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arXiv:1403.5224 · Journal of Physics A Mathematical and Theoretical

Hypercontractivity of quasi-free quantum semigroups

Kristan Temme, Fernando Pastawski, Michael J. Kastoryano

Published Sep 23, 2014• 49 citations
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Abstract

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 $L_p$ spaces to obtain a general hypercontractive inequality from a particular $p \rightarrow q$-norm bound. Then, we derive a bound on the $2 \rightarrow 4$-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an $N$-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 ${\rm poly}(\log(N))$ 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.

DOI
10.1088/1751-8113/47/40/405303

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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 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.
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.
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.
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.
We derive rates of exponential relative-entropy decay for a broad class of infinite-dimensional quantum Gibbs samplers. A key technical ingredient is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities. For suitably chosen dissipative couplings, we obtain exponential entropy decay for full multimode Bose-Hubbard samplers with arbitrary hopping and repulsive density interactions. For harmonically confined quantum Coulomb gases and molecular systems, we prove exponential entropy decay for every fixed finite-rank low-energy truncation of the interaction. Finally, we exhibit a periodic charging model with a positive spectral gap but no finite-time L2L^2L2-to-LpL^pLp bound for any p>2p>2p>2.
We prove an almost optimal hypercontractive inequality for products of quantum erasure channels, generalizing the hypercontractivity for classical binary erasure channels. To our knowledge, this is the first tensorization-type hypercontractivity bound for quantum channels with no fixed states. The traditional inductive arguments for classical hypercontractivity cannot be generalized to the quantum setting due to the nature of the non-commutativity of matrices. To overcome the difficulty, we establish a novel quantum log-Sobolev inequality for Bernoulli entropy, which includes the classical log-Sobolev inequality and the quantum log-Sobolev inequality as one-partite cases. To our knowledge, its classical counterpart is also unknown prior to this work. We establish a connection between our quantum log-Sobolev inequality and the hypercontractivity bound for quantum erasure channels via a refined quantum Gross’ lemma, extending the analogous connection between the quantum log- Sobolev inequality and the hypercontractivity for qubit unital channels. As an application, we prove an almost tight bound (up to a constant factor) on the classical communication complexity of two-party common randomness generation assisted with erasednoisy EPR states, generalizing the tight bound on the same task assisted with erased-noisy random strings due to Guruswami and Radhakrishnan.