Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings

• URL: http://arxiv.org/abs/2009.11866v2
• Date: Thu, 1 Apr 2021 03:22:57 GMT
• Title: Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
• Authors: Marius Junge and Nicholas LaRacuente
• Abstract summary: We show that the physics of quantum field theory and holography motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras.
• Score: 4.56877715768796
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of univeral recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that non-decrease of relative entropy is equivalent to existence of an $L_1$-isometry implementing the channel on both input states.

Related papers

• Mappings preserving quantum Renyi's entropies in von Neumann algebras [0.0]
  A normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
  arXiv  Detail & Related papers  (2023-02-05T02:41:39Z)
• Correspondence between open bosonic systems and stochastic differential equations [77.34726150561087]
  We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
  arXiv  Detail & Related papers  (2023-02-03T19:17:37Z)
• Integral formula for quantum relative entropy implies data processing inequality [0.0]
  We prove the monotonicity of quantum relative entropy under trace-preserving positive linear maps. For a simple application of such monotonicities, we consider any divergence' that is non-increasing under quantum measurements. An argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a divergence' on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states.
  arXiv  Detail & Related papers  (2022-08-25T16:32:02Z)
• Improved Quantum Algorithms for Fidelity Estimation [77.34726150561087]
  We develop new and efficient quantum algorithms for fidelity estimation with provable performance guarantees. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation. We prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general.
  arXiv  Detail & Related papers  (2022-03-30T02:02:16Z)
• Average-case Speedup for Product Formulas [69.68937033275746]
  Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states. Our results open doors to the study of quantum algorithms in the average case.
  arXiv  Detail & Related papers  (2021-11-09T18:49:48Z)
• Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields [53.31927549039624]
  We show that a piecewise discretization preserves better contrast from existing discretization problems. We apply this theory to the problem of matching two images.
  arXiv  Detail & Related papers  (2021-07-13T12:31:06Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• Trace- and improved data processing inequalities for von Neumann algebras [0.0]
  We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III.
  arXiv  Detail & Related papers  (2021-02-15T11:36:44Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Approximate recoverability and relative entropy II: 2-positive channels of general v. Neumann algebras [0.0]
  We prove the approximate recoverability of states which undergo a small change in relative entropy through a quantum channel. We derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two Neumann algebras.
  arXiv  Detail & Related papers  (2020-10-12T08:09:17Z)
• Approximate tensorization of the relative entropy for noncommuting conditional expectations [3.4376560669160385]
  We derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto finite-dimensional von Neumann algebras. We show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
  arXiv  Detail & Related papers  (2020-01-22T12:20:53Z)

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.