Sequential scheme for locally discriminating bipartite unitary operations without inverses

Local distinguishability of bipartite unitary operations has recently received much attention. A nontrivial and interesting question concerning this subject is whether there is a sequential scheme for locally discriminating between two bipartite unitary operations, because a sequential scheme usually represents the most economic strategy for discrimination. An affirmative answer to this question was given in the literature, however with two limitations: (i) the unitary operations to be discriminated were limited to act on $d\otimes d$, i.e., a two-qudit system, and (ii) the inverses of the unitary operations were assumed to be accessible, although this assumption may be unrealizable in experiment. In this paper, we improve the result by removing the two limitations. Specifically, we show that any two bipartite unitary operations acting on $d_A\otimes d_B$ can be locally discriminated by a sequential scheme, without using the inverses of the unitary operations. Therefore, this paper enhances the applicability and feasibility of the sequential scheme for locally discriminating unitary operations.

Figure 1

A mixed scheme for perfectly discriminating bipartite unitary operations $X\in \{U,V\}$ by LOCC. $u_i$ and $v_i$ are single-particle unitary operations. $|{\mathrm{\Phi }}_X\rangle$ denotes the output state of the circuit. A perfect discrimination between $U$ and $V$ means that there exists an input state ${|\psi \rangle }_A{|\phi \rangle }_B$ such that the output states $|{\mathrm{\Phi }}_X\rangle$ corresponding to different $X$ are orthogonal (that is, $|{\mathrm{\Phi }}_U\rangle \perp |{\mathrm{\Phi }}_V\rangle$), and then $|{\mathrm{\Phi }}_U\rangle$ and $|{\mathrm{\Phi }}_V\rangle$ can be perfectly discriminated by LOCC [8]. Note that one of ${|\psi \rangle }_A$ and ${|\phi \rangle }_B$ held by Alice and Bob, respectively, must be a multipartite entangled state.

Figure 2

A sequential scheme for locally discriminating unitary operations $U$ and $V$ acting on $d_A\otimes d_B$. Here $X$ represents the unknown unitary operation $U$ or $V,N+1$ is the finite number of applying $X,{\left\{u_i\right\}}_{i=1}^N$ and ${\left\{v_i\right\}}_{i=1}^N$ are unitary operations acting on ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$, respectively, and ${|\phi \rangle }_A{|\varphi \rangle }_B$ is the input state. The output states $|{\mathrm{\Phi }}_U\rangle$ and $|{\mathrm{\Phi }}_V\rangle$ are orthogonal, and thus can be perfectly discriminated by LOCC [8].