Not Two Hidden Answers - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

The nonzero determinant shows this joint table cannot be split into two hidden single-particle answer lists. That check is exact only for the stated amplitude table; real Bell-style evidence depends on measurement settings, selection assumptions, detector efficiency, noise, and statistical confidence.

The entanglement claim is checked by a nonzero determinant, not a caption. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A product check uses a determinant

For a two-by-two amplitude table, a product state would have determinant 0.

\det_{\text{product}}=0

The determinant scan separates product from Bell state

A product table would have zero determinant; this checked table does not.

\begin{array}{c|c|c}\text{row}&\det&\text{claim}\\\text{product test}&0&\text{product}\\\text{Bell table}&\frac{1}{2}&\text{not product}\\\text{audit}&\frac{1}{2}&\text{entangled}\\\end{array}

This determinant is not zero

The determinant is 1/2, so this checked state is not two separate one-particle states.

\det=\frac{1}{2}\neq 0