Mutually Unbiased Measurements, Hadamard Matrices, and Superdense Coding
Bibliographic record
Abstract
Mutually unbiased bases (MUBs) are highly symmetric bases on complex Hilbert spaces, and the corresponding rank-1 projective measurements are ubiquitous in quantum information theory. In this work, we study a recently introduced generalization of MUBs called mutually unbiased measurements (MUMs). These measurements inherit the essential property of complementarity from MUBs, but the Hilbert space dimension is no longer required to match the number of outcomes. This operational complementarity property renders MUMs highly useful for device-independent quantum information processing. It has been shown that MUMs are strictly more general than MUBs. In this work we provide a complete proof of the characterization of MUMs that are direct sums of MUBs. We then proceed to construct new examples of MUMs that are not direct sums of MUBs. A crucial technical tool for these construction is a correspondence with quaternionic Hadamard matrices, which allows us to map known examples of such matrices to MUMs that are not direct sums of MUBs. Furthermore, we show that -- in stark contrast with MUBs -- the number of MUMs for a fixed outcome number is unbounded. Next, we focus on the use of MUMs in quantum communication. We demonstrate how any pair of MUMs with d outcomes defines a d-dimensional superdense coding protocol. Using MUMs that are not direct sums of MUBs, we disprove a recent conjecture due to Nayak and Yuen on the rigidity of superdense coding for infinitely many dimensions. The superdense coding protocols arising in the refutation reveal how shared entanglement may be used in a manner heretofore unknown.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".