Tensor Products

Canonical treatment: Tensor Products of Hilbert Spaces maintains the physical construction, examples, operator rules, exercises, and references.

This bridge is intentionally limited to the composition postulate and minimum tensor-product notation.

Composition postulate

For distinguishable subsystems,

$$\mathcal H_{AB}=\mathcal H_A\otimes\mathcal H_B.$$

Product vectors have the form $\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B$, often abbreviated $\lvert\psi\rangle_A\lvert\phi\rangle_B$. The construction is bilinear, for example

$$(a\lvert0\rangle+b\lvert1\rangle)\otimes\lvert\phi\rangle =a\lvert0\rangle\lvert\phi\rangle+b\lvert1\rangle\lvert\phi\rangle.$$

If the factor dimensions are $d_A$ and $d_B$, the composite dimension is $d_A d_B$. Fix a basis-ordering convention before forming Kronecker coordinates or matrices; changing factor order requires an explicit swap map.

General vectors in $\mathcal H_A\otimes\mathcal H_B$ need not be product vectors. Continue to the canonical page for product bases, position–spin examples, operator tensor products, regrouping, and completion issues.