For which measure spaces X is the k-fold tensor-product of L^2(X) isomorphic to L^2(X^k)?

Question

For example, if $X=\mathbb{R}$ with the usual Lebesuge measure, then there is a natural map $$\Phi:L^2(X)\otimes L^2(X)\ni f\otimes g\mapsto((x,y)\mapsto f(x)g(y))\in L^2(X^2)$$ which is also unitary map between Hilbert spaces, by Fubinis theorem (which requires $X$ and $Y$ to be complete and $\sigma$-finite).