Local Distinguishability of Multipartite Unitary Operations

We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguished by local operations and classical communication when a finite number of runs is allowed. Intuitively, this result indicates that the lost identity of a nonlocal unitary operation can be recovered locally. No entanglement between distant parties is required.

Figure 1

Illustration of LOCC discrimination of unitary operations. Here $U\in \{U_1,U_2\}$ represents the unknown unitary operation. A general scheme for identifying $U$ is as follows: (1) Prepare suitable input states $|\psi {⟩}^{A_1{A}_1^{\prime }}$ and $|\phi {⟩}^{A_2{A}_2^{\prime }}$, where $A_1^{\prime }$ and $A_2^{\prime }$ are auxiliary systems of Alice and Bob, respectively; (2) perform a finite number of runs of $U$ and insert appropriate local unitary operations $u_k$ and $v_k$ between every two successive runs; (3) distinguish the final output states $|{\Phi }_U⟩$ by LOCC. $U_1$ and $U_2$ can be locally distinguished if and only if the output states $|{\Phi }_{U_1}⟩$ and $|{\Phi }_{U_2}⟩$ can be orthogonal [9].