A conjecture on a tight norm inequality in the finite-dimensional l_p

• URL: http://arxiv.org/abs/2603.24017v1
• Date: Wed, 25 Mar 2026 07:24:55 GMT
• Title: A conjecture on a tight norm inequality in the finite-dimensional l_p
• Abstract summary: We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove.<n>We give a proof for $d=3$ and provide a detailed numerical check for $dleq 200$ confirming the conjecture.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We suggest a tight inequality for norms in $d$-dimensional space $l_p $ which has simple formulation but appears hard to prove. We give a proof for $d=3$ and provide a detailed numerical check for $d\leq 200$ confirming the conjecture. We conclude with a brief survey of solutions for kin problems which anyhow concern minimization of the output entropy of certain quantum channel and rely upon the symmetry properties of the problem. Key words and phrases: $l_p $-norm, Rényi entropy, tight inequality, maximization of a convex function.

Related papers

• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• An Optimal Analysis of the Product Test [42.29010058743949]
  Product testing decides whether a pure multipartite quantum state is fully unentangled across a specified tensor decomposition.<n>Despite being a fundamental property testing task and having many applications, the product test's exact (worst-case) acceptance probability curve has yet to be fully determined.<n>As a complexity-theoretic application, our results improve the one-shot soundness parameter in the Harrow-Montanaro reduction from $mathsfQMA(k)$ to $mathsfQMA(2)$.
  arXiv  Detail & Related papers  (2026-07-23T16:17:44Z)
• Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for $p>3/4$ and $0\leq p<1/4$ [1.9116784879310027]
  Additivity of minimum output entropies is a central problem in quantum information theory.<n>We prove that for every Rényi order $p$ satisfying either $p>3/4$ or $0leq p1/4$, there exist finite-dimensional projection-induced quantum channels.
  arXiv  Detail & Related papers  (2026-07-16T17:08:47Z)
• Resolving the Edge of a Quantum Pyramid [0.0]
  We prove the entropy inequalities of Holevo and Utkin, which certify optimality for obtuse and flat pyramids.<n>For obtuse pyramids, our key contribution is a rigorous proof that local minimizers of the corresponding entropy inequality cannot have three distinct coordinate values.<n>For flat pyramids, we prove a tight $ellp$ inequality for zero-sum vectors that was recently conjectured.
  arXiv  Detail & Related papers  (2026-06-12T17:59:33Z)
• 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)
• The Wehrl-type entropy conjecture for symmetric $SU(N)$ coherent states: cases of equality and stability [0.0]
  We prove that for symmetric $SU(N)$ representations, the corresponding Wehrl-type entropy is minimized by coherent states.<n>A sharp quantitative form of the bound by Lieb and Solovej is also proved.
  arXiv  Detail & Related papers  (2024-12-14T19:33:08Z)
• Physical proof of the topological entanglement entropy inequality [0.0]
  Recently it was shown that the topological entanglement entropy (TEE) of a two-dimensional gapped ground state obeys the universal inequality $gamma geq log mathcalD$. Here we present an alternative, more direct proof of this inequality.
  arXiv  Detail & Related papers  (2024-08-08T17:06:23Z)
• A simple and improved algorithm for noisy, convex, zeroth-order optimisation [59.51990161522328]
  We construct an algorithm that returns a point $hat xin barmathcal X$ such that $f(hat x)$ is as small as possible. We prove that this method is such that the $f(hat x) - min_xin barmathcal X f(x)$ is of smaller order than $d2/sqrtn$ up to poly-logarithmic terms.
  arXiv  Detail & Related papers  (2024-06-26T18:19:10Z)
• Dimension Independent Disentanglers from Unentanglement and Applications [55.86191108738564]
  We construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show that to capture NEXP, it suffices to have unentangled proofs of the form $| psi rangle = sqrta | sqrt1-a | psi_+ rangle where $| psi_+ rangle has non-negative amplitudes.
  arXiv  Detail & Related papers  (2024-02-23T12:22:03Z)
• Dimension-free discretizations of the uniform norm by small product sets [45.85600902330814]
  A classical inequality of Bernstein compares the supremum norm of $f$ over the unit circle to its supremum norm over the sampling set of the $K$-th roots of unity.<n>We show that dimension-free discretizations are possible with sampling sets whose cardinality is independent of $deg(f)$ and is instead governed by the maximum individual degree of $f$.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• Tip of the Quantum Entropy Cone [1.1606619391009658]
  Relations among von Neumann entropies of different parts of an $N$-partite quantum system have direct impact on our understanding of diverse situations. We show that while it is always possible to up-scale an entropy vector to arbitrary integer multiples it is not always possible to down-scale it to arbitrarily small size.
  arXiv  Detail & Related papers  (2023-05-31T21:37:24Z)
• A new operator extension of strong subadditivity of quantum entropy [12.547444644243544]
  Weak monotonicity asserts that $S(rho_AB) - S(rho_A) + S(rho_BC) - S(rho_C)geq 0$ for any tripartite density matrix $rho_ABC$. We prove an operator inequality, which, upon taking an expectation value with respect to the state $rho_ABC$, reduces to the weak monotonicity inequality.
  arXiv  Detail & Related papers  (2022-11-24T01:45:44Z)
• Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel [2.9443230571766845]
  The hypercontractivity inequality for the qubit depolarizing channel $Psi_t$ states that $|Psi_totimes n(X)|_pleq |X|_q$ provided that $pgeq q> 1$ and $tgeq ln sqrtfracp-1q-1$. We first prove an improved quantum logarithmic-Sobolev inequality and then use the well-known equivalence of logarithmic-Sobolev hypercontractivity inequalities to obtain our main result
  arXiv  Detail & Related papers  (2021-05-02T13:04:17Z)
• Locally Private Hypothesis Selection [96.06118559817057]
  We output a distribution from $mathcalQ$ whose total variation distance to $p$ is comparable to the best such distribution. We show that the constraint of local differential privacy incurs an exponential increase in cost. Our algorithms result in exponential improvements on the round complexity of previous methods.
  arXiv  Detail & Related papers  (2020-02-21T18:30:48Z)

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.