Post Undeleted by DaftWullie
DaftWullie
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If we write $$U=\sum_{i,j}U_{ij}|i\rangle\langle j|\quad V=\sum_{kl}V_{kl}|k\rangle\langle l|,$$ and $$|x\rangle=\sum_jx_j|j\rangle\quad |y\rangle=\sum_ly_l|l\rangle,$$ then we can evaluate both sides of the equation $$(U\otimes V)(|x\rangle\otimes|y\rangle)=(U|x\rangle)\otimes(V|y\rangle)$$ using the definition of the tensor product as $$U\otimes V=\sum_{ijkl}U_{ij}V_{kl}|ik\rangle\langle jl|.$$

So, the left-hand side is \begin{align*} (U\otimes V)(|x\rangle\otimes|y\rangle)&=\left(\sum_{ijkl}U_{ij}V_{kl}|ik\rangle\langle jl|\right)\left(\sum_{jl}x_jy_l|jl\rangle\right) \end{align*}\begin{align*} (U\otimes V)(|x\rangle\otimes|y\rangle)&=\left(\sum_{ijkl}U_{ij}V_{kl}|ik\rangle\langle jl|\right)\left(\sum_{jl}x_jy_l|jl\rangle\right) \\ &=\sum_{ijkl}U_{ij}x_jV_{kl}y_l|ik\rangle. \end{align*} Similarly, the right-hand side is \begin{align*} (U|x\rangle)\otimes(V|y\rangle)&=\left(\sum_{ij}U_{ij}x_j|i\rangle\right)\otimes\left(\sum_{kl}V_{kl}y_l|k\rangle\right) \\ &=\sum_{ijkl}U_{ij}x_jV_{kl}y_l|ik\rangle \end{align*} The two are the same.

You may worry that there's a little bit of trickery going on with the kets, that contained within the "definition" of the tensor product is already hiding an implicit use o f the tensor product because I'm going from $$|i\rangle\otimes|k\rangle$$ to $$|ik\rangle$$, and that makes the definition rather circular. However, remember that the text in a ket is just a label, so you can really think about what I'm doing as defining a new composite label in some different Hilbert space.

Post Deleted by DaftWullie
DaftWullie
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• 92

If we write $$U=\sum_{i,j}U_{ij}|i\rangle\langle j|\quad V=\sum_{kl}V_{kl}|k\rangle\langle l|,$$ and $$|x\rangle=\sum_jx_j|j\rangle\quad |y\rangle=\sum_ly_l|l\rangle,$$ then we can evaluate both sides of the equation $$(U\otimes V)(|x\rangle\otimes|y\rangle)=(U|x\rangle)\otimes(V|y\rangle)$$ using the definition of the tensor product as $$U\otimes V=\sum_{ijkl}U_{ij}V_{kl}|ik\rangle\langle jl|.$$

So, the left-hand side is \begin{align*} (U\otimes V)(|x\rangle\otimes|y\rangle)&=\left(\sum_{ijkl}U_{ij}V_{kl}|ik\rangle\langle jl|\right)\left(\sum_{jl}x_jy_l|jl\rangle\right) \end{align*}