Metadata
bibkey: elliott2025complex
title: Strict advantage of complex quantum theory in a communication task
authors: Thomas J. Elliott
year: 2025
url: https://arxiv.org/abs/2505.16523v2
claim: A four-input communication task admits a complex-qubit strategy but no real-qubit strategy; non-real Bargmann invariants witness the ensemble's basis-independent imaginarity. Appendix D shows that a rebit strategy reaches average trace distance d = 1/12 from the target distribution and weakly conjectures that 1/12 is the minimum.
strata_touched:
- D5/S3/QuantumBounds/RebitExclusionDistance
license: citation-only
triage: anchor
doi: 10.1103/PhysRevA.111.062401
Complex dimension advantage and relational witnesses
Elliott, arXiv:2505.16523v2 (version dated 2 June 2025; manuscript dated 11 June 2025), studies the target conditional distribution on four symbols. Section III proves that complex quantum theory attains communication cost , corresponding to a qubit, while classical and real quantum strategies require a larger cost under the paper’s task assumptions. This is an existing operational separation between real and complex quantum state spaces.
Section II.B, equation (2), and Appendix B define Bargmann invariants as traces of products of density matrices. Their invariance under a common unitary and the reality of products of real matrices imply that a non-real invariant rules out one basis in which the entire ensemble is real. First- and second-order invariants are always real; third or higher order is needed for this witness. Section III.E, equation (6), finds for the task’s optimal strategies.
The communication task does not impose a continuum of programmable Schur channels, prescribed Fisher-information equality and optimal curvature, or one analytic preparation curve through specified nodes. The quantum volume’s program-support dimension question requires those additional constraints and its own rigidity and continuation proof. A strict real/complex dimension difference, or the use of an imaginary Bargmann invariant by itself, should not be presented as a new general principle.
The paper attributes relational-state measurements to Oszmaniec, Brod and Galvão, Measuring relational information between quantum states, and applications, New Journal of Physics 26, 013053 (2024), and set imaginarity witnesses to Fernandes et al., arXiv:2403.15066. These pointers do not substitute for inspecting the source when using a stronger realification or measurement theorem.
Verified locator
DOI: 10.1103/PhysRevA.111.062401 (Phys. Rev. A 111, 062401, from Crossref metadata; the journal full text was not read). Primary version read: https://arxiv.org/abs/2505.16523v2 (the latest arXiv version). Source: https://arxiv.org/pdf/2505.16523v2 (text read from the PDF):
- §II and §III.A, the task;
- §IV, the robustness question;
- Appendix D, “Bounding the distance for rebit strategies”.
The rebit distance
§III.A: “X = Y = {0, 1, 2, 3} … The target distribution for the game is P(Y|X) = (1 − δ_XY)/3”.
§IV: “For the trace distance, this would be d = (1/|X|) Σ_{x∈X} (1/2) Σ_{y∈Y} |P(y|x) − P̃(y|x)|, where P and P̃ are the target and realised distributions respectively. … Through an explicit construction we show that d = 1/12 is achievable for a rebit strategy. … Based on a rudimentary attempt at optimisation, we weakly conjecture that this is the optimal rebit strategy according to the average trace distance”.
Appendix D: “Through a rudimentary search over rebit strategies with gradient descent and Bayesian optimisation approaches, we were unable to find a rebit strategy with smaller error than this. … We therefore only weakly conjecture that d = 1/12 the the minimum distance for a rebit strategy.”
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