[Paper Review] The structure of Renyi entropic inequalities

Noah Linden, Milán Mosonyi+1 more|arXiv (Cornell University)|Dec 2, 2012

Statistical Mechanics and Entropy4 citations

TL;DR

This paper investigates Rényi entropic inequalities for multipartite quantum states, revealing that for $0 < \alpha < 1$, the only universal inequality is non-negativity—any non-negative entropy vector can be arbitrarily well approximated by marginals of a quantum state. For $\alpha > 1$, no non-trivial homogeneous inequalities exist, though non-linear constraints still apply, contrasting sharply with the rich structure of von Neumann entropy ($\alpha = 1$) inequalities like strong subadditivity.

ABSTRACT

We investigate the universal inequalities relating the alpha-Renyi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropy (alpha=1) which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 01 we show analogously that there are no non-trivial homogeneous (in particular no linear) inequalities. On the other hand, it is known that there are further, non-linear and indeed non-homogeneous, inequalities delimiting the alpha-entropies of a general quantum state. Finally, we also treat the case of Renyi entropies restricted to classical states (i.e. probability distributions), which in addition to non-negativity are also subject to monotonicity. For alpha different from 0 and 1 we show that this is the only other homogeneous relation.

Motivation & Objective

• To determine the complete set of universal inequalities governing Rényi $\alpha$-entropies of marginals in multipartite quantum states.
• To contrast the structure of Rényi entropic inequalities with the well-known, rich set of constraints (e.g., strong subadditivity) that govern von Neumann entropy ($\alpha = 1$).
• To investigate whether Rényi entropies for $\alpha \neq 1$ satisfy any non-trivial homogeneous or linear inequalities.
• To examine the role of monotonicity in classical states and separable quantum states, and to determine the full set of homogeneous inequalities in this case.

Proposed method

• The authors analyze the set of attainable Rényi entropy vectors for $n$-partite quantum states across different $\alpha$ regimes.
• For $0 < \alpha < 1$, they prove that the only constraint is non-negativity by constructing quantum states whose marginal entropies approximate any given non-negative vector arbitrarily closely.
• For $\alpha > 1$, they show that no non-trivial homogeneous inequalities exist, using perturbation and tensor product constructions to demonstrate density of scaled entropy vectors.
• They extend the analysis to classical states (probability distributions), proving that for $\alpha \neq 0,1$, the only homogeneous inequalities are non-negativity and monotonicity under subset inclusion.
• The proof techniques involve spectral analysis of density operators, direct sum and tensor product constructions of states, and asymptotic approximation arguments.
• They leverage known results on classical entropy inequalities (e.g., Zhang-Yeung) and extend them to the quantum Rényi setting, particularly for $\alpha > 1$.

Experimental results

Research questions

• RQ1 For $0 < \alpha < 1$, are there any non-trivial universal inequalities constraining the Rényi entropies of marginals of a quantum state beyond non-negativity?
• RQ2 For $\alpha > 1$, do any non-trivial homogeneous (e.g., linear) inequalities constrain the Rényi entropy vectors of quantum states?
• RQ3 In the classical case, are non-negativity and monotonicity the only homogeneous inequalities for Rényi entropies when $\alpha \neq 0,1$?
• RQ4 Does the set of attainable Rényi entropy vectors for $\alpha > 1$ form a convex cone, or are there additional non-homogeneous constraints?
• RQ5 Can the rich structure of von Neumann entropy inequalities (e.g., strong subadditivity) be generalized or recovered in the Rényi framework for $\alpha \neq 1$?

Key findings

• For $0 < \alpha < 1$, the only universal Rényi entropic inequality is non-negativity: any vector of non-negative real numbers assigned to the non-empty subsets of $n$ parties can be arbitrarily well approximated by the $\alpha$-entropies of the marginals of some quantum state.
• For $\alpha > 1$, there are no non-trivial homogeneous inequalities constraining Rényi entropy vectors of quantum states, though the set of attainable vectors is not a cone, indicating the presence of non-linear constraints.
• The set of Rényi entropy vectors for $\alpha > 1$ is dense in the positive orthant when scaled appropriately, meaning any positive vector can be approximated by a positive multiple of such entropy vectors.
• In the classical case, for $\alpha \neq 0,1$, the only homogeneous inequalities are non-negativity and monotonicity under subset inclusion, which reflects the structure of classical information measures.
• For $\alpha = 0$, the Rényi entropy is the logarithm of the rank of the density matrix, which is discontinuous and does not satisfy strong subadditivity, though it does satisfy subadditivity.
• The paper identifies that non-homogeneous inequalities exist for $\alpha > 1$, such as the one proved by Audenaert, indicating a complex structure beyond homogeneous relations.