[Paper Review] On some entropy inequalities

Lin Zhang|arXiv (Cornell University)|Mar 26, 2014

Mathematical Inequalities and Applications8 references3 citations

TL;DR

This paper establishes new entropy inequalities using Rényi relative entropy and operator inequalities, extending known results via trace-based bounds on quantum relative entropy. The key contribution is a sufficient condition—trace normalization of the exponential of log-density combinations—under which a tripartite quantum state becomes a Markov state, i.e., has zero conditional mutual information.

ABSTRACT

In this short report, we give some new entropy inequalities based on Rényi relative entropy and the observation made by Berta {\em et al} [arXiv:1403.6102]. These inequalities obtained extends some well-known entropy inequalities. We also obtain a condition under which a tripartite operator becomes a Markov state.

Motivation & Objective

• To extend well-known quantum entropy inequalities using Rényi relative entropy and matrix norm inequalities.
• To derive tighter lower bounds on quantum relative entropy using trace and Frobenius norms.
• To characterize when a tripartite quantum state has vanishing conditional mutual information (i.e., is a Markov state).
• To provide a sufficient condition under which the exponential of a combination of reduced density operators forms a valid quantum state, implying Markovianity.

Proposed method

• Derives a chain of inequalities involving Rényi relative entropy, trace of geometric mean of density operators, and Frobenius norms of square root and density operator differences.
• Applies the monotonicity of Rényi relative entropy and matrix inequalities involving ∥√M − √N∥²₂ ≤ ∥M − N∥₁ ≤ ∥√M − √N∥₂∥√M + √N∥₂.
• Uses the Golden-Thompson inequality to show that exp(logρ_AC + logρ_BC) is a substate (trace ≤ 1), enabling application of prior relative entropy bounds.
• Reformulates conditional mutual information as a relative entropy: I(A:B|C) = S(ρ_ABC || exp(logρ_AC + logρ_BC − logρ_C)).
• Establishes that if Tr[exp(logρ_AC + logρ_BC − logρ_C)] = 1, then the exponential operator is a valid quantum state and ρ_ABC is Markov.
• Applies Proposition 1.2 (entropy inequality under quantum channels) to derive bounds on relative entropy in terms of trace and norm expressions.

Experimental results

Research questions

• RQ1 Under what conditions does the exponential of the sum of log-reduced densities form a valid quantum state?
• RQ2 When does a tripartite state satisfy I(A:B|C) = 0, i.e., become a Markov state?
• RQ3 How can Rényi relative entropy be used to derive tighter bounds on quantum entropy inequalities?
• RQ4 What is the relationship between the trace of exp(logρ_AC + logρ_BC − logρ_C) and the Markovianity of ρ_ABC?
• RQ5 Can the structure of the operator exp(logρ_AC + logρ_BC − logρ_C) be characterized when its trace is 1?

Key findings

• The inequality S(ρ||σ) ≥ −2logTr(√ρ√σ) holds for any quantum state ρ and substate σ, extending the standard relative entropy bound.
• A chain of inequalities links relative entropy to Frobenius norm and trace norm: S(ρ||σ) ≥ ∥√ρ − √σ∥²₂ ≥ (1/4)∥ρ − σ∥²₁.
• For tripartite states, I(A:B|C) ≥ −2logTr(√ρ_ABC√exp(logρ_AC + logρ_BC − logρ_C)) ≥ ∥√ρ_ABC − √exp(·)∥²₂ ≥ (1/4)∥ρ_ABC − exp(·)∥²₁.
• If Tr[exp(logρ_AC + logρ_BC − logρ_C)] = 1, then exp(logρ_AC + logρ_BC − logρ_C) is a valid quantum state and ρ_ABC is a Markov state.
• The operator exp(logρ_AC + logρ_BC − logρ_C) equals both ρ^{1/2}_{AB}ρ^{-1/2}_Bρ_{BC}ρ^{-1/2}_Bρ^{1/2}_{AB} and ρ^{1/2}_{BC}ρ^{-1/2}_Bρ_{AB}ρ^{-1/2}_Bρ^{1/2}_{BC}, confirming its self-consistent structure.
• The trace of exp(logρ_AC + logρ_BC) is bounded by Tr(ρ²_C) ≤ 1, ensuring it is a substate, which enables the application of relative entropy bounds.