Add PPT measurement support to state exclusion SDP

[vprusso]

toqito.state_opt.state_exclusion currently supports "min_error" and "unambiguous" strategies, but unlike its counterpart state_distinguishability, it has no support for PPT (positive partial transpose) measurement constraints.

PPT measurements are a key relaxation in the hierarchy: SEP ⊆ PPT ⊆ global. For state distinguishability, PPT bounds are already implemented via measurement="ppt" in state_distinguishability. The exclusion side has no such support, which means there is currently no way to compute PPT bounds on antidistinguishability in |toqito⟩ — a quantity that appears directly in research on quantum state exclusion and antidistinguishability (see arXiv:2501.03987 and arXiv:2308.00691).

What should be added

Add a measurement="ppt" option to state_exclusion analogous to the one already present in state_distinguishability, along with the required subsystems and dimensions parameters.

Relevant code

• toqito/state_opt/state_exclusion.py — the function to extend
• toqito/state_opt/state_distinguishability.py — reference for how PPT constraints are added (_ppt_primal, _ppt_dual)
• toqito/state_props/is_antidistinguishable.py — wraps state_exclusion; may benefit from a PPT variant

SDP formulation

The min-error PPT exclusion primal should mirror the standard exclusion primal:

$$\min \sum_i p_i \langle M_i, \rho_i \rangle \quad \text{s.t.} \quad \sum_i M_i = \mathbb{I},; M_i \geq 0,; \Gamma_j(M_i) \geq 0 ;\forall, i, j$$

where $\Gamma_j$ denotes partial transpose on subsystem $j$. The dual follows by standard SDP duality with the additional PPT constraints generating dual variables.

Important: The exclusion SDP minimizes overlap (Bob wants to identify which state was not sent), whereas the distinguishability SDP maximizes overlap. The PPT constraints apply to the measurement operators in both cases, but the objective and its dual are structurally different. A correct implementation must handle this distinction — simply copying the distinguishability PPT code with a flipped objective will not yield correct dual variables.

Acceptance criteria

• state_exclusion(..., measurement="ppt", subsystems=..., dimensions=...) works for both "min_error" and "unambiguous" strategies, for both primal and dual
• Existing tests continue to pass (no regression)
• New tests cover:
  • PPT exclusion of Bell states tensored with a resource state (compare against known values from the symmetric extension hierarchy)
  • PPT exclusion value is bounded between the global exclusion value and the separable exclusion value for a known example
  • At least one case where PPT exclusion gives a strictly different value than global exclusion
• Docstring updated with the PPT SDP formulation and an example

Out of scope

• Modifying is_antidistinguishable to accept a measurement type (fine as a follow-up)
• Adding SEP exclusion (that is a separate issue)