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arXiv:1803.05379

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

Ivan Bardet, Cambyse Rouzé

Published Mar 14, 2018• 3 citations
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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.

DOI
10.48550/arxiv.1803.05379

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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 Lp\mathbb{L}_pLp​ 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 Lp\mathbb{L}_pLp​ norms are not uniformly convex for 1≤p≤21\leq p \leq 21≤p≤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.
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 Lp\mathbb {L}_pLp​ 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 Lp\mathbb {L}_pLp​ L p norms are not uniformly convex for 1≤p≤21\le p \le 21≤p≤2 1 ≤ p ≤ 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.
We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup (Φt)t≥0(Φ_t)_{t\ge 0}(Φt​)t≥0​ and every tensor power nnn, we show that the log-Sobolev constant of the product semigroup Φt⊗nΦ_t^{\otimes n}Φt⊗n​ is at least 2/(3ln⁡2)≈0.962/(3\ln 2)\approx 0.962/(3ln2)≈0.96 times the log-Sobolev constant of the single-site semigroup ΦtΦ_tΦt​, independently of nnn and the local dimension ddd. The proof first establishes an exact tensorization of the (q,2)(q,2)(q,2)-hypercontractive inequality for integer qqq, in particular q=3q=3q=3, and then extends the estimate to all real q>2q>2q>2 by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp (q,2)(q,2)(q,2)-hypercontractivity estimates for generalized qubit depolarizing channels.
Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroup's hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative LpL_pLp​ spaces to obtain a general hypercontractive inequality from a particular p→qp \rightarrow qp→q-norm bound. Then, we derive a bound on the 2→42 \rightarrow 42→4-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an NNN-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time poly(log⁡(N)){\rm poly}(\log(N))poly(log(N)) by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.
We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state σσσ, we obtain the modified Log-Sobolev bound α1≥λ(2+log⁡∥σ−1∥∞)α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}α1​≥(2+log∥σ−1∥∞​)λ​ where λλλ is the spectral gap. This removes the assumption of LpL_pLp​-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.
We generalize the notions of the non-commutative Poincaré and modified log-Sobolev inequalities for primitive quantum Markov semigroups (QMS) to not necessarily primitive ones. These two inequalities provide estimates on the decoherence time of the evolution. More precisely, we focus on an algebraic definition of environment-induced decoherence in open quantum systems which happens to be generic on finite dimensional systems and describes the asymptotic behavior of any QMS. An essential tool in our analysis is the explicit structure of the decoherence-free algebra generated by the QMS, a central object in the study of passive quantum error correction schemes. The Poincaré constant corresponds to the spectral gap of the QMS, which implies its positivity, while we prove that the modified log-Sobolev constant is positive under the L1\mathbb L_1L1​-regularity of the Dirichlet form, a condition that also appears in the primitive case. We furthermore prove that strong Lp\mathbb L_pLp​-regularity holds for QMS that satisfy a strong form of detailed balance condition for p≥1p\geq1p≥1. The latter condition includes all known cases where this strong regularity was proved. Finally and to emphasize the mathematical interest of this study compared to the classical case, we focus on two truly quantum scenarios, one exhibiting quantum coherence, and the other, quantum correlations.
In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the ppp-logarithmic Sobolev constant vanishes at p=1/2p=1/2p=1/2 and a correlated graph-based example for which the ppp-logarithmic Sobolev constant tends to −∞-\infty−∞ as p→0+p\to 0^+p→0+. These examples obstruct a positive uniform reverse-hypercontractivity rate on (0,1)(0,1)(0,1). By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for 1/2<p≤q<11/2< p\leq q < 11/2<p≤q<1 with explicit sufficient times. For each fixed δ>0δ>0δ>0, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on [δ,1][δ, 1][δ,1] and hence quantum reverse hypercontractivity for δ<p≤q<1δ< p\leq q<1δ<p≤q<1 with fixed 0<δ<1/20<δ<1/20<δ<1/2. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for δ<p≤q<1δ< p\leq q< 1δ<p≤q<1 with fixed 0<δ<10<δ<10<δ<1 for connected graphs when n≥3n \geq 3n≥3.