analiza aiqbench
This research establishes dimension-free approximate tensorization for hypercontractivity and logarithmic-Sobolev constants across a broad class of quantum Markov semigroups, including generalized depolarizing channels. By proving that these stability bounds hold independently of system size and local dimension, the work provides rigorous theoretical foundations essential for analyzing error propagation in large-scale quantum information processing tasks.
Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups
arXiv:2606.17729v3 Announce Type: replace Abstract: 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 $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, 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. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.
Abstract: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 $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, 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. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.
30 pages, 1 figure. Revised with strengthened results and minor corrections
Quantum Physics (quant-ph); Operator Algebras (math.OA)
arXiv:2606.17729 [quant-ph]
(or arXiv:2606.17729v3 [quant-ph] for this version)
Text integral sub licența deschisă a publicației · 181 cuvinte
Subiecte conexe
Dimension-Free Approximate Tensorization of Quantum Hypercontractivity for Qudit Depolarizing Semigroups
arXiv · 17 iun. 2026
Scalable Quantum Machine Learning via Multi-layer Fully-Connected Variational Quantum Circuits
arXiv · 30 sept. 2026
Qudits and Swap Operators Solve Hard Graph Problem
Quantum Zeitgeist · 3 sept. 2026
Qudit stabilizers beyond the free case and the twisted Kitaev model
arXiv · 17 sept. 2026