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Given a specific density matrix $\rho$ that corresponds to an entangled quantum state, I would like to find a class of operators (that might be $\rho$ dependent) that tranform (with high probability) $\rho$ into other (possibly close) quantum states that have the same (or similar) entanglement. How can I characterize those operators?

Thank you

Fabio

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The entanglement of a density matrix $\rho$ is defined relative to a bipartition of the Hilbert space $H=H_A\otimes H_B$. This is invariant under local unitary operations, $\rho\mapsto U\rho U^\ast$ with $U=U_A\otimes U_B$ acting separately on $H_A$ and $H_B$.

More general entanglement preserving operations exist, see Entanglement-Saving Channels (2015).

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  • $\begingroup$ That's what I thought, but it seems not true. Meaning that given a separable state you cannot entangle it using local operations but given an entangled state you can destroy or decrease its entanglement with local operations. $\endgroup$
    – Fabio
    Commented May 12, 2023 at 10:20
  • $\begingroup$ not with local unitary operations; these just amount to a change of basis, any meaningful measure of entanglement should be basis independent. $\endgroup$
    – Carlo Beenakker
    Commented May 12, 2023 at 10:20
  • $\begingroup$ Thank you, the answer is simpler then what I thught then. Is this true even in the case of multipartite entanglement? So a entanglement preserving channel/operator can be always written as a convex combination of local unitaries (possibly random). $\endgroup$
    – Fabio
    Commented May 12, 2023 at 10:25
  • $\begingroup$ no, there exist more general entanglement-preserving channels, I added a reference. $\endgroup$
    – Carlo Beenakker
    Commented May 12, 2023 at 10:28

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