Reliable Simulation of Quantum Channels: the Error Exponent

Abstract

The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel's entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel's sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.

V2: presentation improved, details on proofs added, references added, minor corrections. V3: minor changes. V4: whole paper rewritten with a new title, more references added, main results and proofs unchanged. V5: close to published version

References in corpus (44)

• Quantum Resource Theories
• On quantum Renyi entropies: a new generalization and some properties
• Entanglement-Assisted Classical Capacity of Noisy Quantum Channels
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Post-selection technique for quantum channels with applications to quantum cryptography
• The mother of all protocols: Restructuring quantum information's family tree
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Quantum Reverse Shannon Theorem
• Monotonicity of a relative Rényi entropy
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
• Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Strong converse exponent for classical-quantum channel coding
• Quantum message compression with applications
• Multiplicativity of completely bounded $p$-norms implies a strong converse for entanglement-assisted capacity
• Entanglement Cost of Quantum Channels
• Application of the Resource Theory of Channels to Communication Scenarios
• Zero-error channel capacity and simulation assisted by non-local correlations
• Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels
• No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number
• Quantum rate distortion, reverse Shannon theorems, and source-channel separation
• Quantum Channel Simulation and the Channel's Smooth Max-Information
• Dynamical Entanglement
• Reliability function of general classical-quantum channel
• Thermodynamic Capacity of Quantum Processes
• Catalytic Decoupling of Quantum Information
• A triangle of dualities: reversibly decomposable quantum channels, source-channel duality, and time reversal
• Quantum rate distortion coding with auxiliary resources
• Quantum Sphere-Packing Bounds with Polynomial Prefactors
• Applications of position-based coding to classical communication over quantum channels
• One-Shot Manipulation of Dynamical Quantum Resources
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Channel Simulation and Coded Source Compression
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
• Precise evaluation of leaked information with universal2 privacy amplification in the presence of quantum attacker
• Universal approximation of multi-copy states and universal quantum lossless data compression
• Moderate deviation expansion for fully quantum tasks
• Thermodynamic Implementations of Quantum Processes
• Reliable Simulation of Quantum Channels: the Error Exponent
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Lower Bounds on Error Exponents via a New Quantum Decoder
• Strong Converse Exponent for Entanglement-Assisted Communication

Cited by in corpus (1)

• Reliable Simulation of Quantum Channels: the Error Exponent