arXiv subjects

Li Gao

Publications and source records attributed to Li Gao.

At least 91 records · Page 5Linked to original sources

arXiv2022

Error Exponent and Strong Converse for Quantum Soft Covering

How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched Rényi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.

quant-ph↗

arXiv2022

Strong Converse for Privacy Amplification against Quantum Side Information

We establish a one-shot strong converse bound for privacy amplification against quantum side information using trace distance as a security criterion. This strong converse bound implies that in the independent and identical scenario, the trace distance exponentially converges to one in every finite blocklength when the rate of the extracted randomness exceeds the quantum conditional entropy. The established one-shot bound has an application to bounding the information leakage of classical-quantum wiretap channel coding and private communication over quantum channels. That is, the trace distance between Alice and Eavesdropper's joint state and its decoupled state vanishes as the rate of randomness used in hashing exceeds the quantum mutual information. On the other hand, the trace distance converges to one when the rate is below the quantum mutual information, resulting in an exponential strong converse. Our result also leads to an exponential strong converse for entropy accumulation, which complements a recent result by Dupuis [arXiv:2105.05342]. Lastly, our result and its applications apply to the moderate deviation regime. Namely, we characterize the asymptotic behaviors of the trace distances when the associated rates approach the fundamental thresholds with speeds slower than $O(1/\sqrt{n})$.

quant-ph↗

arXiv2022

Rapid thermalization of spin chain commuting Hamiltonians

We prove that spin chains weakly coupled to a large heat bath thermalize rapidly at any temperature for finite-range, translation-invariant commuting Hamiltonians, reaching equilibrium in a time which scales logarithmically with the system size. Our main result is a generalization to the quantum setting of a seminal result of Holley and Stroock for classical spin chains and represents an exponential improvement over bounds based on the non-closure of the spectral gap. From a physical point of view, our result rigorously establishes the absence of dissipative phase transition for Davies evolutions over translation-invariant spin chains. The result also applies in the case of Symmetry Protected Topological phases where the evolution is respecting the symmetry of the phase. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems.

quant-ph↗

arXiv2021

Entropy decay for Davies semigroups of a one dimensional quantum lattice

Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.

quant-ph↗

arXiv2021

Recoverability for optimized quantum $f$-divergences

The optimized quantum $f$-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized $f$-divergence. In particular, the refinements state that the absolute difference between the optimized $f$-divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel. Not only do these results lead to physically meaningful refinements of the data-processing inequality for the sandwiched Rényi relative entropy, but they also have implications for perfect reversibility (i.e., quantum sufficiency) of the optimized $f$-divergences. Along the way, we improve upon previous physically meaningful refinements of the data-processing inequality for the standard $f$-divergence, as established in recent work of Carlen and Vershynina [arXiv:1710.02409, arXiv:1710.08080]. Finally, we extend the definition of the optimized $f$-divergence, its data-processing inequality, and all of our recoverability results to the general von Neumann algebraic setting, so that all of our results can be employed in physical settings beyond those confined to the most common finite-dimensional setting of interest in quantum information theory.

quant-ph↗

arXiv2021

Ricci curvature of quantum channels on non-commutative transportation metric spaces

Following Ollivier's work, we introduce the coarse Ricci curvature of a quantum channel as the contraction of non-commutative metrics on the state space. These metrics are defined as a non-commutative transportation cost in the spirit of [N. Gozlan and C. Léonard. 2006], which gives a unified approach to different quantum Wasserstein distances in the literature. We prove that the coarse Ricci curvature lower bound and its dual gradient estimate, under suitable assumptions, imply the Poincaré inequality (spectral gap) as well as transportation cost inequalities. Using intertwining relations, we obtain positive bounds on the coarse Ricci curvature of Gibbs samplers, Bosonic and Fermionic beam-splitters as well as Pauli channels on n-qubits.

math-ph↗

arXiv2021

DSP: Dual Soft-Paste for Unsupervised Domain Adaptive Semantic Segmentation

Unsupervised domain adaptation (UDA) for semantic segmentation aims to adapt a segmentation model trained on the labeled source domain to the unlabeled target domain. Existing methods try to learn domain invariant features while suffering from large domain gaps that make it difficult to correctly align discrepant features, especially in the initial training phase. To address this issue, we propose a novel Dual Soft-Paste (DSP) method in this paper. Specifically, DSP selects some classes from a source domain image using a long-tail class first sampling strategy and softly pastes the corresponding image patch on both the source and target training images with a fusion weight. Technically, we adopt the mean teacher framework for domain adaptation, where the pasted source and target images go through the student network while the original target image goes through the teacher network. Output-level alignment is carried out by aligning the probability maps of the target fused image from both networks using a weighted cross-entropy loss. In addition, feature-level alignment is carried out by aligning the feature maps of the source and target images from student network using a weighted maximum mean discrepancy loss. DSP facilitates the model learning domain-invariant features from the intermediate domains, leading to faster convergence and better performance. Experiments on two challenging benchmarks demonstrate the superiority of DSP over state-of-the-art methods. Code is available at \url{https://github.com/GaoLii/DSP}.

cs.CV↗

arXiv2021

Complete entropic inequalities for quantum Markov chains

We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All the three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or $L_2$-estimates. As applications, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive bounds for examples of both theoretical and practical relevance, including representation of sub-Laplacians on $\operatorname{SU}(2)$ and various classes of local quantum Markov semigroups such as quantum Kac generators and continuous time approximate unitary designs. For the latter, our bounds imply the existence of local continuous time Markovian evolutions on $nk$ qudits forming $ε$-approximate $k$-designs in relative entropy for times scaling as $\widetilde{\mathcal{O}}(n^2 \operatorname{poly}(k))$.

quant-ph↗

arXiv2021

Geometric Approach Towards Complete Logarithmic Sobolev Inequalities

In this paper, we use the Carnot-Caratheodory distance from sub-Riemanian geometry to prove entropy decay estimates for all finite dimensional symmetric quantum Markov semigroups. This estimate is independent of the environment size and hence stable under tensorization. Our approach relies on the transference principle, the existence of $t$-designs, and the sub-Riemannian diameter of compact Lie groups and implies estimates for the spectral gap.

quant-ph↗

arXiv2020

Complete Logarithmic Sobolev inequalities via Ricci curvature bounded below

We prove that for a symmetric Markov semigroup, Ricci curvature bounded from below by a non-positive constant combined with a finite $L_\infty$-mixing time implies the modified log-Sobolev inequality. Such $L_\infty$-mixing time estimates always hold for Markov semigroups that have spectral gap and finite Varopoulos dimension. Our results apply to non-ergodic quantum Markov semigroups with noncommutative Ricci curvature bounds recently introduced by Carlen and Maas. As an application, we prove that the heat semigroup on a compact Riemannian manifold admits a uniform modified log-Sobolev inequality for all its matrix-valued extensions.

math.OA↗

arXiv2020

Complete Logarithmic Sobolev Inequalities via Ricci Curvature Bounded Below II

Using a non-negative curvature condition, we prove the complete version of modified log-Sobolev inequalities for central Markov semigroups on various compact quantum groups, including group von Neumann algebras, free orthogonal group and quantum automorphism groups. We also prove that the "geometric Ricci curvature lower bound" introduced by Junge-Li-LaRacuente is stable under tensor products and amalgamated free products. As an application, we obtain the geometric Ricci curvature lower bound and complete modified logarithmic Sobolev inequality for word-length semigroups on free group factors and amalgamated free product algebras.

math.OA↗

arXiv2019

Quantum Majorization on Semifinite von Neumann Algebras

We extend Gour et al's characterization of quantum majorization via conditional min-entropy to the context of semifinite von Neumann algebras. Our method relies on a connection between conditional min-entropy and operator space projective tensor norm for injective von Neumann algebras. This approach also connects the tracial Hahn-Banach theorem of Helton, Klep and McCullough to noncommutative vector-valued $L_1$-space.

math.OA↗

arXiv2019

Relative entropy for von Neumann subalgebras

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.

math.OA↗

arXiv2019

Quantum Euclidean Spaces with Noncommutative Derivatives

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semi-finite spectral triple with non-flat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a simplified local index formula (even case) that is similar to the commutative setting.

math.OA↗

arXiv2019

Unifying Entanglement with Uncertainty via Symmetries of Observable Algebras

Strong subadditivity goes beyond the tensored subsystem and commuting operator models. As previously noted by Petz and later by Araki and Moriya, two subalgebras of observables satisfy a generalized SSA-like inequality if they form a commuting square. We explore the interpretation and consequences in finite dimensions, connecting various entropic uncertainty relations for mutually unbiased bases with the positivity of a generalized conditional mutual information (CMI), and with inequalities on relative entropies of coherence and asymmetry. We obtain a bipartite resource theory of operations under which the two subalgebras are respectively invariant and covariant, with CMI as a monotone, and generalized non-classical monotones based on squashed entanglement and entanglement of formation. Free transformations support conversion between entanglement and uncertainty-based configurations, as "EPR <-> 2UCR." Our theory quantifies the common non-classicality in entanglement and uncertainty, implying a strong conceptual link between these fundamentally quantum phenomena.

quant-ph↗

arXiv2018

Fisher Information and Logarithmic Sobolev Inequality for Matrix Valued Functions

We prove a version of Talagrand's concentration inequality for subordinated sub-Laplacian on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.

math.FA↗

arXiv2018

A New Uzawa-exact Type Algorithm for Nonsymmetric Saddle Point Problems

Saddle point problems have been attracting people's attention in recent years. To solve large and sparse saddle point problems, Uzawa type algorithms were proposed. The main contribution of this paper is to present a new Uzawa-exact type algorithm from the aspect of optimization method to solve nonsymmetric saddle point problems, which often arise from linear variational inequalities and finite element discretization of Navier-Stokes equations. In the paper, convergence of the new algorithm is analysed and numerical experiments are presented.

math.OC↗

arXiv2018

Quantum Teleportation and Super-dense Coding in Operator Algebras

Let $\mathcal{B}_d$ be the unital $C^*$-algebra generated by the elements $u_{jk}, \, 0 \le i, j \le d-1$, satisfying the relations that $[u_{j,k}]$ is a unitary operator, and let $C^*(\mathbb{F}_{d^2})$ be the full group $C^*$-algebra of free group of $d^2$ generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two $*$-isomorphisms $M_d(C^*(\mathbb{F}_{d^2}))\cong \mathcal{B}_d\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$ and $M_d(\mathcal{B}_d)\cong C^*(\mathbb{F}_{d^2})\rtimes \mathbb{Z}_d\rtimes \mathbb{Z}_d$, for certain actions of $\mathbb{Z}_d$. As an application, we show that for any $d,m\ge 2$ with $(d,m)\neq (2,2)$, the matrix-valued generalization of the (tensor product) quantum correlation set of $d$ inputs and $m$ outputs is not closed.

math.OA↗