Quantum Directional Signalling

Exact diamond-norm evaluations of directional signalling for continuous bipartite unitary families.

For a unitary $U$ on $A\otimes B$, the project studies

$$ \delta_A(U)=\inf_{\mathcal E\ \mathrm{CPTP}} \left|\operatorname{Tr}_B\operatorname{Ad}_U -\mathcal E\operatorname{Tr}_B\right|_\diamond . $$

This asks how well the reduced output on $A$ can be predicted from the reduced input on $A$ alone, even when the input is correlated with hidden system $B$ and an arbitrary reference.

Plain-language picture

Imagine two quantum objects, A and B, interact. You then discard B and try to describe A using only what was initially known about A. That description can fail because B carried hidden information that affected A.

$\delta_A(U)$ is the smallest unavoidable worst-case error, even after choosing the best possible A-only rule. Zero means A has autonomous dynamics; a larger value means B can signal more strongly into A. The diamond norm makes this an adversarial test that also permits correlations with an external reference.

The main qualitative result is that the best simplified rule can change as the interaction gets stronger. For partial SWAP it eventually becomes better to add deliberate noise than to pretend A evolved unchanged. For XY the exact curve has three regimes:

Results

• Two-qubit Ising: for $U_\theta=e^{-i\theta Z\otimes Z}$, $\delta_A(U_\theta)=|\sin 2\theta|$.
• Partial SWAP in dimension $d$: the optimization reduces exactly to one depolarizing parameter. For $\sin\phi\le (d^2-3)/(d^2-1)$, $\delta_A(U_\phi)=2\sin\phi$; at SWAP, $\delta_A=2(1-1/d^2)$.
• Two-qubit XY/iSWAP: the weak branch is $\delta_A=\sin4\theta$, while near iSWAP it is $\delta_A=\sin2\theta+\tfrac12\sin^2 2\theta$, giving $\delta_A(\mathrm{iSWAP})=3/2$. Together with the universal bound of Barsse et al., this makes iSWAP maximally signalling among two-qubit unitaries.
• Intermediate XY regime: an exact quartic and unique physical root are proved. Convex KKT analysis establishes the $q_Z=0$ active facet, and an exact resultant plus root isolation selects the physical algebraic sheet.

The signalling measure itself and the exact isolated CNOT and SWAP values are prior work by Barsse, Perinotti, Tosini, and Vaglini. The contribution here is the continuous-family evaluation, thresholds, optimal effective channels, and matching witnesses.

Their convention measures signalling from $A$ to $B$; this repository measures the reversed direction $B$ to $A$. After exchanging subsystem labels, the definitions and unhalved diamond-norm normalization coincide.

Repository layout

• quantum_coarse_graining/ — exact Gaussian-rational certificate library.
• tests/ — focused theorem and regression certificates.
• paper/aqc_signalling.tex — manuscript source.
• paper/aqc_signalling.pdf — compiled manuscript.
• paper/*_phase.dat — generated optimal-channel and witness trajectories.
• tools/derive_xy_middle.py — exact symbolic eliminations; part of the proof of record for the intermediate XY branch.
• tools/verify_block_reductions.py — independent exact physical-Choi spectra.
• tools/check_numerical_sdp.py — independent unreduced numerical SDP.
• AUDIT.md — independent-review checklist.
• RESEARCH_NOTES.md — claims ledger, proof boundaries, and open questions.

Reproduce

The exact certificate suite has no third-party Python dependencies:

python3 -m unittest discover -s tests -p 'test_aqc_*.py' -v

Verify the load-bearing XY eliminations. The intermediate-branch result is a computer-assisted proof: the conjugate-sheet exclusion, the resultant factorization, the cofactor positivity, and the Sturm root count are checked here in exact rational arithmetic, with no floating point. SymPy is an optional dependency, but this step is not evidentially optional.

python3 -m pip install -e '.[proof]'
python3 tools/derive_xy_middle.py
python3 tools/verify_block_reductions.py

The block-reduction checker constructs the unreduced physical Choi operators and verifies their exact characteristic polynomials for partial SWAP in dimensions 2, 3, and 4 and for the XY reduction. It does not import the certificate package. These are spot checks of a hand calculation that is valid for all d; they are not the source of the all-dimensions claim.

Independently solve the original, unreduced channel/diamond-norm SDP:

python3 -m pip install -e '.[numerical]'
python3 tools/check_numerical_sdp.py

This numerical checker does not import the exact certificate package. It optimizes over an arbitrary CPTP effective channel and compares the result with representative Ising, partial-SWAP, and XY formulas in every regime. Use --full to include the slower qutrit partial-SWAP checks.

Build the manuscript with:

sudo apt-get install texlive-latex-extra texlive-fonts-recommended lmodern
cd paper
pdflatex -interaction=nonstopmode -halt-on-error aqc_signalling.tex
pdflatex -interaction=nonstopmode -halt-on-error aqc_signalling.tex

Regenerate the defect curve and optimizer phase diagrams with:

python3 paper/generate_xy_plot.py

Scope

This is mathematical quantum information, not a proposal for new fundamental particles, constants, or cosmology. Code supplies exact finite certificates and regression protection; analytical identities are proved in the manuscript.

The optional numerical SDP is deliberately redundant with the analytic development: it checks the unreduced optimization and is not used as proof.

Publication status

The repository contains a journal-style draft, not a peer-reviewed publication. The novelty statements are therefore phrased as "to our knowledge" and remain subject to independent literature review. The archived v0.1.0 release is available at https://doi.org/10.5281/zenodo.21706516. Use that version-specific DOI when citing this software and reproducibility archive.

Outstanding pre-publication items are tracked in the "Outstanding before publication" section of AUDIT.md; after the v0.1.0 archive, these are an ORCID iD, an arXiv deposit, and a completed independent review.

Provenance

The work was developed on the approximate-quantum-coarse-graining branch of richorama/cho, through source commit a81742b, then extracted into this focused repository.

License

Copyright 2026 Richard Astbury.

• Source code and software tooling: MIT License.
• Paper, documentation, figures, and data: CC BY 4.0.