Unambiguous identification of coherent states: Searching a quantum database

We consider an unambiguous identification of an unknown coherent state with one of two unknown coherent reference states. Specifically, we consider two modes of an electromagnetic field prepared in unknown coherent states $\mid {\alpha }_1⟩$ and $\mid {\alpha }_2⟩$, respectively. The third mode is prepared either in the state $\mid {\alpha }_1⟩$ or in the state $\mid {\alpha }_2⟩$. The task is to identify (unambiguously) which of the two modes are in the same state. We present a scheme consisting of three beam splitters capable to perform this task. Although we do not prove the optimality, we show that the performance of the proposed setup is better than the generalization of the optimal measurement known for a finite-dimensional case. We show that a single beam splitter is capable to perform an unambiguous quantum state comparison for coherent states optimally. Finally, we propose an experimental setup consisting of $2N-1$ beam splitters for unambiguous identification among $N$ unknown coherent states. This setup can be considered as a search in a quantum database. The elements of the database are unknown coherent states encoded in different modes of an electromagnetic field. The task is to specify the two modes that are excited in the same, though unknown, coherent state.

Figure 1

The beam splitter setup designed for an unambiguous identification of coherent states.

Figure 2

(Color online) The probability of identification $P(\mid {\alpha }_1⟩,\mid {\alpha }_2⟩)$ as a function of the scalar product (given by $\mid {\alpha }_1-{\alpha }_2\mid$) for four UI strategies applied on coherent states $\mid {\alpha }_1⟩$, $\mid {\alpha }_2⟩$. Starting from the bottom the two lowest lines correspond to universal UI measurements (the SWAP-based is in magenta (lowest line) and the optimal strategy is in black, respectively). The next two lines are associated with the UI measurements designed for coherent states (the SWAP-like measurement is in green while the three beam splitters setup is in solid blue, respectively). The top (red) curve corresponds to the optimal discrimination probability among two known states.

Figure 3

Unambiguous identification measurement setup identifying among $N$ coherent states.