Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups

• URL: http://arxiv.org/abs/2606.17729v2
• Date: Thu, 18 Jun 2026 14:36:54 GMT
• Title: Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups
• Abstract summary: We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of reversible Markov semigroups.<n>The proof first establishes an exact tensorization of the ((q,2))-hypercontractive inequality for integer (q), in particular (q), and then extends the estimate to all real (q>2) by complex.
• Score: 20.643914876632866
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of 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 \((Φ_t)_{t\ge 0}\) and every tensor power \(n\), we show that the log-Sobolev constant of the product semigroup \(Φ_t^{\otimes n}\) is at least \(2/(3\ln 2)\approx 0.96\) times the log-Sobolev constant of the single-site semigroup \(Φ_t\), independently of \(n\) and the local dimension \(d\). The proof first establishes an exact tensorization of the \((q,2)\)-hypercontractive inequality for integer \(q\), in particular \(q=3\), and then extends the estimate to all real \(q>2\) by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. As an application of the same method, we also obtain sharp \((q,2)\)-hypercontractivity estimates for qubit depolarizing channels.

Related papers

• Relative Entropy Decay via BKM coercivity for Quantum Markov Semigroups [12.387975598917158]
  We show that the positivity of the BKM coercivity constant governs the exponential relative entropy decay of the semigroup.<n>We prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the conditional expectation.
  arXiv  Detail & Related papers  (2026-09-14T17:19:43Z)
• Separable Counterexamples to Complementary Quantum Correlations, and Why Random Search Missed Them [0.0]
  We show that CQC relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements.<n>We also give a state-dependent corrected inequality that is universal, and show it is with CQC already at (2times2)
  arXiv  Detail & Related papers  (2026-08-14T18:21:29Z)
• Maximal Classicalization of Finite-Group Quantum Reference-Frame Noise [0.0]
  A finite quantum reference token with group-valued misalignment induces a random-unitary channel.<n>We prove that the following conditions are equivalent for a unitary representation U of a finite group G.
  arXiv  Detail & Related papers  (2026-07-14T17:07:57Z)
• Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization [33.64818873746132]
  We present a rigorous theoretical analysis of TV regularization for tensor completion under exponential-family noise.<n>We propose a family of dual-TV (DTV) regularizers based on the transformed L1 function, which simultaneously capture sparsity and low-rank structures in the gradient tensor.
  arXiv  Detail & Related papers  (2026-06-29T22:32:19Z)
• Optimal Dimension-Free Sampling for Regularized Classification [56.72526267755301]
  We prove optimal sampling bounds achieving $(1pmvarepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions.<n>This includes important functions such as logistic and sigmoid loss, hinge loss, and ReLU loss, as prominent and popular representative examples.
  arXiv  Detail & Related papers  (2026-05-22T15:05:33Z)
• The Collapse of Unentangled Stoquastic Merlin-Arthur Proof Systems [46.657699628769336]
  Entanglement and interference are among the most fundamental properties of quantum mechanics.<n>We prove that entanglement gives no additional power to stoquastic verification.
  arXiv  Detail & Related papers  (2026-05-15T17:52:33Z)
• Two-Indexed Schatten Quasi-Norms with Applications to Quantum Information Theory [9.048611509540079]
  We define 2-indexed $(q,p)$-Schatten quasi-norms for any $q,p > 0$ on operators on a tensor product of Hilbert spaces.<n>For linear maps between spaces of such quasi-norms, we introduce completely bounded quasi-norms and co-quasi-norms.<n>Our results imply a tensorizing notion of reverse hypercontractivity, additivity of the completely bounded minimum output Rényi-$$-entropy for $geqfrac12$ or the Sandwiched Rényi Umlaut information
  arXiv  Detail & Related papers  (2026-04-15T16:35:46Z)
• Typical entanglement in anyon chains: Page curves beyond Lie group symmetries [41.99844472131922]
  We study bipartite entanglement statistics in one-dimensional anyon chains constrained by fusion rules of unitary pre-modular categories.<n>Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(sqrtL)$ or $O(1)$ symmetry-type corrections.<n> Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions.
  arXiv  Detail & Related papers  (2026-03-26T18:00:03Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Transmutation based Quantum Simulation for Non-unitary Dynamics [35.35971148847751]
  We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=Ldagger L$.<n>Our main tool is the Kannai transform, which represents the diffusion semigroup $e-TA$ as a Gaussian-weighted superposition of unitary wave propagators.
  arXiv  Detail & Related papers  (2026-01-07T05:47:22Z)
• Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes [5.00389879175348]
  We estimate the individual $beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $mathcal O(log(n) n-1/2)$.
  arXiv  Detail & Related papers  (2024-02-11T20:17:10Z)
• Sample Complexity Bounds for Estimating Probability Divergences under Invariances [31.946304450935628]
  Group-invariant probability distributions appear in many data-generative models in machine learning. In this work, we study how the inherent invariances, with respect to any smooth action of a Lie group on a manifold, improve sample complexity. Results are completely new for groups of positive dimension and extend recent bounds for finite group actions.
  arXiv  Detail & Related papers  (2023-11-06T04:45:21Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Inequivalent $Z_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics [0.0]
  Inequivalent brackets of Lie-type which are compatible with the grading and satisfy graded Jacobi identities is $b_n= n+lfloor n/2rfloor+1$. The inequivalent brackets, recovered from $Zntimes Znrightarrow Z$ mappings, are defined by consistent sets of commutators/anticommutators describing particles accommodated into an $n$-bit para.
  arXiv  Detail & Related papers  (2023-09-02T15:18:31Z)
• Relative entropy decay and complete positivity mixing time [11.225649178057697]
  We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. Our results apply to GNS-symmetric semigroups on general von Neumanns.
  arXiv  Detail & Related papers  (2022-09-22T17:40:22Z)
• Classical shadows of fermions with particle number symmetry [0.0]
  We provide an estimator for any $k$-RDM with $mathcalO(k2eta)$ classical complexity. Our method, in the worst-case of half-filling, still provides a factor of $4k$ advantage in sample complexity.
  arXiv  Detail & Related papers  (2022-08-18T17:11:12Z)
• Monogamy of entanglement between cones [43.57338639836868]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones.<n>Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Stochastic behavior of outcome of Schur-Weyl duality measurement [45.41082277680607]
  We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. We derive various types of distribution including a kind of central limit when $n$ goes to infinity.
  arXiv  Detail & Related papers  (2021-04-26T15:03:08Z)
• Complete entropic inequalities for quantum Markov chains [17.21921346541951]
  We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional algebra satisfies a modified log-Sobolev inequality. We also establish the first general approximateization property of relative entropy.
  arXiv  Detail & Related papers  (2021-02-08T11:47:37Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.