[Paper Review] Hypercontractivity and logarithmic Sobolev Inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates

Ivan Bardet, Cambyse Rouzé|arXiv (Cornell University)|Mar 14, 2018

Random Matrices and Applications4 citations

TL;DR

This paper introduces a novel framework for hypercontractivity and logarithmic Sobolev inequalities (LSI) in non-primitive quantum Markov semigroups (QMS) using amalgamated $ L_p$ norms, generalizing prior results from the primitive case. It establishes that weak hypercontractivity implies weak LSI, proves strong LSI and hypercontractivity do not hold for non-trivially primitive QMS, and derives universal bounds on decoherence rates via completely bounded (CB) norms, extending results to tensorized systems.

ABSTRACT

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated $\mathbb{L}_p$ norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated $\mathbb{L}_p$ norms are not uniformly convex for $1\leq p \leq 2$. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

Motivation & Objective

• To generalize weak hypercontractivity and weak logarithmic Sobolev inequality (LSI) to non-primitive quantum Markov semigroups (QMS), where the dynamics converge to a non-trivial decoherence-free algebra.
• To establish a version of Gross’ integration lemma in the non-primitive setting, showing that weak hypercontractivity implies weak LSI.
• To demonstrate that strong LSI and strong hypercontractivity cannot hold for non-trivially primitive QMS, implying non-uniform convexity of amalgamated $ L_p$ norms for $1 \leq p \leq 2$.
• To derive universal upper bounds on weak logarithmic Sobolev constants for finite-dimensional QMS using a method inspired by Diaconis and Saloff-Coste and Temme et al., leading to estimates on decoherence rates.
• To extend the theory of completely bounded (CB) log-Sobolev inequalities to general primitive QMS and provide estimates on weak constants, including tensorization properties.

Proposed method

• Introduces a new notion of hypercontractivity based on amalgamated $ L_p$ norms, which depend on the semigroup’s decoherence-free algebra $ N( P)$ and the trace-normalized invariant state $\sigma_{\operatorname{Tr}}$.
• Applies Kosaki’s theory of non-commutative weighted $ L_p$ spaces and recent operator space techniques from Junge and Parcet to define the amalgamated norms.
• Proves a generalized Gross’ integration lemma: weak hypercontractivity implies weak LSI, with the constant depending on the cardinality of the center of the decoherence-free algebra.
• Uses the spectral gap and the inverse norm $\|\sigma^{-1}\|_{\infty}$ to bound the weak CB-log-Sobolev constant, leading to explicit upper bounds on decoherence rates.
• Applies the theory to tensorized QMS by leveraging the multiplicativity of completely bounded (CB) norms, proving that the weak CB-LSI constant is additive under tensorization.
• Establishes that strong LSI and strong hypercontractivity fail for non-primitive QMS, due to the non-uniform convexity of amalgamated $ L_p$ norms for $1 \leq p \leq 2$.

Experimental results

Research questions

• RQ1 Can the concepts of weak hypercontractivity and weak logarithmic Sobolev inequality be generalized to non-primitive quantum Markov semigroups?
• RQ2 Does weak hypercontractivity imply weak LSI in the non-primitive case, and what is the role of the decoherence-free algebra’s center in the constant?
• RQ3 Can strong logarithmic Sobolev inequalities and strong hypercontractivity hold for non-primitive QMS, and what structural properties of the amalgamated $ L_p$ norms prevent this?
• RQ4 What universal upper bounds can be derived for the weak logarithmic Sobolev constant of a finite-dimensional QMS, and how do they relate to decoherence rates?
• RQ5 How do the completely bounded (CB) norms extend the tensorization of log-Sobolev inequalities to general primitive QMS, and what are the implications for multi-partite systems?

Key findings

• Weak hypercontractivity implies weak logarithmic Sobolev inequality (LSI) for non-primitive QMS, with the constant bounded by the cardinality of the center of the decoherence-free algebra.
• Strong LSI and strong hypercontractivity do not hold for non-trivially primitive QMS, which implies that the amalgamated $ L_p$ norms are not uniformly convex for $1 \leq p \leq 2$.
• A universal upper bound on the weak CB-log-Sobolev constant is derived: $ c \leq \frac{\ln\|\sigma^{-1}\|_{\infty} + 2}{2\lambda(\mathcal{L})} $, where $\lambda(\mathcal{L})$ is the spectral gap.
• The weak CB-log-Sobolev inequality tensorizes with additive constants: for $n$ tensorized primitive QMS, the constant $c$ is the maximum of individual $c_i$, and $d = \sum d_i$.
• The framework extends the CB-log-Sobolev inequality to general primitive QMS, generalizing results of Beigi and King, and provides estimates on the weak constants in tensorized settings.
• The results lead to new, explicit bounds on decoherence rates for non-primitive QMS, particularly in the context of dissipative state preparation and quantum memory stability.