Fidelity of entanglement and quantum entropies: unveiling their relationship in quantum states and channels

TL;DR

This paper investigates the relationship between the fidelity of entanglement and quantum entropies in states and channels, providing new characterizations, bounds, and insights into their interplay in quantum information processing.

Contribution

It introduces the concept of fidelity annihilating channels, analyzes their properties, and establishes bounds linking entanglement fidelity with various quantum entropies for states and channels.

Findings

Characterization of channels reducing entanglement fidelity beyond thresholds

Bounds on quantum entropies in terms of entanglement fidelity

Connections between fidelity and negative conditional entropies

Abstract

Entanglement serves as a fundamental resource for various quantum information processing tasks. Fidelity of entanglement (which measures the proximity to a maximally entangled state) and various quantum entropies are key indicators for certifying entanglement in a quantum state. Quantum states with high fidelity are particularly useful for numerous information-theoretic applications. Similarly, states possessing negative conditional entropy provide significant advantages in several quantum information processing protocols. In this work, we examine the relationship between these two indicators of entanglement, both in state and channel regimes. First, we present a comprehensive analysis and characterization of channels that reduce fidelity of entanglement beyond a threshold limit of bipartite composite systems. In this context, we introduce the notion of fidelity annihilating channel and discuss its topological characterization, along with various information-theoretic properties. We then provide a comparison between channels that diminish the fidelity of entanglement and negative conditional entropies, using the depolarizing channel as an illustrative example. In particular, we determine the parameter regimes in which the depolarizing channel belongs to a given family and establish connections among these families of channels. Extending our analysis from channels to the state level, we further examine the relationship between the fidelity of entanglement and various quantum entropies for general two-qubit states. We derive the upper bound on R\'enyi 2-entropy, conditional R\'enyi 2-entropy, Tsallis 2-entropy, and conditional Tsallis 2-entropy, in terms of the fidelity of entanglement. Finally, we explore the relationship between relative entropy and the fidelity of entanglement of a two qudit quantum state.