Optimal sequential state discrimination between two mixed quantum states

Recently, sequential state discrimination, as a quantum-key distribution protocol, has been proposed for multiple receivers. A previous study [J. A. Bergou et al., Phys. Rev. Lett. 111, 100501 (2013)] showed that every receiver could successfully perform a sequential state discrimination of two pure states with identical prior probabilities. In this study, we extend the sequential state discrimination to mixed states with arbitrary prior probability. First, we analytically obtain the condition of the receiver's optimal measurement. In addition, we show that the optimal probability for every receiver to share the mixed state prepared by the sender is not zero. Furthermore, we compare the sequential state discrimination to the strategies of quantum reproducing and quantum broadcasting. We find that there are cases in which, unlike that of the pure state, the sequential state discrimination of mixed states shows a better performance than the other strategies.

Figure 1

Sequential state discrimination of Bob and Charlie. First, Alice prepares a quantum state ${\rho }_i$ with prior probability $q_i$ and sends it to Bob. By a nonoptimal POVM $\left\{M_i\right\}$, Bob performs unambiguous discrimination on Alice's quantum state. If Bob correctly discriminates the quantum state of Alice, Bob sends his postmeasurement state ${\sigma }_i$ to Charlie. Then, Charlie discriminates ${\sigma }_i$, with an optimal POVM.

Figure 2

Bob's POVM condition that optimizes sequential state discrimination. The solid lines display the points satisfying Eq. (10). The dotted lines display the points satisfying the equality condition of Eq. (11).

Figure 3

Quantum reproducing scenario of Bob and Charlie for a quantum state prepared by Alice. Bob performs the optimal POVM $\left\{M_i\right\}$ on quantum state ${\rho }_i$ prepared by Alice with prior probability $q_i$. If Bob correctly discriminates Alice's quantum state, he reproduces ${\rho }_i$ and sends it to Charlie. Using an optimal POVM, Charlie discriminates the quantum state ${\rho }_i$ received by Bob. When Bob's measurement fails to discriminate Alice's mixed state, Bob informs Charlie of the fact that his measurement fails.

Figure 4

Quantum broadcasting scenario of Bob and Charlie for a quantum state prepared by Alice. Bob broadcasts a quantum state ${\rho }_i$ prepared by Alice with prior probability $q_i$. When Bob succeeds at broadcasting it, Bob can share the bipartite state $\rho$ with Charlie. Then, Bob and Charlie perform an optimal POVM on the partial state. When Bob's measurement fails to discriminate Alice's mixed state, Bob informs Charlie of the fact that his measurement fails.

Figure 5

Geometric condition for Charlie's optimal POVM. If in $0\le {\beta }_1,{\beta }_2\le 1$, the line $f={\mathrm{\Lambda }}_1{\beta }_1+{\mathrm{\Lambda }}_2{\beta }_2$ becomes a tangent to the boundary of the set satisfying Charlie's POVM condition, Charlie's POVM is optimal.