3D local qupit quantum code without string logical operator

Recently Haah introduced a new quantum error correcting code embedded on a cubic lattice. One of the defining properties of this code is the absence of string logical operator. We present new codes with similar properties by relaxing the condition on the local particle dimension. The resulting code is well-defined when the local Hilbert space dimension is prime. These codes can be divided into two different classes: the local stabilizer generators are either symmetric or antisymmetric with respect to the inversion operation. These is a nontrivial correspondence between these two classes. For any symmetric code without string logical operator, there exists a complementary antisymmetric code with the same property and vice versa. We derive a sufficient condition for the absence of string logical operator in terms of the algebraic constraints on the defining parameters of the code. Minimal number of local particle dimension which satisfies the condition is 5. These codes have logarithmic energy barrier for any logical error.Comment: 9 pages, 7 figure

Modulus of convexity for operator convex functions

Given an operator convex function $f(x)$, we obtain an operator-valued lower bound for $cf(x) + (1-c)f(y) - f(cx + (1-c)y)$, $c \in [0,1]$. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.Comment: 5 pages, change of title. The new version shows that the main result of the original paper cannot be extended to convex functions that are not operator convex

Precursor problem and holographic mutual information

The recent proposal of Almheiri et al.http://arxiv.org/abs/1411.7041, together with the Ryu-Takayanagi formula, implies the entanglement wedge hypothesis for certain choices of boundary subregions. This fact is derived in the pure AdS space. A similar conclusion holds in the presence of quantum corrections, but in a more restricted domain of applicability. We also comment on http://arxiv.org/abs/1601.05416 and some similarities and differences with this workComment: 13 pages, 1 figur

Localization from superselection rules in translation invariant systems

We study a translation invariant spin model in a three-dimensional regular lattice, called the cubic code model, in the presence of arbitrary extensive perturbations. Below a critical perturbation strength, we show that most states with finite energy are localized; the overwhelming majority of such states have energy concentrated around a finite number of defects, and remain so for a time that is near-exponential in the distance between the defects. This phenomenon is due to an emergent superselection rule and does not require any disorder. An extensive number of local integrals of motion for these finite energy sectors are identified as well. Our analysis extends more generally to systems with immobile topological excitations.Comment: 7.5+1pages, 2 figure

Quantum codes on Hurwitz surfaces

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2007.Includes bibliographical references (p. 41-43).Ever since the birth of the first quantum error correcting code, many error correcting techniques and formalism has been constructed so far. Among those, generating a quantum code on a locally planar geometry have lead to some interesting classes of codes. Main idea of this thesis stems from Kitaev's Toric code, which was the first surface code, yet it suffered from having a asymptotically vanishing encoding rate. In this paper, we propose a quantum surface code on a more complicated closed surface which has large genus, namely the Hurwitz surface. This code admits a constant encoding rate in the asymptotic limit that the number of genus goes to infinity. However, we give evidence that t/n, where n is the number of qubits and t is the number of correctible errors, converges to 0 asymptotically. This is based on numerically generating many Hurwitz surfaces and observing the corresponding quantum code in the limit that genus number goes to infinity.by Isaac H. Kim.S.B