7,341 research outputs found
Quantum Capacities of Channels with small Environment
We investigate the quantum capacity of noisy quantum channels which can be represented by coupling a system to an effectively small environment. A capacity formula is derived for all cases where both system and environment are two-dimensional--including all extremal qubit channels. Similarly, for channels acting on higher dimensional systems we show that the capacity can be determined if the channel arises from a sufficiently small coupling to a qubit environment. Extensions to instances of channels with larger environment are provided and it is shown that bounds on the capacity with unconstrained environment can be obtained from decompositions into channels with small environment
Unbounded violations of bipartite Bell Inequalities via Operator Space theory
In this work we show that bipartite quantum states with local Hilbert space
dimension n can violate a Bell inequality by a factor of order (up
to a logarithmic factor) when observables with n possible outcomes are used. A
central tool in the analysis is a close relation between this problem and
operator space theory and, in particular, the very recent noncommutative
embedding theory. As a consequence of this result, we obtain better Hilbert
space dimension witnesses and quantum violations of Bell inequalities with
better resistance to noise
Matrix Product State Representations
This work gives a detailed investigation of matrix product state (MPS)
representations for pure multipartite quantum states. We determine the freedom
in representations with and without translation symmetry, derive respective
canonical forms and provide efficient methods for obtaining them. Results on
frustration free Hamiltonians and the generation of MPS are extended, and the
use of the MPS-representation for classical simulations of quantum systems is
discussed.Comment: Minor changes. To appear in QI
Matrix Product States: Symmetries and Two-Body Hamiltonians
We characterize the conditions under which a translationally invariant matrix
product state (MPS) is invariant under local transformations. This allows us to
relate the symmetry group of a given state to the symmetry group of a simple
tensor. We exploit this result in order to prove and extend a version of the
Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in
the context of MPS. We illustrate the results with an exhaustive search of
SU(2)--invariant two-body Hamiltonians which have such MPS as exact ground
states or excitations.Comment: PDFLatex, 12 pages and 6 figure
Undecidability of the Spectral Gap
We construct families of translationally-invariant, nearest-neighbour
Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an
undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either
have continuous spectrum above the ground state in the thermodynamic limit,
or its spectral gap is lower-bounded by a constant. Moreover, this constant
can be taken equal to the operator norm of the local operator that generates
the Hamiltonian (the local interaction strength). The result still holds true if
one restricts to arbitrarily small quantum perturbations of classical Hamiltonians. The proof combines a robustness analysis of Robinson’s aperiodic
tiling, together with tools from quantum information theory: the quantum
phase estimation algorithm and the history state technique mapping Quantum
Turing Machines to Hamiltonians
The Un(solv)able Problem
After a years-long intellectual journey, three mathematicians have discovered that a problem of central importance in physics is impossible to solve—and that means other big questions may be undecidable, too.
In Brief:
Kurt Gödel famously discovered in the 1930s that some statements are impossible to prove true or false—they will always be “undecidable.”
Mathematicians recently set out to discover whether a certain fundamental problem in quantum physics—the so-called spectral gap question—falls into this category. The spectral gap refers to the energy difference between the lowest energy state a material can occupy and the next state up.
After three years of blackboard brainstorming, midnight calculating and much theorizing over coffee, the mathematicians produced a 146-page proof that the spectral gap problem is, in fact, undecidable. The result raises the possibility that other important questions may likewise be unanswerable
Strings, Projected Entangled Pair States, and variational Monte Carlo methods
We introduce string-bond states, a class of states obtained by placing
strings of operators on a lattice, which encompasses the relevant states in
Quantum Information. For string-bond states, expectation values of local
observables can be computed efficiently using Monte Carlo sampling, making them
suitable for a variational abgorithm which extends DMRG to higher dimensional
and irregular systems. Numerical results demonstrate the applicability of these
states to the simulation of many-body sytems.Comment: 4 pages. v2: Submitted version, containing more numerical data.
Changed title and renamed "string states" to "string-bond states" to comply
with PRL conventions. v3: Accepted version, Journal-Ref. added (title differs
from journal
Fundamental limitations in the purifications of tensor networks
We show a fundamental limitation in the description of quantum many-body
mixed states with tensor networks in purification form. Namely, we show that
there exist mixed states which can be represented as a translationally
invariant (TI) matrix product density operator (MPDO) valid for all system
sizes, but for which there does not exist a TI purification valid for all
system sizes. The proof is based on an undecidable problem and on the
uniqueness of canonical forms of matrix product states. The result also holds
for classical states.Comment: v1: 11 pages, 1 figure. v2: very minor changes. About to appear in
Journal of Mathematical Physic
Assessing non-Markovian dynamics
We investigate what a snapshot of a quantum evolution - a quantum channel
reflecting open system dynamics - reveals about the underlying continuous time
evolution. Remarkably, from such a snapshot, and without imposing additional
assumptions, it can be decided whether or not a channel is consistent with a
time (in)dependent Markovian evolution, for which we provide computable
necessary and sufficient criteria. Based on these, a computable measure of
`Markovianity' is introduced. We discuss how the consistency with Markovian
dynamics can be checked in quantum process tomography. The results also clarify
the geometry of the set of quantum channels with respect to being solutions of
time (in)dependent master equations.Comment: 5 pages, RevTex, 2 figures. (Except from typesetting) version to be
published in the Physical Review Letter
Gapless Hamiltonians for the toric code using the PEPS formalism
We study Hamiltonians which have Kitaev's toric code as a ground state, and
show how to construct a Hamiltonian which shares the ground space of the toric
code, but which has gapless excitations with a continuous spectrum in the
thermodynamic limit. Our construction is based on the framework of Projected
Entangled Pair States (PEPS), and can be applied to a large class of
two-dimensional systems to obtain gapless "uncle Hamiltonians".Comment: 8 pages, 2 figure
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