11,748 research outputs found
Multi-Error-Correcting Amplitude Damping Codes
We construct new families of multi-error-correcting quantum codes for the
amplitude damping channel. Our key observation is that, with proper encoding,
two uses of the amplitude damping channel simulate a quantum erasure channel.
This allows us to use concatenated codes with quantum erasure-correcting codes
as outer codes for correcting multiple amplitude damping errors. Our new codes
are degenerate stabilizer codes and have parameters which are better than the
amplitude damping codes obtained by any previously known construction.Comment: 5 pages. Submitted to ISIT 201
Note on disjoint faces in simple topological graphs
We prove that every -vertex complete simple topological graph generates at
least pairwise disjoint -faces. This improves upon a recent
result by Hubard and Suk. As an immediate corollary, every -vertex complete
simple topological graph drawn in the unit square generates a -face with
area at most . This can be seen as a topological variant of Heilbronn's
problem for -faces. We construct examples showing that our result is
asymptotically tight. We also discuss the similar problem for -faces with
arbitrary .Comment: fixed some gap
Quantum Capacities for Entanglement Networks
We discuss quantum capacities for two types of entanglement networks:
for the quantum repeater network with free classical
communication, and for the tensor network as the rank of the
linear operation represented by the tensor network. We find that
always equals in the regularized case for the samenetwork graph.
However, the relationships between the corresponding one-shot capacities
and are more complicated, and the min-cut upper
bound is in general not achievable. We show that the tensor network can be
viewed as a stochastic protocol with the quantum repeater network, such that
is a natural upper bound of . We analyze the
possible gap between and for certain networks,
and compare them with the one-shot classical capacity of the corresponding
classical network
Symmetric Extension of Two-Qubit States
Quantum key distribution uses public discussion protocols to establish shared
secret keys. In the exploration of ultimate limits to such protocols, the
property of symmetric extendibility of underlying bipartite states
plays an important role. A bipartite state is symmetric extendible
if there exits a tripartite state , such that the marginal
state is identical to the marginal state, i.e. .
For a symmetric extendible state , the first task of the public
discussion protocol is to break this symmetric extendibility. Therefore to
characterize all bi-partite quantum states that possess symmetric extensions is
of vital importance. We prove a simple analytical formula that a two-qubit
state admits a symmetric extension if and only if
\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}. Given the
intimate relationship between the symmetric extension problem and the quantum
marginal problem, our result also provides the first analytical necessary and
sufficient condition for the quantum marginal problem with overlapping
marginals.Comment: 10 pages, no figure. comments are welcome. Version 2: introduction
rewritte
Minimum Entangling Power is Close to Its Maximum
Given a quantum gate acting on a bipartite quantum system, its maximum
(average, minimum) entangling power is the maximum (average, minimum)
entanglement generation with respect to certain entanglement measure when the
inputs are restricted to be product states. In this paper, we mainly focus on
the 'weakest' one, i.e., the minimum entangling power, among all these
entangling powers. We show that, by choosing von Neumann entropy of reduced
density operator or Schmidt rank as entanglement measure, even the 'weakest'
entangling power is generically very close to its maximal possible entanglement
generation. In other words, maximum, average and minimum entangling powers are
generically close. We then study minimum entangling power with respect to other
Lipschitiz-continuous entanglement measures and generalize our results to
multipartite quantum systems.
As a straightforward application, a random quantum gate will almost surely be
an intrinsically fault-tolerant entangling device that will always transform
every low-entangled state to near-maximally entangled state.Comment: 26 pages, subsection III.A.2 revised, authors list updated, comments
are welcom
Quantum state reduction for universal measurement based computation
Measurement based quantum computation (MBQC), which requires only single
particle measurements on a universal resource state to achieve the full power
of quantum computing, has been recognized as one of the most promising models
for the physical realization of quantum computers. Despite considerable
progress in the last decade, it remains a great challenge to search for new
universal resource states with naturally occurring Hamiltonians, and to better
understand the entanglement structure of these kinds of states. Here we show
that most of the resource states currently known can be reduced to the cluster
state, the first known universal resource state, via adaptive local
measurements at a constant cost. This new quantum state reduction scheme
provides simpler proofs of universality of resource states and opens up plenty
of space to the search of new resource states, including an example based on
the one-parameter deformation of the AKLT state studied in [Commun. Math. Phys.
144, 443 (1992)] by M. Fannes et al. about twenty years ago.Comment: 5 page
Forest formulas of discrete Green's functions
The discrete Green's functions are the pseudoinverse (or the inverse) of the
Laplacian (or its variations) of a graph. In this paper, we will give
combinatorial interpretations of Green's functions in terms of enumerating
trees and forests in a graph that will be used to derive further formulas for
several graph invariants. For example, we show that the trace of the Green's
function associated with the combinatorial Laplacian of a
connected simple graph on vertices satisfies
, where denotes the eigenvalues of
the combinatorial Laplacian, denotes the number of spanning trees and
denotes the set of rooted spanning -forests in . We
will prove forest formulas for discrete Green's functions for directed and
weighted graphs and apply them to study random walks on graphs and digraphs. We
derive a forest expression of the hitting time for digraphs, which gives
combinatorial proofs to old and new results about hitting times, traces of
discrete Green's functions, and other related quantities.Comment: minor changes and fixed typo
On Higher Dimensional Point Sets in General Position
A finite point set in ?^d is in general position if no d + 1 points lie on a common hyperplane. Let ?_d(N) be the largest integer such that any set of N points in ?^d with no d + 2 members on a common hyperplane, contains a subset of size ?_d(N) in general position. Using the method of hypergraph containers, Balogh and Solymosi showed that ??(N) < N^{5/6 + o(1)}. In this paper, we also use the container method to obtain new upper bounds for ?_d(N) when d ? 3. More precisely, we show that if d is odd, then ?_d(N) < N^{1/2 + 1/(2d) + o(1)}, and if d is even, we have ?_d(N) < N^{1/2 + 1/(d-1) + o(1)}.
We also study the classical problem of determining the maximum number a(d,k,n) of points selected from the grid [n]^d such that no k + 2 members lie on a k-flat. For fixed d and k, we show that a(d,k,n)? O(n^{d/{2?(k+2)/4?}(1- 1/{2?(k+2)/4?d+1})}), which improves the previously best known bound of O(n^{d/?(k + 2)/2?}) due to Lefmann when k+2 is congruent to 0 or 1 mod 4
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