bibkey: watrous2018theory authors: John Watrous year: 2018 title: The Theory of Quantum Information doi: 10.1017/9781316848142 claim: Positive trace-preserving maps on finite-dimensional complex matrix spaces admit invariant density operators. strata_touched:
- D5/S3/Quantum/ChannelFixedState license: citation-only triage: anchor
The Theory of Quantum Information
John Watrous’s book supplies the literature anchor for fixed points of
finite-dimensional quantum channels in Section 4.4. The result represented by
D5/S3/Quantum/ChannelFixedState.channel_fixed_state_exists is the standard
invariant-density-operator existence statement for a positive trace-preserving
complex-linear map on a nonempty finite-dimensional matrix algebra.
The repository proof uses Cesaro averages and finite-dimensional compactness. This note does not attribute that proof method, the pure-fixed-point premise of Theorem 4.5, complete positivity, the tangent factor, or equivalence with an interior faithful invariant state to the cited section.
Search log
- 2026-08-07: The review correction identified John Watrous, The Theory of
Quantum Information, Cambridge University Press (2018), DOI
10.1017/9781316848142, Section 4.4, as the literature source for the finite-dimensional channel fixed-point setting and invariant states. - 2026-08-07: The scope was cross-checked against the Lean declaration: the repository theorem assumes a complex-linear map, positivity, and trace preservation on matrices indexed by a finite nonempty type. It does not assume complete positivity.
- 2026-08-07: No specific theorem number is attributed because none was verified in this worktree.
Verified locator
- DOI: https://doi.org/10.1017/9781316848142
- Cambridge University Press, Section 4.4