Unambiguous comparison of ensembles of quantum states

We present a solution of the problem of the optimal unambiguous comparison of two ensembles of unknown quantum states ${|{\psi }_1⟩}^{\otimes k}$ and ${|{\psi }_2⟩}^{\otimes l}$. We consider two cases: (1) The two unknown states $|{\psi }_1⟩$ and $|{\psi }_2⟩$ are arbitrary states of qudits. (2) Alternatively, they are coherent states of a harmonic oscillator. For the case of coherent states we propose a simple experimental realization of the optimal “comparison” machine composed of a finite number of beam splitters and a single photodetector.

Figure 1

The probability of revealing the difference between the compared states $|{\psi }_1⟩$, $|{\psi }_2⟩$. The gray dashed lines are valid for the optimal state comparison among all pure states. Each line corresponds to a different number of copies of the compared states. The solid black lines indicate the performance of the optimal comparison if we are restricted to coherent states only.

Figure 2

The mean probability of the detection of a difference between the compared states $|{\psi }_1⟩$, $|{\psi }_2⟩$ as a function of the dimension of the Hilbert space of the compared systems.

Figure 3

The beam-splitter setup for the comparison of two finite-size ensembles composed of $k$ copies of the coherent state $|{\alpha }_1⟩$ and $l$ copies of the coherent state $|{\alpha }_2⟩$, respectively.