bibkey: graydon2022stochastic
authors: Matthew A. Graydon and Joshua Skanes-Norman and Joel J. Wallman
year: 2022
title: Designing Stochastic Channels
doi: null
url: https://arxiv.org/abs/2201.07156v1
claim: "Section II asks whether all stochastic channels are unital for d = 3; Eq. (11) defines stochasticity by a positive identity-vector eigenvalue of the normalized Choi state, and Appendix A proves qubit unitality."
strata_touched:
  - D5/S3/QuantumChannels/QutritStochasticNonunitalRefutation
license: citation-only
triage: anchor

Stochastic channels and qutrit unitality

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Source: https://arxiv.org/abs/2201.07156v1

The arXiv preprint has no journal DOI. Section II, p. 2, defines the normalized Choi state in Eq. (2), $J(\Phi)=d^{-1}(\Phi\otimes\mathrm{id}_d) (\operatorname{col}(1_d)\operatorname{col}(1_d)^\dagger)$.

The stochastic-channel definition, Section II, p. 2, is:

A stochastic channel is a CPTP map $\Phi \in B(B({\mathbb{C}}^d))$ such that for some $0<\lambda \le 1$ $J(\Phi )\mathrm{col}(1_d)=\lambda \mathrm{col}(1_d).$ (11)

The qutrit question, Section II, p. 2, is:

We leave open the question of whether or not all stochastic channels are unital for d = 3.

The source’s construction in Eqs. (14)–(17) gives a nonunital stochastic channel in dimension four and extends to every even dimension greater than two. Appendix A, pp. 5–6, proves that every stochastic qubit channel is unital:

Theorem 5. Let $\Xi \in B(B({\mathbb{C}}^2))$ be a stochastic quantum channel. Then $\Xi$ is unital.

The qutrit instance uses zero-based Fin 3 indices and output-first Choi coordinates. Its four Kraus matrices are $I_3/\sqrt{2}$, $\mathrm{diag}(1,-1/2,-1/2)/\sqrt{2}$, $\sqrt{3/8}{E}_{01}$ and $\sqrt{3/8}{E}_{02}$. The associated channel has normalized Choi eigenvalue $1/2$ along the identity vector and sends $I_3$ to $\mathrm{diag}(7/4,5/8,5/8)$.