The apex of the family tree of protocols: Optimal rates and resource inequalities

Abstract

We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the Fully Quantum Slepian-Wolf protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in Quantum Information Theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic i.i.d. scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results, yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previously bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.

31 pages, 2 figures. Published version

References in corpus (12)

• Distillation of secret key and entanglement from quantum states
• Quantum information can be negative
• Coding Theorem and Strong Converse for Quantum Channels
• One-Shot Classical-Quantum Capacity and Hypothesis Testing
• The mother of all protocols: Restructuring quantum information's family tree
• Relating quantum privacy and quantum coherence: an operational approach
• The decoupling approach to quantum information theory
• Distilling entanglement from arbitrary resources
• Quantum state redistribution based on a generalized decoupling
• State redistribution as merging: introducing the coherent relay
• Public and private communication with a quantum channel and a secret key
• Beyond i.i.d. in Quantum Information Theory

Cited by in corpus (35)

• Exponential Decay of Correlations Implies Area Law
• Quantum message compression with applications
• Leading order corrections to the quantum extremal surface prescription
• Quantum Coding with Finite Resources
• Quantifying resource in catalytic resource theory
• Resource convertibility and ordered commutative monoids
• Smooth Max-Information as One-Shot Generalization for Mutual Information
• Second-order asymptotics for source coding, dense coding and pure-state entanglement conversions
• One-shot entanglement-assisted quantum and classical communication
• Catalytic Decoupling of Quantum Information
• One-shot lossy quantum data compression
• Unitary $2$-designs from random $X$- and $Z$-diagonal unitaries
• Strong converse theorems using Rényi entropies
• Smooth Entropy Bounds on One-Shot Quantum State Redistribution
• Decoding quantum information via the Petz recovery map
• "Pretty strong" converse for the quantum capacity of degradable channels
• Sequential decoding of a general classical-quantum channel
• Channel Simulation and Coded Source Compression
• An upper bound on the second order asymptotic expansion for the quantum communication cost of state redistribution
• General theory of environment-assisted entanglement distillation
• One-shot quantum error correction of classical and quantum information
• Decoupling with random diagonal unitaries
• Carrying an arbitrarily large amount of information using a single quantum particle
• One-Shot Hybrid State Redistribution
• Quantum State Merging for Arbitrarily Small-Dimensional Systems
• Entropy in Quantum Information Theory -- Communication and Cryptography
• Noisy quantum state redistribution with promise and the Alpha-bit
• Direct Sum Theorem for Bounded Round Quantum Communication Complexity
• One-Shot Triple-Resource Trade-Off in Quantum Channel Coding
• One-shot Marton inner bound for classical-quantum broadcast channel
• Optimal one-shot entanglement sharing
• Efficient achievability for quantum protocols using decoupling theorems
• Source Compression with a Quantum Helper
• Fully Quantum Source Compression with a Quantum Helper
• One-shot source coding with coded side information available at the decoder