Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to figure out what A ⊗ B ⊗ C - A ⊗ B ⊗ C is.
By bilinearity, A ⊗ B ⊗ C - A ⊗ B ⊗ C = (A - A) ⊗ B ⊗ C = A ⊗ (B - B) ⊗ C = A ⊗ B ⊗ (C - C) = 0 ⊗ B ⊗ C = A ⊗ 0 ⊗ C = A ⊗ B ⊗ 0. But I'm sure how to conclude that A ⊗ B ⊗ C - A ⊗ B ⊗ C = 0 ⊗ 0 ⊗ 0.
Wolfram Mathworld says x ⊗ 0 = 0 ⊗ y = 0, but I'm not sure what the 0 stands for (does it mean 0 ⊗ 0?).
Any help would be great, and could responses please include references preferably to books that address this specific question? Thanks!
A key property of tensor product is that $$(\lambda A)\otimes B=A\otimes(\lambda B)$$ for any scalar $\lambda\in\Bbb C$.
Now, apply it for $\lambda=0$, we have $${\bf0}\otimes B=(0\cdot {\bf0})\otimes B={\bf0}\otimes(0\cdot B)={\bf0}\otimes{\bf0}\,.$$ Similarly for more terms.