Union bound for quantum information processing

Area of Science:

• Quantum Information Theory
• Quantum Measurement Theory

Background:

• The Hayashi-Nagaoka inequality is crucial for analyzing error probabilities in quantum information theory, particularly for square-root measurements.
• Existing methods for analyzing sequences of quantum measurements can be complex.

Purpose of the Study:

• To introduce and prove a novel quantum union bound applicable to sequential binary-outcome quantum measurements.
• To demonstrate the bound's utility and elementary proof method.

Main Methods:

• The proof relies on fundamental concepts: properties of projectors, the Pythagorean theorem, and the Cauchy-Schwarz inequality.
• A tunable parameter in the bound is optimized, analogous to parameters in existing inequalities.

Main Results:

• A new quantum union bound is established, featuring an optimizable parameter.
• The bound is applied to show that a sequential decoding strategy achieves a lower bound on a quantum channel's second-order coding rate.

Conclusions:

• The derived quantum union bound offers an elementary and advantageous approach for analyzing quantum measurements.
• The bound has potential applications in quantum communication, quantum algorithms, and quantum complexity theory, especially in the non-asymptotic regime.