Cross-Checks of Quantum Budgets

Bell Parity Cross-Check

same and different joint tables

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=0P_{\text{same}}=1,\quad P_{\text{different}}=0
Same-parity Bell tableThe diagonal cells carry the whole budget.001/sqrt(2)P=1/2010P=0100P=0111/sqrt(2)P=1/2det = 1/2not product

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.

statePsamePdifferentdet⁡Φ1012Ψ01−12product100\begin{array}{c|c|c|c}\text{state}&P_{\text{same}}&P_{\text{different}}&\det\\\Phi&1&0&\frac{1}{2}\\\Psi&0&1&\frac{-1}{2}\\\text{product}&1&0&0\\\end{array}
Different-parity Bell tableThe off-diagonal cells carry the whole budget.000P=0011/sqrt(2)P=1/2101/sqrt(2)P=1/2110P=0det = -1/2not product

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.

det⁡Φ=12,det⁡Ψ=−12,det⁡product=0\det_{\Phi}=\frac{1}{2},\quad \det_{\Psi}=\frac{-1}{2},\quad \det_{\text{product}}=0
Product contrastThe zero determinant is computed from the table.001P=1010P=0100P=0110P=0det = 0