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Given two density matrix $\rho_1,\rho_2$ with the same size, how to get a mix state of the two matrix, $$ \rho = \frac12 (\rho_1+\rho_2)? $$ e.g. there are two quantum channel both of them have 4 qubits, with density matrix $\rho_1,\rho_2$. Now, we combine two channels, and there are 8 qubits in channel.
My objection get density matrix $\frac12 (\rho_1+\rho_2)$ from $\rho_1,\rho_2$.
With given state $\rho_1,\rho_2$, is it theoretically permissible to construct state $\frac12 (\rho_1+\rho_2)$ after some operation? Any operation is permissible.

Naïvely, I can tomography both $\rho_1$ and $\rho_2$, and add them in classic computer. Then prepare it in quantum circuit. However, I think this method is so stupid.
Can we get $\frac12 (\rho_1+\rho_2)$ in quantum channel without classic computer or at least no using tomography?

Boyuan Wang
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1 Answers1

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Your question is a bit confusing because it is not clear what you mean by "combine two channels into an 8 qubit channel", but maybe the following suffices?:

  1. Prepare $\rho_1$ on qubits $q_0$ to $q_3$.
  2. Prepare $\rho_2$ on qubits $q_4$ to $q_7$.
  3. Prepare $|+\rangle$ on qubit $q_8$.
  4. Apply $\text{CSWAP}$ gates between $q_i$ and $q_{i+4}$ conditioned on $q_8$.
  5. Discard (trace out) $q_8$.
  6. You effectively have $\rho = \frac{1}{2}(\rho_1 + \rho_2)$ in both $q_0$ thru $q_3$, and $q_4$ thru $q_7$.
diemilio
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