Union bound for quantum information processing

Abstract

In this paper, we prove a quantum union bound that is relevant when performing a sequence of binary-outcome quantum measurements on a quantum state. The quantum union bound proved here involves a tunable parameter that can be optimized, and 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. We expect that the bound will find a range of applications in quantum communication theory, quantum algorithms, and quantum complexity theory.

arXiv commentv2: 23 pages, includes proof, based on arXiv:1208.1400 and arXiv:1510.04682, for a lower bound on the second-order asymptotics of hypothesis testing for i.i.d. quantum states acting on a separable Hilbert space

References in corpus (28)

• Entanglement-Assisted Classical Capacity of Noisy Quantum Channels
• One-Shot Classical-Quantum Capacity and Hypothesis Testing
• A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks
• The quantum capacity of channels with arbitrarily correlated noise
• Quantum channels and their entropic characteristics
• Second-order asymptotics for quantum hypothesis testing
• Converse bounds for private communication over quantum channels
• On entanglement-assisted classical capacity
• Gaussian hypothesis testing and quantum illumination
• Quantum Coding with Finite Resources
• Entanglement assisted capacity of the broadband lossy channel
• Second-Order Asymptotics for the Classical Capacity of Image-Additive Quantum Channels
• On the Second-Order Asymptotics for Entanglement-Assisted Communication
• Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions
• Position-based coding and convex splitting for private communication over quantum channels
• Quantum union bounds for sequential projective measurements
• Achieving the Holevo bound via sequential measurements
• Applications of position-based coding to classical communication over quantum channels
• Upper bounds on secret key agreement over lossy thermal bosonic channels
• A simple proof of the detectability lemma and spectral gap amplification
• Sequential decoding of a general classical-quantum channel
• Second-order asymptotics for quantum hypothesis testing in settings beyond i.i.d. - quantum lattice systems and more
• Divide and conquer method for proving gaps of frustration free Hamiltonians
• Belief propagation decoding of quantum channels by passing quantum messages
• The Classical-Quantum Channel with Random State Parameters Known to the Sender
• Second-order coding rates for pure-loss bosonic channels
• Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions
• Entanglement-assisted capacity of constrained quantum channel

Cited by in corpus (7)

• Practical route to entanglement-assisted communication over noisy bosonic channels
• Entanglement-Assisted Communication Surpassing the Ultimate Classical Capacity
• On the near-optimality of one-shot classical communication over quantum channels
• Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels
• Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications
• Transceiver designs to attain the entanglement assisted communications capacity
• Polar Codes for Quantum Reading