no individual points to rebut or revise at this stage, but we will address any minor issues in the updated version.

Circularity Check

0 steps flagged

No circularity: results follow from standard definitions of causal nonseparability and dephasing maps

full rationale

The paper derives its bipartite and multipartite (QC-QC-restricted) results directly from the definitions of causal nonseparability for process matrices and the explicit action of the dephasing CPTP maps on each system. No step reduces a claimed prediction or theorem to a fitted parameter, self-referential definition, or load-bearing self-citation; the restriction to QC-QCs is stated explicitly rather than smuggled in. The derivation chain is therefore self-contained against external benchmarks in quantum process theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard background definitions from quantum information theory rather than introducing new free parameters or invented entities.

axioms (1)
  • standard math Standard definitions and properties of causally nonseparable processes and dephasing maps in quantum theory.
    Invoked throughout the abstract to frame the results on causal separability.

pith-pipeline@v0.9.0 · 5732 in / 1243 out tokens · 35808 ms · 2026-05-22T05:07:21.917270+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 10 internal anchors

  1. [1]

    quantum switch

    Bipartite case A matrixW∈ L(H P AIO BIO F )is the process matrix of a bipartite QC-QC (withN= 2input operations A, B) if and only ifWis PSD and there exist PSD matricesW(A,B) ∈ L(H P AIO BI ),W (B,A) ∈ L(H P BIO AI ), W(A) ∈ L(H P AI ),W (B) ∈ L(H P BI )such that TrF W=W (A,B) ⊗1 BO +W (B,A) ⊗1 AO ,(10) TrBI W(A,B) =W (A) ⊗1 AO ,Tr AI W(B,A) =W (B) ⊗1 BO ...

    work page

  2. [2]

    already dephased

    Bipartite case A matrixW∈ L(H P AIO BIO F )is the process matrix of a bipartite QC-CC if and only ifWis PSD and there exist PSD matricesW(A,B,F) ∈ L(H P AIO BIO F ),W (B,A,F) ∈ L(H P AIO BIO F ),W (A,B) ∈ L(H P AIO BI ), W(B,A) ∈ L(H P BIO AI ),W (A) ∈ L(H P AI ),W (B) ∈ L(H P BI )such that W=W (A,B,F) +W (B,A,F) ,(16) TrF W(A,B,F) =W (A,B) ⊗1 BO ,Tr F W(...

    work page

  3. [3]

    Quantum computations without definite causal structure

    G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Phys. Rev. A88, 022318 (2013), arXiv:0912.0195 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  4. [4]

    Quantum correlations with no causal order

    O. Oreshkov, F. Costa, and Č. Brukner, Nat. Commun.3, 1092 (2012), arXiv:1105.4464 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  5. [5]

    Appearance of causality in process matrices when performing fixed-basis measurements for two parties

    V. Baumann and Č. Brukner, Phys. Rev. A93, 062324 (2016), arXiv:1601.06620 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  6. [6]

    van der Lugt, J

    T. van der Lugt, J. Barrett, and G. Chiribella, Nat. Commun.14, 5811 (2023), arXiv:2208.00719 [quant-ph]

    work page arXiv 2023

  7. [7]

    [16] also showed an advantage for the quantum switch with dephased slots (as in that case, fully depolarising channels were plugged into these)

    Note that Ref. [16] also showed an advantage for the quantum switch with dephased slots (as in that case, fully depolarising channels were plugged into these). However, the advantage was not with respect to all possible causally definite processes (but only when comparing to a simple sequential composition of channels in a fixed order), hence that advanta...

    work page

  8. [8]

    2, certain systems in the quantum switch are dephased in theXbasis

    As illustrated on Fig. 2, certain systems in the quantum switch are dephased in theXbasis. Alternatively, one could first rotate these systems (by applying a Hadamard gate) and then dephase all systems in the computational basis. As one can see, the specific choice of bases is irrelevant for the claim thatthere existsa process that remains causally indefi...

    work page

  9. [9]

    Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios

    A. Baumeler, A. Feix, and S. Wolf, Phys. Rev. A90, 042106 (2014), arXiv:1403.7333 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  10. [10]

    The space of logically consistent classical processes without causal order

    Ä. Baumeler and S. Wolf, New J. Phys.18, 013036 (2016), arXiv:1507.01714 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  11. [11]

    Wechs, H

    J. Wechs, H. Dourdent, A. A. Abbott, and C. Branciard, PRX Quantum2, 030335 (2021), arXiv:2101.08796 [quant-ph]

    work page arXiv 2021

  12. [12]

    Transforming quantum operations: quantum supermaps

    G. Chiribella, G. M. D’Ariano, and P. Perinotti, EPL83, 30004 (2008), arXiv:0804.0180 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  13. [13]

    Choi, Linear Algebra Its Appl.10, 285 (1975)

    M.-D. Choi, Linear Algebra Its Appl.10, 285 (1975)

    work page 1975

  14. [14]

    Araújo, A

    M. Araújo, A. Feix, M. Navascués, and Č. Brukner, Quantum1, 10 (2017), arXiv:1611.08535 [quant-ph]

    work page arXiv 2017

  15. [15]

    Witnessing causal nonseparability

    M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, New J. Phys.17, 102001 (2015), arXiv:1506.03776

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    On the definition and characterisation of multipartite causal (non)separability

    J. Wechs, A. A. Abbott, and C. Branciard, New J. Phys.21, 013027 (2019), arXiv:1807.10557 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  17. [17]

    Causal and causally separable processes

    O. Oreshkov and C. Giarmatzi, New J. Phys.18, 093020 (2016), arXiv:1506.05449 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  18. [18]

    Enhanced communication with the assistance of indefinite causal order

    D. Ebler, S. Salek, and G. Chiribella, Phys. Rev. Lett.120, 120502 (2018), arXiv:1711.10165 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018