Area law for the maximally mixed ground state in degenerate 1D gapped systems

• URL: http://arxiv.org/abs/2310.19028v1
• Date: Sun, 29 Oct 2023 14:36:30 GMT
• Title: Area law for the maximally mixed ground state in degenerate 1D gapped systems
• Authors: Itai Arad, Raz Firanko, Rahul Jain
• Abstract summary: We show an area law with logarithmic correction for the maximally mixed state $Omega$ in the (degenerate) ground space of a 1D gapped local Hamiltonian $H$. We also get an area law for the mutual information of the form $mathrmI(L:R)_Omega leq O(log |L|)$.
• Score: 5.088702935364181
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show an area law with logarithmic correction for the maximally mixed state $\Omega$ in the (degenerate) ground space of a 1D gapped local Hamiltonian $H$, which is independent of the underlying ground space degeneracy. Formally, for $\varepsilon>0$ and a bi-partition $L\cup L^c$ of the 1D lattice, we show that $$\mathrm{I}^{\varepsilon}_{\max}(L:L^c)_{\Omega} \leq O(\log(|L|)+\log(1/\varepsilon)),$$ where $|L|$ represents the number of qudits in $L$ and $\mathrm{I}^{\epsilon}_{\max}(L:L^c)_{\Omega}$ represents the $\varepsilon$- 'smoothed maximum mutual information' with respect to the $L:L^c$ partition in $\Omega$. As a corollary, we get an area law for the mutual information of the form $\mathrm{I}(L:R)_\Omega \leq O(\log |L|)$. In addition, we show that $\Omega$ can be approximated up to an $\varepsilon$ in trace norm with a state of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• $\ell_p$-Regression in the Arbitrary Partition Model of Communication [59.89387020011663]
  We consider the randomized communication complexity of the distributed $ell_p$-regression problem in the coordinator model. For $p = 2$, i.e., least squares regression, we give the first optimal bound of $tildeTheta(sd2 + sd/epsilon)$ bits. For $p in (1,2)$,we obtain an $tildeO(sd2/epsilon + sd/mathrmpoly(epsilon)$ upper bound.
  arXiv  Detail & Related papers  (2023-07-11T08:51:53Z)
• $\boldsymbol{\alpha_{>}(\epsilon) = \alpha_{<}(\epsilon)}$ For The Margolus-Levitin Quantum Speed Limit Bound [0.0]
  I show that $alpha_>(epsilon)$ is indeed equal to $alpha_(epsilon)$. I also point out a numerical stability issue in computing $alpha_>(epsilon)$.
  arXiv  Detail & Related papers  (2023-05-17T10:07:31Z)
• Krylov Methods are (nearly) Optimal for Low-Rank Approximation [8.017116107657206]
  We show that any algorithm requires $Omegaleft(log(n)/varepsilon1/2right)$ matrix-vector products, exactly matching the upper bound obtained by Krylov methods. Our lower bound addresses Open Question 1WooWoo14, providing evidence for the lack of progress on algorithms for Spectral LRA.
  arXiv  Detail & Related papers  (2023-04-06T16:15:19Z)
• Sparse Dimensionality Reduction Revisited [13.170012290527017]
  The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. We revisit sparse embeddings and identify a loophole in the lower bound. We also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.
  arXiv  Detail & Related papers  (2023-02-13T08:01:25Z)
• Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products [58.05771390012827]
  We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Our main result is an algorithm that uses only $tildeO(k/sqrtepsilon)$ matrix-vector products.
  arXiv  Detail & Related papers  (2022-02-10T16:10:41Z)
• Learning low-degree functions from a logarithmic number of random queries [77.34726150561087]
  We prove that for any integer $ninmathbbN$, $din1,ldots,n$ and any $varepsilon,deltain(0,1)$, a bounded function $f:-1,1nto[-1,1]$ of degree at most $d$ can be learned.
  arXiv  Detail & Related papers  (2021-09-21T13:19:04Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Approximate Maximum Halfspace Discrepancy [6.35821487778241]
  We consider the range space $(X, mathcalH_d)$ where $X subset mathbbRd$ and $mathcalH_d$ is the set of ranges defined by $d$ halfspaces. For each halfspace $h in mathcalH_d$ define a function $Phi(h)$ that measures the "difference" between the fraction of red and fraction of blue points which fall in the range $h$.
  arXiv  Detail & Related papers  (2021-06-25T19:14:45Z)
• Infinite-Horizon Offline Reinforcement Learning with Linear Function Approximation: Curse of Dimensionality and Algorithm [46.36534144138337]
  In this paper, we investigate the sample complexity of policy evaluation in offline reinforcement learning. Under the low distribution shift assumption, we show that there is an algorithm that needs at most $Oleft(maxleft fracleftVert thetapirightVert _24varepsilon4logfracddelta,frac1varepsilon2left(d+logfrac1deltaright)right right)$ samples to approximate the
  arXiv  Detail & Related papers  (2021-03-17T18:18:57Z)
• Average Case Column Subset Selection for Entrywise $\ell_1$-Norm Loss [76.02734481158458]
  It is known that in the worst case, to obtain a good rank-$k$ approximation to a matrix, one needs an arbitrarily large $nOmega(1)$ number of columns. We show that under certain minimal and realistic distributional settings, it is possible to obtain a $(k/epsilon)$-approximation with a nearly linear running time and poly$(k/epsilon)+O(klog n)$ columns. This is the first algorithm of any kind for achieving a $(k/epsilon)$-approximation for entrywise
  arXiv  Detail & Related papers  (2020-04-16T22:57:06Z)

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.