Programmable quantum-state discriminators with simple programs

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as an input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, and the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to $n$ copies. In comparison to other works we find that providing $n>1$ copies of the data state yields higher success probabilities than providing $n>1$ copies of the program states.

Figure 1

(Color online) Comparison of the optimum performances of the error minimizing strategy and the unambiguous discrimination strategy for the discrimination of two unknown pure states. The minimum error probabilities $P_E$ resulting from a three-qubit measurement (full line) and $P_E^{AB}$, $P_E^{BC}$ resulting from two-qubit measurements (dashed-dotted and dashed-double dotted line, respectively) are compared to the failure probabilities for unambiguous discrimination $Q_F^{AB}$, $Q_F^{BC}$ (dotted lines) and $Q_F^{\mathrm{POVM}}$ (dashed line). The error and failure probabilities are plotted vs the prior probability ${\eta }_1$ that the state of the data qubit $B$ matches the state of the program qubit $A$.

Figure 2

(Color online) Failure and error probabilities for the various strategies of discriminating between one known and one unknown state vs the prior probability ${\eta }_1$. The minimum error probabilities $P_E^{\prime }$ resulting from a joint measurement on the qubits $B$ and $C$ (full line) and from a single-qubit measurement, $P_E^{″}$ (dashed-dotted line) are compared to the failure probabilities for unambiguous identification $Q_F^{\prime B}$, $Q_F^{\prime BC}$ (dotted lines), and $Q_F^{\prime \mathrm{POVM}}$ (dashed line).

Figure 3

(Color online) Comparison of the optimum performances of the failure probabilities of the projective measurements (upper curve) and the failure probability of the POVM (lower curve) vs the number of copies $n$ for the value of ${\eta }_1=2∕(n+4)$, when their difference is the largest for the unambiguous discrimination of one known state from one unknown state when $n$ copies of the data state are provided.

Figure 4

(Color online) The difference between the upper curve and the lower curve in Fig. 3 as a function of $n$. The difference between the performance of the PVM and POVM is maximum for $n=5$.

Figure 5

(Color online) The ratio of the upper and lower curve in Fig. 3 as a function of $n$. Asymptotically the failure rate of the PVM is 50% higher than that of the POVM.