Optimize Measurements

BellScenario.Nonlocality.optimize_measurement — Function

LocalSignaling scenario:

optimize_measurement(
    scenario :: LocalSignaling,
    game :: BellGame,
    ρ_states :: Vector{<:AbstractMatrix}
)

Finds the measurement that optimizes the score of the BellGame against the set of quantum states ρ_states. The optimization is performed with the following semi-definite program:

\[\begin{aligned} &\max_{\{\Pi_y\}} \sum_{x,y} G_{x,y} \text{Tr}[\Pi_y \rho_x] \\ &s.t. \quad \sum_y \Pi_y = \mathbb{I}, \quad \Pi_y \geq 0 \end{aligned}\]

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BipartiteNonSignaling scenario:

optimize_measurement(
    game :: BellGame,
    scenario :: BipartiteNonSignaling,
    ρ_AB :: AbstractMatrix;
    A_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},
) :: Dict

Find Bob's measurement which optimizes a BellGame's score for the shared quantum state ρ_AB and POVM measurement applied by Alice. The following semi-definite program optimizes the Bob's POVM:

\[\begin{aligned} &\max_{\{\Pi_b^y\}} \sum_{a,b,x,y} G_{a,b,x,y}\text{Tr}[(\Pi_a^x \otimes \Pi_b^y)\rho_{AB}] \\ &s.t. \quad \sum_b \Pi_b^y = \mathbb{I},\quad \Pi_b^y \geq 0 \quad \forall\; y \end{aligned}\]

optimize_measurement(
    game :: BellGame,
    scenario :: BipartiteNonSignaling,
    ρ_AB :: AbstractMatrix;
    B_POVMs :: Vector{<:AbstractVector{<:AbstractMatrix}},
) :: Dict

Find Alice's measurement which optimizes a BellGame's score for the shared quantum state ρ_{AB} and POVM measurement applied by Bob. The following semi-definite program optimizes the Alice's POVM:

\[\begin{aligned} &\max_{\{\Pi_a^x\}} \sum_{a,b,x,y} G_{a,b,x,y}\text{Tr}[(\Pi_a^x \otimes \Pi_b^y)\rho_{AB}] \\ &s.t. \quad \sum_a \Pi_a^x = \mathbb{I},\quad \Pi_a^x \geq 0 \quad \forall \;x \end{aligned}\]

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