bibkey: braun2014universalqubitchannels authors: Daniel Braun, Olivier Giraud, Ion Nechita, Clément Pellegrini, Marko Žnidarič year: 2014 title: A universal set of qubit quantum channels doi: 10.1088/1751-8113/47/13/135302 url: https://arxiv.org/abs/1306.0495v2 claim: “Theorem IV.9 classifies pure output sets of qubit channels; three distinct pure outputs force unitary conjugation.” strata_touched:
- D5/S3/Quantum/QuantumChannels/PureProgramQubitObstruction license: citation-only triage: anchor
Verified locator
DOI: https://doi.org/10.1088/1751-8113/47/13/135302
Source: https://arxiv.org/abs/1306.0495v2
Journal of Physics A: Mathematical and Theoretical 47, 135302. The arXiv DOI metadata and Crossref record agree on the DOI, title and authors.
Pure-output rigidity
Theorem IV.9, arXiv v2 PDF pp. 21–22, states that the pure output set of a qubit channel is empty, a singleton, a pair, or all pure qubit states. In the last case the channel is unitary conjugation. Consequently three distinct pure outputs force unitary conjugation, which sends every pure input to a pure output.
Covariant pure programs
Fixing a normalized basis input of a processor with a qubit program register
and qubit output gives a qubit channel from programs to outputs. The two basis
replacement targets and their equal-superposition target are three distinct
pure outputs. The classification forces unitary conjugation, so no pure
program can produce the fourth replacement target .
This is the literature ingredient applied to the GBW-5 pure-program question.
PureProgramQubitObstruction.no_four_replacements proves the selected-target
Gram-matrix obstruction even without program normalization. The classification
itself is cited literature; the delivered theorem does not formalize its full
classification statement.