bibkey: lovas2016volume authors: Attila Lovas, Attila Andai year: 2016 title: Volume of the space of qubit channels and some new results about the distribution of the quantum Dobrushin coefficient doi: 10.48550/arXiv.1607.01215 url: https://arxiv.org/abs/1607.01215 claim: The paper computes volumes of spaces of qubit channels, shows that the trace-distance contraction coefficient takes every value strictly between |a - f| and a stated upper value over the qubit channels above a fixed classical channel, and conjectures that its infimum there is |a - f|. strata_touched:

• D5/S3/Quantum/QuantumChannels/LovasAndaiDobrushinInfimum license: citation-only triage: anchor

Volume of the space of qubit channels and the quantum Dobrushin coefficient

Lovas and Andai represent a qubit channel Q : M_2 → M_2, a completely positive trace-preserving map, by its Choi block matrix: the action of Q is (a b; c d) ↦ a Q_11 + b Q_12 + c Q_21 + d Q_22, and the underlying classical channel is P = (dg(Q_11); dg(Q_22)). The general element is parametrized as

Q = ((a, b, c, d), (b̄, 1−a, e, −c), (c̄, ē, f, g), (d̄, −c̄, ḡ, 1−f)),

with a, f ∈ [0,1] (eq:matQ). For a CPT map the trace-distance contraction coefficient is

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

and the paper computes it as the largest singular value of the Bloch matrix T (eq:matT), whose lower-right entry is a − f. The set of qubit channels over the classical channel ((1−a, a), (1−f, f)) with respect to the parametrization (eq:matQ) is denoted Q_C(a,f). The theorem of the section on the distribution of η^Tr over classical channels gives, for all x ∈ (|a−f|, √((1−a)f) + √(a(1−f))), a channel Q ∈ Q_R(a,f) ⊂ Q_C(a,f) with η^Tr(Q) = x, and the section 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.

The paper writes channel positivity as Q > 0; Choi’s theorem characterizes complete positivity by a positive semidefinite Choi matrix, and this reading is used for Q_C(a,f). By (eq:matT), the member of the theorem’s family with d = e = 0, whose Choi matrix is diag(a, 1−a, f, 1−f), has η^Tr = |a − f|; the theorem is stated for the open interval only.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.1607.01215
• URL: https://arxiv.org/abs/1607.01215
• Version and location: arXiv:1607.01215 [math-ph, quant-ph] (2016-07-05), source file volume04.tex: the definition of η^Tr in the preliminaries, the parametrization (eq:matQ) and the Bloch matrix (eq:matT) in the section on the volume of qubit channels, and the theorem and conjecture (\label{conj}) of the subsection on the distribution of η^Tr over classical channels.