slug: lovas-andai-2016-dobrushin-infimum
bibkey: lovas2016volume
doi: 10.48550/arXiv.1607.01215
url: https://arxiv.org/abs/1607.01215
triage: theorem
motivation_gids:
  - D5/S3/Quantum/QuantumChannels/LovasAndaiDobrushinInfimum.result

Lovas and Andai’s conjecture on the least trace-distance contraction over a classical channel

Problem

Lovas and Andai represent a qubit channel Q : M_2 → M_2, a completely positive trace-preserving map, by its Choi block matrix, whose blocks Q_11 = Q(|0⟩⟨0|) and Q_22 = Q(|1⟩⟨1|) have diagonals (a, 1−a) and (f, 1−f) in the parametrization (eq:matQ). The trace-distance contraction coefficient is

η^Tr(Q) = sup { Tr|Q(ρ) − Q(σ)| / Tr|ρ − σ| : ρ, σ ∈ M_2 },

and Q_C(a,f) denotes the set of qubit channels over the classical channel ((1−a, a), (1−f, f)) with respect to that parametrization. The subsection on the distribution of η^Tr over classical channels states

We conjecture that inf{η(Q) : Q ∈ Q_C(a,f)} = |a−f| which is equal to the trace-distance contraction coefficient of the underlying classical channel.

Issue #10033 fixes the readings: M_2 is the set of qubit states and a qubit channel is a completely positive trace-preserving map; Q_C(a,f) consists of the channels for which Q(|0⟩⟨0|) and Q(|1⟩⟨1|) have diagonals (a, 1−a) and (f, 1−f); the source writes positivity of the Choi matrix as Q > 0, and complete positivity is read as a positive semidefinite Choi matrix; the supremum defining η^Tr runs over all pairs of states, the ratio of trace norms Tr|·| being the ratio of trace distances (the factors ½ cancel), and a pair ρ = σ counts as 0/0 = 0, which does not change the supremum; a, f ∈ [0,1].

Motivation

The frozen declaration D5/S3/Quantum/QuantumChannels/LovasAndaiDobrushinInfimum.result proves that, for all a, f ∈ [0,1], |a − f| is the least value of η^Tr over Q_C(a,f): the infimum equals |a − f| and is attained. The trace-distance contraction coefficient is the quantum analogue of the Dobrushin coefficient of ergodicity and bounds mixing times of quantum Markov processes; the theorem says that fixing the classical channel of a qubit channel forces no contraction beyond that of the classical channel itself.

Gap

Issue #10033 preregisters this published conjecture and its literature check. MathDB /p/333535 has status open with zero solutions; Semantic Scholar lists one citation of arXiv:1607.01215, a paper of the same authors on locally diagonalizable bipartite states that does not address the conjecture. The source proves that every value strictly between |a − f| and √((1−a)f) + √(a(1−f)) is attained, and its simulations estimate the infimum numerically.

These readings are not-found-in-searched-scope; they do not establish an exhaustive worldwide literature search, priority, or the absence of an independent proof.

Route

Lower bound: for Q ∈ Q_C(a,f), the matrix A = Q(|0⟩⟨0|) − Q(|1⟩⟨1|) has A_00 = a − f and, by trace preservation, A_11 = f − a. The variational formula Tr|A| = max_U re Tr(UA) over unitaries, tested with U = diag(s, −s) for the sign s of a − f, gives Tr|A| ≥ 2|a − f|, while |0⟩⟨0| and |1⟩⟨1| are at trace distance one. So the ratio for this pair is at least |a − f|, and so is η^Tr(Q).

Attainment: the measure-and-prepare channel Q_0(X) = Σ_{i,j} p_j(i) X_jj |i⟩⟨i|, with p_0 = (a, 1−a) and p_1 = (f, 1−f), has Kraus operators √p_j(i) |i⟩⟨j| and lies in Q_C(a,f). For states ρ, σ with t = ρ_00 − σ_00, it sends ρ − σ to diag((a−f)t, (f−a)t). Every unitary has diagonal entries of modulus at most one, so the variational formula gives Tr|Q_0ρ − Q_0σ| ≤ 2|a − f||t|, while the same test unitary as above gives Tr|ρ − σ| ≥ 2|t|. Every ratio is therefore at most |a − f|, and η^Tr(Q_0) = |a − f|.

Falsifier

A counterexample would be a channel in Q_C(a,f) with η^Tr < |a − f|; under every such channel the images of the pair |0⟩⟨0|, |1⟩⟨1|, which are at trace distance one, have trace distance at least |a − f|. A failure of attainment would require every channel in Q_C(a,f) to exceed |a − f|; the measure-and-prepare channel does not.

Evidence

For 3000 random Stinespring channels with a four-dimensional environment, η^Tr computed as the largest singular value of the Bloch matrix of (eq:matT) never falls below |a − f|; the least excess is 0.00152. For (a,f) = (.3,.8), (.1,.15), (.5,.5), (.9,.2), (0,1), (1,1) the measure-and-prepare channel has η^Tr = |a − f|, and direct sampling over pure-state pairs agrees within 7·10^{-4}; the positive control η^Tr(Q_0) ≥ |a − f| + 0.01 fails in all six cases. The Lean proof has only the standard axiom closure propext, Classical.choice, and Quot.sound. These finite checks support the reading but do not establish the universal theorem.

Triage

First-tier external named open problem: Lovas and Andai, arXiv:1607.01215 (2016), Conjecture \label{conj} of the subsection on the distribution of η^Tr over classical channels, preregistered in issue #10033. Resolution: proved, with the infimum attained.

The public surface is exactly classicalFiber, dobrushin, claim, and result. This is a uniform theorem, not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies.

ASSUMED-UNVERIFIED

Whether the journal version, Rev. Math. Phys. 30 (2018) 1850019 (DOI 10.1142/S0129055X18500198), retains or changes the conjecture has not been checked. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The Lean kernel does not authenticate the external source, its version history, or the literature-check coverage. The finite checks do not establish the universal theorem.