POVMs are equivalent to projections for perfect state exclusion of three pure states in three dimensions
Bibliographic record
Abstract
Performing perfect/conclusive quantum state exclusion means to be able to discard with certainty at least one out of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>possible quantum state preparations by performing a measurement of the resulting state. This task of state exclusion has recently been studied at length in \cite{bandyopadhyay2014conclusive}, and it is at the heart of the celebrated PBR thought experiment \cite{pusey2012reality}. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is mentioned in \cite{caves2002conditions} as unsuccessfully tackled. In this paper, we give an analytical proof that in this case considering POVMs does indeed not give any additional power with respect to projective measurements. To do so, we first make without loss of generality some assumptions about the structure of an optimal POVM. The justification of these assumptions involves arguments based on convexity, rank and symmetry properties. We show then that any pure states perfectly excluded by such a POVM meet the conditions identified in \cite{caves2002conditions} for perfect exclusion by a projective measurement of three pure states in three dimensions. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".