bibkey: briet2009jensenshannon
authors: Jop Briët, Peter Harremoës
year: 2009
title: "Properties of classical and quantum Jensen-Shannon divergence"
doi: 10.1103/PhysRevA.79.052311
url: https://arxiv.org/abs/0806.4472v4
claim: "However, based on a large amount of numerical evidence, we conjecture the function A → Tr(e^A) to be exponentially convex for density matrices A."
strata_touched:
  - D5/S3/Divergence/TraceExpExponentialConvexityRefutation
license: citation-only
triage: anchor

Properties of classical and quantum Jensen-Shannon divergence

Jop Briët and Peter Harremoës, Properties of classical and quantum Jensen-Shannon divergence, Physical Review A 79, 052311 (2009), arXiv:0806.4472v4.

Definition 4 (Exponential convexity):

Let X be a convex set and φ : X → R a mapping. Then φ is said to be exponentially convex if the function X × X → R given by (x, y) → φ((x+y)/2) is positive definite.

§VI (Conclusions and open problems):

Whether (QJD1)^{1/2} is a metric for all mixed states remains unknown. However, based on a large amount of numerical evidence, we conjecture the function A → Tr(e^A) to be exponentially convex for density matrices A. Proving this would imply that QJDα is negative definite for α ∈ (0, 2], and hence the square of a metric that can be embedded in a real Hilbert space.

The centre-plus-trine qubit configuration refutes the trace-exponential conjecture. The kernel $\mathrm{Tr}\exp (t(\rho +\sigma )/2)$ fails positive definiteness for every real $t\ne 0$. This closes the proposed argument through exponential convexity without deciding the Hilbert-space embedding question for quantum Jensen–Shannon divergence.

Locator

Author text: https://arxiv.org/abs/0806.4472v4.

• arXiv v4, Definition 4 (Exponential convexity).
• arXiv v4, §VI (Conclusions and open problems).
• arXiv v4, Lemma 5 (the pure-state restriction).
• DOI: https://doi.org/10.1103/PhysRevA.79.052311