Pure and mixed state

Tsuyoshi Sasaki Usuda

The optimum quantum measurements that minimize the error probability for mixed‐state signals are considered. The square‐root measurement (SRM) is known to be the optimal measurement for many pure‐state signals, e.g., symmetric states, linear codes, etc. However, when the states are mixed, the SRM is no longer guaranteed to be optimal even if the states are symmetric. Here, we define the ‘pure’ square‐root measurement (pure‐SRM) which is the SRM for pure states and its optimality for mixed states. It is shown that the pure‐SRM is satisfied the necessary and sufficient conditions of the quantum Bayes strategy when the signals are mixed by transmission errors, and therefore the pure‐SRM is the optimal measurement.

J. C. A. Barata, Marcos Brum, Victor Chabu +1 more

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Jin-Hua Zhang, Fulin Zhang, Zhi‐Xi Wang +2 more

We investigate the discrimination of pure-mixed (quantum filtering) and mixed-mixed states and compare their optimal success probability with the one for discriminating other pairs of pure states superposed by the vectors included in the mixed states. We prove that under the equal-fidelity condition, the pure-pure state discrimination scheme is superior to the pure-mixed (mixed-mixed) one. With respect to quantum filtering, the coherence exists only in one pure state and is detrimental to the state discrimination for lower dimensional systems; while it is the opposite for the mixed-mixed case with symmetrically distributed coherence. Making an extension to infinite-dimensional systems, we find that the coherence which is detrimental to state discrimination may become helpful and vice versa.

Di Yao-Min, Baolin Hu, Liu Dong-Dong +2 more

In this article an obvious expression of concurrence of the mixed state of two non-orthogonal pure states in two qubits is derived. The result reveals that the expression of concurrence of the mixed state of two orthogonal states can be directly generalized to the case of the mixed state of two non-orthogonal states. Then the maximally entangled states mixed by two pure states and the concurrences in some important special cases are discussed.

Peter Mittelstaedt