If Alice and Bob share the state: $$\left| {{\psi _{AB}}} \right\rangle = \sin \theta \left| {10} \right\rangle + \cos \theta \left| {01} \right\rangle $$ then $\rho_{AB}$ can be obtained as: $${\rho _{AB}} = \left| {{\psi _{AB}}} \right\rangle \left\langle {{\psi _{AB}}} \right|.$$ Is there a way to get $\rho_{BA}$ instead?

New contributor
Bekaso is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • 3
    $\begingroup$ Crossposted from Mathematics. I'm not sure why though, I already answered it on MSE. $\endgroup$ – Rammus Jul 21 at 11:34
  • $\begingroup$ what do you mean with "get". A quantum circuit sending one to the other? Also, what's your definition of $\rho_{BA}$ here? Is it the same state after swapping the spaces or something else? $\endgroup$ – glS Jul 21 at 13:04
  • $\begingroup$ @glS By $\rho_{BA}$ I mean the density matrix after permutation of the subsystems. I want to get the formula that enables me to calculate $\rho_{BA}$. $\endgroup$ – Bekaso Jul 21 at 14:55

$A$ and $B$ are labels for the Hilbert spaces in which each subsystem exists. There is no different physical content between $\mathcal{H}_A\otimes \mathcal{H}_B$ and $\mathcal{H}_B\otimes \mathcal{H}_A$, they are just different ways of bookkeeping.

As such, we can immediately trade all of the information about subspaces $A$ and $B$ and write $$|\psi_{BA}\rangle=\sin\theta |0\rangle_B\otimes|1\rangle_A+\cos\theta |1\rangle_B\otimes|0\rangle_A$$ and $$\rho_{BA}=|\psi_{BA}\rangle\langle \psi_{BA}|.$$

  • $\begingroup$ What about the coefficients $\sin \theta $ and $\cos \theta $? Won't they have any change? $\endgroup$ – Bekaso 2 days ago
  • $\begingroup$ Nope, no change! We are simply changing the labels - I'll make it more explicit $\endgroup$ – Quantum Mechanic 2 days ago

Your Answer

Bekaso is a new contributor. Be nice, and check out our Code of Conduct.

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.