Union bound for quantum information processing
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Gao's quantum union bound is a generalization of the union bound from probability theory and finds a range of applications in quantum communication theory, quantum algorithms, and quantum complexity theory [Phys. Rev. A, 92(5):052331, 2015]. It is relevant when performing a sequence of binary-outcome quantum measurements on a quantum state, giving the same bound that the classical union bound would, except with a scaling factor of four. In this paper, we improve upon Gao's quantum union bound, by proving a quantum union bound that involves a tunable parameter that can be optimized. This tunable parameter plays a similar role to a parameter involved in the Hayashi-Nagaoka inequality [IEEE Trans. Inf. Theory, 49(7):1753 (2003)], used often in quantum information theory when analyzing the error probability of a square-root measurement. An advantage of the proof delivered here is that it is elementary, relying only on basic properties of projectors, the Pythagorean theorem, and the Cauchy--Schwarz inequality. As a non-trivial application of our quantum union bound, we prove that a sequential decoding strategy for classical communication over a quantum channel achieves a lower bound on the channel's second-order coding rate. This demonstrates the advantage of our quantum union bound in the non-asymptotic regime, in which a communication channel is called a finite number of times.
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