Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps

TL;DR

The paper proves a universal bound on the trace of any 2-positive, trace-preserving map in terms of its smallest eigenvalue: $\mathrm{tr}(\Phi) \le d \min \Re(\sigma(\Phi)) + (d^2 - d)$. It shows the bound is tight and that 2-positivity is necessary in general, with a transpose map example at $d=2$ illustrating the limitation of weaker positivity. The authors present a purely algebraic proof based on transition matrices and a Hermitian-averaging trick, then extend the bound to generators of one-parameter semigroups, obtaining $\mathrm{tr}(L) \le d \min \Re(\sigma(L))$ for 2-positive TP generators. This generalizes known completely positive results on relaxation rates, highlighting a common linear constraint across positivity classes and motivating open questions about spectral distinctions between CP and weaker positivity notions and the role of trace-preservation.

Abstract

We prove an upper bound on the trace of any 2-positive, trace-preserving map in terms of its smallest eigenvalue. We show that this spectral bound is tight, and that 2-positivity is necessary for this inequality to hold in general. Moreover, we use this to infer a similar bound for generators of one-parameter semigroups of 2-positive trace-preserving maps. With this approach we generalize known results for completely positive trace-preserving dynamics while providing a significantly simpler proof that is entirely algebraic.

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Theorem 2
• Lemma 1
• proof
• Remark 2
• proof : Proof of Thm. \ref{['thm_main']}
• proof : Proof of Theorem \ref{['thm_main2']}
• Example 1
• Example 2