bibkey: mitra2021layers authors: A. Mitra year: 2021 title: “Layers of classicality in the compatibility of measurements” doi: 10.1103/PhysRevA.104.022206 url: https://arxiv.org/abs/2101.05752v4 claim: “Section III.A asks whether the channels broadcasting an observable have a highest element or a lowest element under precomposition by nondisturbing channels.” strata_touched:
- D5/S3/Quantum/Measurements/BroadcastingChannelExtremaRefutation license: citation-only triage: anchor
Mitra, layers of classicality
A. Mitra, Layers of classicality in the compatibility of measurements, Phys. Rev. A 104, 022206 (2021).
Verified locator
DOI: https://doi.org/10.1103/PhysRevA.104.022206
Source: https://arxiv.org/abs/2101.05752v4
The page numbers below are the printed page numbers of the arXiv v4 PDF.
Broadcasting channels and observables
Section II.B, p. 2: “A CPTP map Θ : S(Hin) → S(Hout) is called a quantum channel, where Hin and Hout are two Hilbert spaces [15].”
Definition 1, p. 2: “A channel Λ : S(H) → S(H_A ⊗ H_B) with H_A = H_B = H is called a broadcasting channel.”
Definition 3, p. 3: “An observable A ∈ O_X^d is broadcast by a broadcasting channel Λ ∈ C_broad^d if for all ρ ∈ S(H), x ∈ X” followed by equation (5):
Nondisturbing channels and the precomposition order
Definition 8, p. 5: “A channel Θ : S(H) → S(H) is nondisturbing with respect to an observable A if for all ρ ∈ S(H),” followed by equation (22):
Its equivalent Heisenberg form is equation (23), Θ*(A(x)) = A(x). The order uses Λ′ = Λ ∘ Θ with Θ nondisturbing for A. Proposition 4 states that this precomposition preserves broadcasting of A.
The two Section III.A questions
Section III.A, p. 5, following Proposition 4:
For an observable, in this way, one can get the set of different broadcasting channels that broadcast the observable. It is an open question whether this set has the highest element i.e., a broadcasting channel in this set from which all other broadcasting channels that can broadcast the observable, can be constructed through concatenation with nondisturbing channels. It is also an open question whether this set has the lowest element i.e., a broadcasting channel in this set which can be constructed from all other broadcasting channels that broadcast the observable through concatenation with nondisturbing channels.
The Conclusions, p. 9, repeat: “Firstly, it is not known whether the set of all broadcasting channels that broadcast a particular observable has the greatest or the lowest element.”
Finite-matrix encoding
The input carrier is Fin d, the output carrier is Fin d × Fin d, and outcomes
are Fin m. Observables use the frozen IsPOVM predicate. States use the
frozen IsDensity predicate. Channels use finite Kraus operators with
completeness Σ KᴴK = I and the frozen MatrixMap.of_kraus K K action.
The universal assertion says that every observable with a nonempty broadcasting
set has a highest or a lowest element. Its refutation supplies one observable
for which both answers are negative.
Refutation and scope
For d = 3 and m = 2, take P = diag(1,1,0), Q = I − P, A₀ = (3/4)P + (1/4)Q and A₁ = I − A₀. Three channels prepare the mutually orthogonal matrix units at (0,0), (1,1) and (0,1) on the P sector, and the matrix unit at (2,2) on the Q sector. Nondisturbing precomposition fixes these channels. Two different fixed channels exclude a lowest element. A common upper bound would distinguish three orthogonal outputs from states supported in the two-dimensional P sector; the positive trace bounds give 3 ≤ 2.
The Lean result concerns this instance. The rank-r family, sharp observables, and the qubit extremum questions remain open here.