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

TL;DR

The paper investigates how fidelity of entanglement and quantum entropies jointly certify entanglement, developing a channel-side and a state-side framework. It introduces fidelity-breaking (FBC) and fidelity-annihilating (FAC) channels, plus negative conditional entropy breaking/annihilating channels, and provides topological characterizations, operational tests (including depolarizing channels), and relationships among resource theories. On states, it derives sharp entropy–fidelity bounds for general two-qubit states and Weyl states, linking F>½ to Rényi and Tsallis entropies (and their conditional forms), and connects these to relative entropy via a maximization-based bound. The results clarify how entropic quantities constrain entanglement and how certain channels diminish entanglement resources, offering insights with teleportation, quantum communication, and resource theory applications. The work also maps how channel behavior (FBC/FAC) interplays with negative conditional entropy resources, highlighting both inclusions and non-subset relations that guide future investigations in higher dimensions and explicit channel representations.

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 Rényi 2-entropy, conditional Rényi 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.

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