Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

Area of Science:

• Quantum Information Science
• Quantum Physics

Background:

• Quantum coherence is a key resource in quantum information science.
• Two primary measures quantify coherence: relative entropy of coherence (Cr(ρ)) and the ℓ1-norm of coherence (Cℓ1(ρ)).

Purpose of the Study:

• To derive a symmetry-like relation for the relative entropy of coherence in n-partite quantum states.
• To establish lower and upper bounds for Cr(ρ).
• To investigate the conjecture Cr(ρ) ≤ Cℓ1(ρ) for general quantum states.

Main Methods:

• Derivation of a symmetry-like relation for the relative entropy of coherence in n-partite systems.
• Analysis of the relationship between multipartite states and their reduced states.
• Exploration of the conjecture Cr(ρ) ≤ Cℓ1(ρ) using mixtures and modified entropy definitions.

Main Results:

• A novel symmetry-like relation for n-partite quantum states is established, providing bounds for the relative entropy of coherence.
• It is shown that an n-partite state is incoherent if and only if all its reduced states are incoherent, provided each individual subsystem state is pure.
• The conjecture Cr(ρ) ≤ Cℓ1(ρ) is confirmed for mixed qubit states and pure states, and new observations are made for general states and modified entropy measures.

Conclusions:

• The derived inequalities offer new insights into quantifying and understanding quantum coherence in complex quantum systems.
• The findings contribute to the ongoing effort to fully characterize and utilize quantum coherence as a resource.
• Further investigation into the conjecture Cr(ρ) ≤ Cℓ1(ρ) is warranted for general quantum states.