6,677 research outputs found
The complexity of antiferromagnetic interactions and 2D lattices
Estimation of the minimum eigenvalue of a quantum Hamiltonian can be
formalised as the Local Hamiltonian problem. We study the natural special case
of the Local Hamiltonian problem where the same 2-local interaction, with
differing weights, is applied across each pair of qubits. First we consider
antiferromagnetic/ferromagnetic interactions, where the weights of the terms in
the Hamiltonian are restricted to all be of the same sign. We show that for
symmetric 2-local interactions with no 1-local part, the problem is either
QMA-complete or in StoqMA. In particular the antiferromagnetic Heisenberg and
antiferromagnetic XY interactions are shown to be QMA-complete. We also prove
StoqMA-completeness of the antiferromagnetic transverse field Ising model.
Second, we study the Local Hamiltonian problem under the restriction that the
interaction terms can only be chosen to lie on a particular graph. We prove
that nearly all of the QMA-complete 2-local interactions remain QMA-complete
when restricted to a 2D square lattice. Finally we consider both restrictions
at the same time and discover that, with the exception of the antiferromagnetic
Heisenberg interaction, all of the interactions which are QMA-complete with
positive coefficients remain QMA-complete when restricted to a 2D triangular
lattice.Comment: 35 pages, 11 figures; v2 added reference
Characterizing common cause closedness of quantum probability theories
We prove new results on common cause closedness of quantum probability
spaces, where by a quantum probability space is meant the projection lattice of
a non-commutative von Neumann algebra together with a countably additive
probability measure on the lattice. Common cause closedness is the feature that
for every correlation between a pair of commuting projections there exists in
the lattice a third projection commuting with both of the correlated
projections and which is a Reichenbachian common cause of the correlation. The
main result we prove is that a quantum probability space is common cause closed
if and only if it has at most one measure theoretic atom. This result improves
earlier ones published in Z. GyenisZ and M. Redei Erkenntnis 79 (2014) 435-451.
The result is discussed from the perspective of status of the Common Cause
Principle. Open problems on common cause closedness of general probability
spaces are formulated, where is an
orthomodular bounded lattice and is a probability measure on
.Comment: Submitted for publicatio
Entanglement and Density Matrix of a Block of Spins in AKLT Model
We study a 1-dimensional AKLT spin chain, consisting of spins in the bulk
and at both ends. The unique ground state of this AKLT model is described
by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a
contiguous block of bulk spins in this ground state. It is shown that the
density matrix is a projector onto a subspace of dimension . This
subspace is described by non-zero eigenvalues and corresponding eigenvectors of
the density matrix. We prove that for large block the von Neumann entropy
coincides with Renyi entropy and is equal to .Comment: Revised version, typos corrected, references added, 31 page
Efficient and feasible state tomography of quantum many-body systems
We present a novel method to perform quantum state tomography for
many-particle systems which are particularly suitable for estimating states in
lattice systems such as of ultra-cold atoms in optical lattices. We show that
the need for measuring a tomographically complete set of observables can be
overcome by letting the state evolve under some suitably chosen random circuits
followed by the measurement of a single observable. We generalize known results
about the approximation of unitary 2-designs, i.e., certain classes of random
unitary matrices, by random quantum circuits and connect our findings to the
theory of quantum compressed sensing. We show that for ultra-cold atoms in
optical lattices established techniques like optical super-lattices, laser
speckles, and time-of-flight measurements are sufficient to perform fully
certified, assumption-free tomography. Combining our approach with tensor
network methods - in particular the theory of matrix-product states - we
identify situations where the effort of reconstruction is even constant in the
number of lattice sites, allowing in principle to perform tomography on
large-scale systems readily available in present experiments.Comment: 10 pages, 3 figures, minor corrections, discussion added, emphasizing
that no single-site addressing is needed at any stage of the scheme when
implemented in optical lattice system
Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices
We investigate the relationship between the gap between the energy of the
ground state and the first excited state and the decay of correlation functions
in harmonic lattice systems. We prove that in gapped systems, the exponential
decay of correlations follows for both the ground state and thermal states.
Considering the converse direction, we show that an energy gap can follow from
algebraic decay and always does for exponential decay. The underlying lattices
are described as general graphs of not necessarily integer dimension, including
translationally invariant instances of cubic lattices as special cases. Any
local quadratic couplings in position and momentum coordinates are allowed for,
leading to quasi-free (Gaussian) ground states. We make use of methods of
deriving bounds to matrix functions of banded matrices corresponding to local
interactions on general graphs. Finally, we give an explicit entanglement-area
relationship in terms of the energy gap for arbitrary, not necessarily
contiguous regions on lattices characterized by general graphs.Comment: 26 pages, LaTeX, published version (figure added
Interest rate models with Markov chains
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Mapping all classical spin models to a lattice gauge theory
In our recent work [Phys. Rev. Lett. 102, 230502 (2009)] we showed that the
partition function of all classical spin models, including all discrete
standard statistical models and all Abelian discrete lattice gauge theories
(LGTs), can be expressed as a special instance of the partition function of a
4-dimensional pure LGT with gauge group Z_2 (4D Z_2 LGT). This provides a
unification of models with apparently very different features into a single
complete model. The result uses an equality between the Hamilton function of
any classical spin model and the Hamilton function of a model with all possible
k-body Ising-type interactions, for all k, which we also prove. Here, we
elaborate on the proof of the result, and we illustrate it by computing
quantities of a specific model as a function of the partition function of the
4D Z_2 LGT. The result also allows one to establish a new method to compute the
mean-field theory of Z_2 LGTs with d > 3, and to show that computing the
partition function of the 4D Z_2 LGT is computationally hard (#P hard). The
proof uses techniques from quantum information.Comment: 21 pages, 21 figures; published versio
Entanglement in Valence-Bond-Solid States
This article reviews the quantum entanglement in Valence-Bond-Solid (VBS)
states defined on a lattice or a graph. The subject is presented in a
self-contained and pedagogical way. The VBS state was first introduced in the
celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki
(abbreviation AKLT is widely used). It became essential in condensed matter
physics and quantum information (measurement-based quantum computation). Many
publications have been devoted to the subject. Recently entanglement was
studied in the VBS state. In this review we start with the definition of a
general AKLT spin chain and the construction of VBS ground state. In order to
study entanglement, a block subsystem is introduced and described by the
density matrix. Density matrices of 1-dimensional models are diagonalized and
the entanglement entropies (the von Neumann entropy and Renyi entropy) are
calculated. In the large block limit, the entropies also approach finite
limits. Study of the spectrum of the density matrix led to the discovery that
the density matrix is proportional to a projector.Comment: Published version, 80 pages, 8 figures; references update
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