Implement channel exclusion (channel antidistinguishability)

[vprusso]

|toqito⟩ supports channel distinguishability via channel_distinguishability, but has no support for channel exclusion — the task where Bob receives the output of one of $n$ channels and must identify which channel was not used. This is the channel-level analogue of state_exclusion and is central to the study of channel antidistinguishability (see arXiv:2309.03649).

Background

Given $n$ quantum channels ${\Phi_1, \ldots, \Phi_n}$ with prior probabilities ${p_1, \ldots, p_n}$, channel exclusion asks: what is the minimum probability that Bob incorrectly excludes a channel, optimized jointly over the input state $\rho$ and the measurement ${M_1, \ldots, M_n}$?

Via the Choi isomorphism, this can be cast as a single SDP. Let $J(\Phi_i)$ denote the Choi matrix of channel $\Phi_i$. The min-error channel exclusion problem is:

$$\min \sum_{i=1}^n p_i \text{Tr}!\big[(M_i \otimes \rho^T), J(\Phi_i)\big] \quad \text{s.t.} \quad \sum_i M_i = \mathbb{I},; M_i \geq 0,; \rho \geq 0,; \text{Tr}(\rho)=1$$

This is not a standard SDP as stated (bilinear in $M_i$ and $\rho$), but it can be linearized. The key insight is that for a fixed input state the problem reduces to state exclusion of the output states, and the joint optimization can be handled by lifting to the Choi picture and optimizing over a single variable $W_i = M_i \otimes \rho^T$ with appropriate marginal constraints.

The specific SDP formulation for channel exclusion and its dual are given explicitly in Section III of arXiv:2309.03649. Contributors should read this section of the paper to obtain the correct constraints — in particular, the constraint that the partial trace over the output system of the sum of measurement-state variables equals the identity on the input space (Equation 6 and surrounding discussion).

What should be added

A new function channel_exclusion in toqito/channel_metrics/ (or toqito/state_opt/, following the pattern of state_exclusion) that computes:

1. Min-error channel exclusion — the minimum probability of incorrectly excluding a channel, for $n \geq 2$ channels
2. Unambiguous channel exclusion — the minimum probability of an inconclusive outcome when Bob must never incorrectly exclude

Relevant code

• toqito/channel_metrics/channel_distinguishability.py — reference for channel discrimination patterns (Choi matrix handling, dimension extraction)
• toqito/state_opt/state_exclusion.py — reference for the state-level exclusion SDP structure
• toqito/channel_ops/kraus_to_choi.py — Kraus-to-Choi conversion
• toqito/channel_props/channel_dim.py — dimension extraction

Acceptance criteria

• Function accepts $n \geq 2$ channels as Choi matrices or Kraus operators
• Both primal and dual formulations implemented for min-error
• Returns the optimal exclusion probability and the optimal measurement operators
• Tests include:
  • Channel exclusion of $n$ depolarizing channels with known noise parameters — verify the exclusion probability interpolates correctly between perfect exclusion (orthogonal channels) and chance (identical channels)
  • Verify that for identity-in, identity-out special cases, channel exclusion reduces to state exclusion of the Choi states (sanity check)
  • At least one test case where the value is verified against a result from arXiv:2309.03649
• Docstring includes the SDP formulation and a worked example

Out of scope

• Adaptive (multi-round) channel discrimination
• PPT or separable measurement constraints on channel exclusion (fine as follow-ups)
• Channel antidistinguishability as a boolean function (can be a thin wrapper later, analogous to is_antidistinguishable)