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How does one actually take the partial trace on a quantum computer/real experiment? Wikipedia says that this is a valid quantum operation but I can't see how to implement it. Given an entangled pure state $\psi_{AB} \in H_{A}\otimes H_{B}$, I wish to do some operations and measurements to obtain $\rho_A = \sum_{i\in H_B} \langle i\vert \rho_{AB} \vert i\rangle$.

Since $\rho_A$ has many possible purifications, this computation is not unitary but $\rho_A$ is unique. Applying a projective measurement $\sum_i \vert i\rangle\langle i\vert$ on $B$, doesn't work. I somehow need to "forget" that the state is actually entangled and "lose" the $H_{B}$ part of the state but this is (correct me if I'm wrong) not allowed in quantum information.

So if I have a single copy of a quantum bipartite state, what quantum circuit should I use that spits out the partial trace? Also, I'd love to know if such a circuit exists, what the computational complexity of it would be.

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The circuit will not "spit out" the partial trace. But what you can do is to just look at the A part of the system, and ignore the B part. The A part will be described by the reduced density matrix, and in particular, any measurement/operation you perform will be.

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  • $\begingroup$ I see. I'm unfamiliar with how one enforces that one should ignore or forget the B part since this seems like actively losing information. Or is the question ill posed? $\endgroup$ – user1936752 Nov 6 '18 at 17:27
  • $\begingroup$ Just don't touch that part any more. Or drop it in the dumpster. $\endgroup$ – Norbert Schuch Nov 6 '18 at 18:06

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