A Bell parity cross-check compares same-basis parity with the determinant of the joint probability table, showing that parity alone cannot distinguish entangled from product states. The Phi and Psi Bell states have opposite parity with determinants 1/2 and -1/2, while the product contrast has the same parity but determinant 0 exactly for the stated toy tables; real Bell tests face detector efficiency, background counts, basis alignment, and finite-shot uncertainty.
Bell-style tables can have opposite parity while both remain non-product joint states. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
One Bell table populates same-outcome cells
The first Bell table puts all probability in zero-zero and one-one. The parity claim is read from the populated cells.
Psame=1,Pdifferent=0
A parity scan distinguishes two Bell tables
The first two rows are non-product Bell-style states with opposite same-basis parity. The third row is a checked product contrast, so every row is a real state budget.
The determinant separates Bell rows from the product contrast
The two Bell rows have nonzero determinant. The product contrast can have same parity, but its determinant is zero, so parity alone is not the entanglement check.