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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.

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*}

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) \\ &=\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.

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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*}