1,326 research outputs found
On multivariate trace inequalities of Sutter, Berta and Tomamichel
We consider a family of multivariate trace inequalities recently derived by
Sutter, Berta and Tomamichel. These inequalities generalize the Golden-Thompson
inequality and Lieb's three-matrix inequality to an arbitrary number of
matrices in a way that features complex matrix powers. We show that their
inequalities can be rewritten as an -matrix generalization of Lieb's
original three-matrix inequality. The complex matrix powers are replaced by
resolvents and appropriate maximally entangled states. We expect that the
technically advantageous properties of resolvents, in particular for
perturbation theory, can be of use in applications of the -matrix
inequalities, e.g., for analyzing the rotated Petz recovery map in quantum
information theory.Comment: 14 pages; comments welcom
New Counterexamples for Sums-Differences
We present new counterexamples, which provide stronger limitations to
sums-differences statements than were previously known. The main idea is to
consider non-uniform probability measures.Comment: 5 pages, to appear in Proc. Amer. Math. So
Finite-size criteria for spectral gaps in -dimensional quantum spin systems
We generalize the existing finite-size criteria for spectral gaps of
frustration-free spin systems to dimensions. We obtain a local gap
threshold of , independent of , for nearest-neighbor
interactions. The scaling persists for arbitrary finite-range
interactions in . The key observation is that there is more
flexibility in Knabe's combinatorial approach if one employs the operator
Cauchy-Schwarz inequality.Comment: 16 page
Gaplessness is not generic for translation-invariant spin chains
The existence of a spectral gap above the ground state has far-reaching
consequences for the low-energy physics of a quantum many-body system. A recent
work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially
random local quantum Hamiltonian is generically gapless. Here we observe that a
gap is more common for translation-invariant quantum spin chains, more
specifically, that these are gapped with a positive probability if the
interaction is of small rank. This is in line with a previous analysis of the
spin- case by Bravyi and Gosset. The Hamiltonians are constructed by
selecting a single projection of sufficiently small rank at random, and then
translating it across the entire chain. By the rank assumption, the resulting
Hamiltonians are automatically frustration-free and this fact plays a key role
in our analysis.Comment: 17 pages; minor changes; comments welcom
Gapped PVBS models for all species numbers and dimensions
Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ
spin model and feature particle species on . The PVBS models
were originally introduced as toy models for the classification of ground state
phases. A crucial ingredient for this classification is the existence of a
spectral gap above the ground state sector. In this work, we derive a spectral
gap for PVBS models at arbitrary species number and in arbitrary dimension
in the perturbative regime of small anisotropy parameters. Instead of using
the more common martingale method, the proof verifies a finite-size criterion
in the spirit of Knabe.Comment: 22 pages; revised version to appear in Rev. Math. Phy
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