20 research outputs found
Area laws for the entanglement entropy - a review
Physical interactions in quantum many-body systems are typically local:
Individual constituents interact mainly with their few nearest neighbors. This
locality of interactions is inherited by a decay of correlation functions, but
also reflected by scaling laws of a quite profound quantity: The entanglement
entropy of ground states. This entropy of the reduced state of a subregion
often merely grows like the boundary area of the subregion, and not like its
volume, in sharp contrast with an expected extensive behavior. Such "area laws"
for the entanglement entropy and related quantities have received considerable
attention in recent years. They emerge in several seemingly unrelated fields,
in the context of black hole physics, quantum information science, and quantum
many-body physics where they have important implications on the numerical
simulation of lattice models. In this Colloquium we review the current status
of area laws in these fields. Center stage is taken by rigorous results on
lattice models in one and higher spatial dimensions. The differences and
similarities between bosonic and fermionic models are stressed, area laws are
related to the velocity of information propagation, and disordered systems,
non-equilibrium situations, classical correlation concepts, and topological
entanglement entropies are discussed. A significant proportion of the article
is devoted to the quantitative connection between the entanglement content of
states and the possibility of their efficient numerical simulation. We discuss
matrix-product states, higher-dimensional analogues, and states from
entanglement renormalization and conclude by highlighting the implications of
area laws on quantifying the effective degrees of freedom that need to be
considered in simulations.Comment: 28 pages, 2 figures, final versio
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
Entanglement between particle partitions in itinerant many-particle states
We review `particle partitioning entanglement' for itinerant many-particle
systems. This is defined as the entanglement between two subsets of particles
making up the system. We identify generic features and mechanisms of particle
entanglement that are valid over whole classes of itinerant quantum systems. We
formulate the general structure of particle entanglement in many-fermion ground
states, analogous to the `area law' for the more usually studied entanglement
between spatial regions. Basic properties of particle entanglement are first
elucidated by considering relatively simple itinerant models. We then review
particle-partitioning entanglement in quantum states with more intricate
physics, such as anyonic models and quantum Hall states.Comment: review, about 20 pages. Version 2 has minor revisions
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
Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules
We present a new class of tensor network states that are specifically
designed to capture the electron correlation of a molecule of arbitrary
structure. In this ansatz, the electronic wave function is represented by a
Complete-Graph Tensor Network (CGTN) ansatz which implements an efficient
reduction of the number of variational parameters by breaking down the
complexity of the high-dimensional coefficient tensor of a
full-configuration-interaction (FCI) wave function. We demonstrate that CGTN
states approximate ground states of molecules accurately by comparison of the
CGTN and FCI expansion coefficients. The CGTN parametrization is not biased
towards any reference configuration in contrast to many standard quantum
chemical methods. This feature allows one to obtain accurate relative energies
between CGTN states which is central to molecular physics and chemistry. We
discuss the implications for quantum chemistry and focus on the spin-state
problem. Our CGTN approach is applied to the energy splitting of states of
different spin for methylene and the strongly correlated ozone molecule at a
transition state structure. The parameters of the tensor network ansatz are
variationally optimized by means of a parallel-tempering Monte Carlo algorithm
Entanglement Entropy dynamics in Heisenberg chains
By means of the time-dependent density matrix renormalization group algorithm
we study the zero-temperature dynamics of the Von Neumann entropy of a block of
spins in a Heisenberg chain after a sudden quench in the anisotropy parameter.
In the absence of any disorder the block entropy increases linearly with time
and then saturates. We analyze the velocity of propagation of the entanglement
as a function of the initial and final anisotropies and compare, wherever
possible, our results with those obtained by means of Conformal Field Theory.
In the disordered case we find a slower (logarithmic) evolution which may
signals the onset of entanglement localization.Comment: 15 pages, 9 figure
Entanglement, combinatorics and finite-size effects in spin-chains
We carry out a systematic study of the exact block entanglement in XXZ
spin-chain at Delta=-1/2. We present, the first analytic expressions for
reduced density matrices of n spins in a chain of length L (for n<=6 and
arbitrary but odd L) of a truly interacting model. The entanglement entropy,
the moments of the reduced density matrix, and its spectrum are then easily
derived. We explicitely construct the "entanglement Hamiltonian" as the
logarithm of this matrix. Exploiting the degeneracy of the ground-state, we
find the scaling behavior of entanglement of the zero-temperature mixed state.Comment: 5 pages, 4 figures, one mathematica file attache
Efficient tomography of a quantum many-body system
Quantum state tomography (QST) is the gold standard technique for obtaining an estimate for the state of small quantum systems in the laboratory [1]. Its application to systems with more than a few constituents (e.g. particles) soon becomes impractical as the e ff ort required grows exponentially with the number of constituents. Developing more e ffi cient techniques is particularly pressing as precisely-controllable quantum systems that are well beyond the reach of QST are emerging in laboratories. Motivated by this, there is a considerable ongoing e ff ort to develop new state characterisation tools for quantum many-body systems [2–11]. Here we demonstrate Matrix Product State (MPS) tomography [2], which is theoretically proven to allow the states of a broad class of quantum systems to be accurately estimated with an e ff ort that increases e ffi ciently with constituent number. We use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually-controlled spins (qubits): a size far beyond the practical limits of QST. Our results reveal the dynamical growth of entanglement and description complexity as correlations spread out during a quench: a necessary condition for future beyond-classical performance. MPS tomography should therefore find widespread use to study large quantum many-body systems and to benchmark and verify quantum simulators and computers
Permutationally invariant state reconstruction
Feasible tomography schemes for large particle numbers must possess, besides
an appropriate data acquisition protocol, also an efficient way to reconstruct
the density operator from the observed finite data set. Since state
reconstruction typically requires the solution of a non-linear large-scale
optimization problem, this is a major challenge in the design of scalable
tomography schemes. Here we present an efficient state reconstruction scheme
for permutationally invariant quantum state tomography. It works for all common
state-of-the-art reconstruction principles, including, in particular, maximum
likelihood and least squares methods, which are the preferred choices in
today's experiments. This high efficiency is achieved by greatly reducing the
dimensionality of the problem employing a particular representation of
permutationally invariant states known from spin coupling combined with convex
optimization, which has clear advantages regarding speed, control and accuracy
in comparison to commonly employed numerical routines. First prototype
implementations easily allow reconstruction of a state of 20 qubits in a few
minutes on a standard computer.Comment: 25 pages, 4 figues, 2 table