Finite-size criteria for spectral gaps in DD-dimensional quantum spin systems

We generalize the existing finite-size criteria for spectral gaps of frustration-free spin systems to $D>2$ dimensions. We obtain a local gap threshold of $\frac{3}{n}$, independent of $D$, for nearest-neighbor interactions. The $\frac{1}{n}$ scaling persists for arbitrary finite-range interactions in $\mathbb Z^3$. 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

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 $n$-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 $n$-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

Information-theoretic limitations on approximate quantum cloning and broadcasting

We prove new quantitative limitations on any approximate simultaneous cloning or broadcasting of mixed states. The results are based on information-theoretic (entropic) considerations and generalize the well known no-cloning and no-broadcasting theorems. We also observe and exploit the fact that the universal cloning machine on the symmetric subspace of $n$ qudits and symmetrized partial trace channels are dual to each other. This duality manifests itself both in the algebraic sense of adjointness of quantum channels and in the operational sense that a universal cloning machine can be used as an approximate recovery channel for a symmetrized partial trace channel and vice versa. The duality extends to give control on the performance of generalized UQCMs on subspaces more general than the symmetric subspace. This gives a way to quantify the usefulness of a-priori information in the context of cloning. For example, we can control the performance of an antisymmetric analogue of the UQCM in recovering from the loss of $n-k$ fermionic particles.Comment: 13 pages; new results on approximate cloning between general subspaces, e.g., cloning of fermion