Mark Wilde will give a short talk at 3:30 in 6-310.
Title: Union bound for quantum information processing
Abstract:
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.
Joint work with Samad Khabbazi Oskouei (Islamic Azad University) and
Stefano Mancini (University of Camerino)
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