toqito.state_props.learnability
===============================

.. py:module:: toqito.state_props.learnability

.. autoapi-nested-parse::

   Evaluate the quantum learnability semidefinite programs.





Module Contents
---------------

.. py:function:: learnability(states, k, *, solver = 'SCS', solver_kwargs = None, verify_reduced = True, verify_tolerance = 0.0001, tol = 1e-08)

   Compute the average error value of the learnability semidefinite program.

   This routine minimizes

   \[
       \frac{1}{n} \sum_{i = 1}^n \left\langle \rho_i,
       \sum_{S: i \notin S} M_S \right\rangle.
   \]

   over POVM elements \((M_S)\) indexed by ``k``-element subsets, subject to
   \(\sum_S M_S = \mathbb{I}\) and \(M_S \succeq 0\).  When all inputs are pure, the
   reduced Gram-matrix SDP

   \[
       \sum_{i = 1}^n \bra{i} \sum_{S: i \notin S} W_S \ket{i}.
   \]

   with constraint \(\sum_S W_S = G\) (Gram matrix) and \(W_S \succeq 0\)
   is also solved as a consistency check.

   .. rubric:: Examples

   ```python exec="1" source="above"
   from toqito.state_props import learnability
   from toqito.states import basis

   e0, e1 = basis(2, 0), basis(2, 1)
   print(learnability(
       [e0, e1],
       k=1,
       solver="SCS",
       solver_kwargs={"eps": 1e-6, "max_iters": 5_000},
   ))
   ```

   :param states: Sequence of state vectors or density matrices acting on the same space.
   :param k: Subset size for the POVM outcomes; must satisfy `1 <= k <= len(states)`.
   :param solver: Optional CVXPY solver name. Defaults to `"SCS"`.
   :param solver_kwargs: Extra keyword arguments forwarded to :meth:`cvxpy.Problem.solve`.
   :param verify_reduced: If `True` and the states are pure, also solve the reduced SDP.
   :param verify_tolerance: Absolute tolerance used when comparing the two optimal values.
   :param tol: Numerical tolerance used when validating positivity and rank-one states.

   :returns: Dictionary with keys `value`, `total_value`, `status`, `measurement_operators`, and optionally `reduced_value`,
             `reduced_total_value`, `reduced_status`, `reduced_operators`.

   :raises ValueError: If the data are inconsistent with valid quantum states or if `k` lies outside the permissible range.
   :raises cvxpy.error.SolverError: If the selected solver reports a failure.