bibkey: watrous2018theory
authors: John Watrous
year: 2018
title: The Theory of Quantum Information
doi: 10.1017/9781316848142
claim: Finite-dimensional quantum channels admit complete Kraus and isometric Stinespring representations; positive trace-preserving maps admit invariant density operators.
strata_touched:
  - D5/S3/Quantum/ChannelFixedState
  - D5/S3/Quantum/Entanglement/FiniteSectorChannelOptimality
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

Complete channel representations

The author-hosted pre-publication copy at https://cs.uwaterloo.ca/~watrous/TQI/TQI.pdf gives the following precise locators. Page numbers are printed pages, followed by one-based PDF pages.

• Theorem 2.22, pp. 82-83 (PDF 90-91): complete positivity is equivalent to positive Choi representation, a Kraus representation with identical left and right operators, and a Stinespring representation with identical left and right dilation operators. The proof applies complete positivity to the unnormalized maximally entangled operator and factors the resulting positive Choi operator spectrally.
• Theorem 2.26, pp. 87-89 (PDF 95-97): trace preservation is equivalent to completeness of every Kraus representation and to the identity A* B = I for every Stinespring representation.
• Corollary 2.27, p. 89 (PDF 97), equations (2.129)-(2.130): a quantum channel admits a complete Kraus family and an isometric Stinespring dilation, each representing its action on every input operator.

The proof of FiniteSectorChannelOptimality.result applies this finite-dimensional representation theorem internally in the canonical matrix channel interface. Its Kraus/environment index has cardinality equal to the product of input and output dimensions; zero Kraus operators pad the spectral representation. They do not assert that this is the minimal Choi-rank environment. The internal representation also includes empty coordinate types; those cases follow from the empty matrix identities, and are not attributed to the book’s nonzero-map statement in Theorem 2.22.

The finite representation theorem alone does not establish the optimum over local channels in the spectral coarse-graining problem, or its stabilized trace-norm error formula.

Instrument composition and predictive intertwining

The book’s finite-dimensional channel and measurement formalism supplies the standard setting below. The displayed simulation lemma is derived here by composition; it is not attributed to a numbered theorem in the source. Let ${\mathcal{I}}_{t,a,y}$ and ${\mathcal{J}}_{t,a,y}$ be CP, trace-nonincreasing instrument branches, whose sums over $y$ are trace preserving for every legal action $a$. Let ${\mathcal{E}}_t$ be CPTP summaries between the respective finite-dimensional matrix or classical–quantum block algebras. Assume the initial target state is ${\hat{\sigma }}_{\varnothing }={\mathcal{E}}_0({\sigma }_{\varnothing })$ and, on all reachable unnormalized states,

$${\mathcal{E}}_{t+1}{\mathcal{I}}_{t,a,y}={\mathcal{J}}_{t,a,y}{\mathcal{E}}_t.$$

Use the same policy $a=\pi (h)$ determined by the recorded history $h$, and preserve action-legality guards. For unnormalized post-record states, induction gives

$${\hat{\sigma }}_h={\mathcal{E}}_{|h|}({\sigma }_h),\underset{\mathcal{J}}{\mathrm{Pr}}(h)=\mathrm{tr}{\hat{\sigma }}_h=\mathrm{tr}{\sigma }_h=\underset{\mathcal{I}}{\mathrm{Pr}}(h).$$

Indeed, apply the branch identity after each step; the trace of an unnormalized branch state is its outcome-history probability. Matched histories select the same next action. At positive probability, dividing by the common trace gives the corresponding conditional-state identity. This establishes all finite transcript laws, not an efficient method to obtain or evaluate a summary. For infinite or continuous classical registers, an additional measurable channel/instrument framework is needed; finite block algebras alone do not supply it.

The project-specific substitution, including program, optimizer memory and native gate choice, is in Self-Generated Predictive Memory §11.