Unambiguous pure-state identification without classical knowledge

We study how to unambiguously identify a given quantum pure state with one of the two reference pure states when no classical knowledge on the reference states is given but a certain number of copies of each reference quantum state are presented. By unambiguous identification, we mean that we are not allowed to make a mistake but our measurement can produce an inconclusive result. Assuming the two reference states are independently distributed over the whole pure state space in a unitary invariant way, we determine the optimal mean success probability for an arbitrary number of copies of the reference states and a general dimension of the state space. It is explicitly shown that the obtained optimal mean success probability asymptotically approaches that of the unambiguous discrimination as the number of the copies of the reference states increases.

Figure 1

Decomposition of the product of three $\mathrm{U}\left(d\right)$ irreducible representations $\left[1\right]\otimes \left[N\right]\otimes \left[N\right]$. The decomposition leads to the three orthogonal subspaces $V_n$ $(n=1,2,3)$ according to the number of rows, $n$, of the Young’s diagram.

Figure 2

The optimal mean unambiguous identification probability $p_{\max }^{\left(N\right)}\left(d\right)$ as a function of the number of the copies $\left(N\right)$ of the reference states. As $N$ increases, $p_{\max }^{\left(N\right)}\left(d\right)$ approaches the mean optimal unambiguous discrimination probability shown by the horizontal lines.