Distinguishing mixed quantum states: Minimum-error discrimination versus optimum unambiguous discrimination

We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.

Figure 1

Minimum probabilities of error in ambiguous discrimination $P_E$ (full line), and of failure in unambiguous discrimination $Q_F$ (dashed line), for distinguishing between ${\rho }_1=\mid \psi ⟩⟨\psi \mid$ and ${\rho }_2=\frac{1}{3}{\sum }_{j=1}^3\mid u_j⟩⟨u_j\mid$, where $⟨u_i\mid u_j⟩={\delta }_{ij}$. The probabilities are plotted vs the norm of the parallel component, $\parallel {\psi }^{\Vert }\parallel ={\left({\sum }_{j=1}^3{\mid ⟨u_j\mid \psi ⟩\mid }^2\right)}^{1∕2}$, and the prior probability of ${\rho }_1$ is assumed to be ${\eta }_1=0.25$.

Figure 2

Minimum-error probability in ambiguous discrimination $P_E$ (full line) and minimum failure probability in unambigous discrimination $Q_F$ (dashed line) for the states ${\rho }_1$ and ${\rho }_2$ specified in Fig. 1. The probabilities are depicted vs the prior probability ${\eta }_1$ of the state ${\rho }_1$, in the special case where $\parallel {\psi }^{\Vert }\parallel =1$.

Figure 3

Same as Fig. 2 but for the case $\parallel {\psi }^{\Vert }\parallel =0.5$.