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I would like to better understand some passages in a paper (Appendix A):

Properties of Tensor Product

  • Bilinearity: $A\otimes(B+ C) = A \otimes B + A \otimes C $
  • Mixed-product property: $(A\otimes B)(C \otimes D) = AC \otimes BD$
  • $ (|a\rangle \otimes|b\rangle) (\langle c| \otimes \langle d|) = |a\rangle \langle c| \otimes |b\rangle \langle d| $

For a unitary $U_p |0\rangle |0\rangle \mapsto \sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle|j\rangle $, we have a matrix $A$ defined as such:

$$A = \Pi U\widetilde{\Pi}=\Pi\left(U_p\otimes I\right)\widetilde{\Pi}=\Big(\sum_{i=1}^{n}I\otimes|i\rangle\langle i|\otimes |i\rangle\langle i| \Big)(U_{p}\otimes I)\big(|0\rangle\langle 0|\otimes |0 \rangle\langle 0|\otimes I\big) $$

We used the mixed-product property

$$=\sum_{i=1}^{n}\Big((I\otimes |i \rangle \langle i|)U_{p}(|0 \rangle \langle 0|\otimes |0 \rangle \langle 0|)\Big)\otimes |i \rangle \langle i| $$

Here we applied the definition of $U_p$.

$$=\sum_{i=1}^{n}\Big((I\otimes |i \rangle \langle i|)\sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle|j\rangle\langle 0| \langle 0|\Big)\otimes |i \rangle \langle i| $$

My steps are: $$=\sum_{i=1}^{n}\Big((I\otimes |i \rangle \langle i|)\sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle \langle 0| \otimes |j\rangle\langle 0|\Big)\otimes |i \rangle \langle i| $$

Because of the third property (name?)

$$=\sum_{i=1}^{n}\left((I\otimes |i \rangle \langle i|) \left( \sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle \langle 0| \otimes |j\rangle\langle 0|\right)\right)\otimes |i \rangle \langle i| $$

Because of associativity property of matrix product

$$=\sum_{i=1}^{n}\left((I\otimes |i \rangle \langle i|) \left( \sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle \langle 0| \otimes |j\rangle\langle 0|\right)\right)\otimes |i \rangle \langle i| $$

$$=\sum_{i=1}^{n}\left((\sum_{j=1}^{n}\sqrt{p_{j}}|\phi_{j}\rangle \langle 0| \otimes |i \rangle \langle i| |j\rangle\langle 0|)\right)\otimes |i \rangle \langle i| $$

As $\langle i | j \rangle = \delta_{ij}$, the final result is:

$$=\sum_{i=1}^{n}\sqrt{p_i}|\phi_i \rangle \langle 0|\otimes |i \rangle \langle 0|\otimes |i \rangle \langle i|.$$

Can you check that my answer is indeed correct and I have made explicit all the properties of the operations that are used in these passages? I think passages are correct, and there are no other properties that I have used.

Thanks!

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    $\begingroup$ The calculation and the steps seems correct to me. $\endgroup$ – vasjain Jul 7 at 0:30

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