toqito.state_opt.npa_hierarchy

Generates the NPA constraints.

Module Contents

class toqito.state_opt.npa_hierarchy.Symbol

Bases: tuple

player
question
answer
toqito.state_opt.npa_hierarchy.IDENTITY_SYMBOL
toqito.state_opt.npa_hierarchy.PLAYERS = ('Alice', 'Bob')
toqito.state_opt.npa_hierarchy.npa_constraints(assemblage, k=1, referee_dim=1, no_signaling=True)[source]

Generate the constraints specified by the NPA hierarchy up to a finite level.

[@Navascues_2008_AConvergent]

You can determine the level of the hierarchy by a positive integer or a string of a form like “1+ab+aab”, which indicates that an intermediate level of the hierarchy should be used, where this example uses all products of 1 measurement, all products of one Alice and one Bob measurement, and all products of two Alice and one Bob measurement.

The commuting measurement assemblage operator must be given as a dictionary. The keys are tuples of Alice and Bob questions (x, y) and the values are cvxpy Variables which are matrices with entries:

[

K_{xy}Big(i + a cdot dim_R, j + b cdot dim_R Big) = langle i| text{Tr}_{mathcal{H}} Big( big(

I_R otimes A_a^x B_b^y big) sigma Big) |j rangle

]

Parameters:

  • assemblage (dict[tuple[int, int], cvxpy.Variable]) – The commuting measurement assemblage operator.

  • k (int | str) – The level of the NPA hierarchy to use (default=1).

  • referee_dim (int) – The dimension of the referee’s quantum system (default=1).

  • no_signaling (bool) – bool

Returns:

A list of cvxpy constraints.

Return type:

list[cvxpy.constraints.constraint.Constraint]

toqito.state_opt.npa_hierarchy.bell_npa_constraints(p_var, desc, k=1)[source]

Generate NPA hierarchy constraints for Bell inequalities [@Navascues_2008_AConvergent].

The constraints are based on the positivity of a moment matrix constructed from measurement operators. This function generates constraints for a CVXPY variable representing probabilities or correlations in the Collins-Gisin notation. [@Collins_2004]

The level of the hierarchy k can be an integer (standard NPA level) or a string specifying intermediate levels (e.g., “1+ab”, “2+aab”).

The input p_var is a CVXPY variable representing the probabilities in the Collins-Gisin (CG) notation. It should have dimensions (((oa-1) times ma+1, (ob-1) times mb+1)), where (oa, ob) are the number of outputs and (ma, mb) are the number of inputs for Alice and Bob, respectively, as specified in desc = [(oa), (ob), (ma), (mb)]. The entries of p_var correspond to: - p_var[0, 0]: The overall probability (should be 1). - p_var[i, 0] (for (i > 0)): Marginal probabilities/correlations for Alice. - p_var[0, j] (for (j > 0)): Marginal probabilities/correlations for Bob. - p_var[i, j] (for (i > 0, j > 0)): Joint probabilities/correlations for Alice and Bob.

The mapping from indices ((i, j)) to specific operators depends on the ordering defined by desc. Specifically, if (i = (oa-1) times x + a + 1) and (j = (ob-1) times y + b + 1)

  • p_var[i, 0] corresponds to the expectation of Alice’s operator

    (A_{a|x}) (using (0) to (oa-2) for (a)).

  • p_var[0, j] corresponds to the expectation of Bob’s operator

    (B_{b|y}) (using (0) to (ob-2) for (b)).

  • p_var[i, j] corresponds to the expectation of the product (A_{a|x} B_{b|y}).

Examples

Consider the CHSH inequality scenario with desc = [2, 2, 2, 2]. We want to generate the NPA level 1 constraints.

`python exec="1" source="above" session="npa_example" import cvxpy import numpy as np from toqito.state_opt.npa_hierarchy import bell_npa_constraints desc = [2, 2, 2, 2] oa, ob, ma, mb = desc p_var_dim = ((oa - 1) * ma + 1, (ob - 1) * mb + 1) p_var = cvxpy.Variable(p_var_dim, name="p_cg") constraints = bell_npa_constraints(p_var, desc, k=1) print(len(constraints)) print(constraints[0]) `

We can also use intermediate levels, like “1+ab”:

`python exec="1" source="above" session="npa_example" constraints_1ab = bell_npa_constraints(p_var, desc, k="1+ab") print(len(constraints_1ab)) print(constraints_1ab[0]) `

For the CGLMP inequality with dim=3, desc = [3, 3, 2, 2], level 1:

`python exec="1" source="above" import cvxpy import numpy as np from toqito.state_opt.npa_hierarchy import bell_npa_constraints desc_cglmp = [3, 3, 2, 2] oa_c, ob_c, ma_c, mb_c = desc_cglmp p_var_dim_c = ((oa_c - 1) * ma_c + 1, (ob_c - 1) * mb_c + 1) p_var_c = cvxpy.Variable(p_var_dim_c, name="p_cglmp") constraints_c = bell_npa_constraints(p_var_c, desc_cglmp, k=1) print(len(constraints_c)) print(constraints_c[0]) `

Raises:

ValueError – If internal identity mapping fails.

Parameters:

  • p_var (cvxpy.Variable) – A CVXPY Variable representing probabilities/correlations in Collins-Gisin notation. Shape: (((oa-1) times ma+1, (ob-1) times mb+1)).

  • desc (list[int]) – A list [(oa), (ob), (ma), (mb)] specifying outputs and inputs for Alice and Bob.

  • k (int | str) – The level of the NPA hierarchy (integer or string like “1+ab”). Default is 1.

Returns:

A list of CVXPY constraints.

Return type:

list[cvxpy.constraints.constraint.Constraint]