467,047 research outputs found
Matrix product states, geometry, and invariant theory
Matrix product states play an important role in quantum information theory to
represent states of many-body systems. They can be seen as low-dimensional
subvarieties of a high-dimensional tensor space. In these notes, we consider
two variants: homogeneous matrix product states and uniform matrix product
states. Studying the linear spans of these varieties leads to a natural
connection with invariant theory of matrices. For homogeneous matrix product
states, a classical result on polynomial identities of matrices leads to a
formula for the dimension of the linear span, in the case of 2x2 matrices.
These notes are based partially on a talk given by the author at the
University of Warsaw during the thematic semester "AGATES: Algebraic Geometry
with Applications to TEnsors and Secants", and partially on further research
done during the semester. This is still a preliminary version; an updated
version will be uploaded over the course of 2023.Comment: 10 pages; comments welcome
Low rank positive partial transpose states and their relation to product vectors
It is known that entangled mixed states that are positive under partial
transposition (PPT states) must have rank at least four. In a previous paper we
presented a classification of rank four entangled PPT states which we believe
to be complete. In the present paper we continue our investigations of the low
rank entangled PPT states. We use perturbation theory in order to construct
rank five entangled PPT states close to the known rank four states, and in
order to compute dimensions and study the geometry of surfaces of low rank PPT
states. We exploit the close connection between low rank PPT states and product
vectors. In particular, we show how to reconstruct a PPT state from a
sufficient number of product vectors in its kernel. It may seem surprising that
the number of product vectors needed may be smaller than the dimension of the
kernel.Comment: 29 pages, 4 figure
Maximal Entanglement, Collective Coordinates and Tracking the King
Maximal entangled states (MES) provide a basis to two d-dimensional particles
Hilbert space, d=prime . The MES forming this basis are product states
in the collective, center of mass and relative, coordinates. These states are
associated (underpinned) with lines of finite geometry whose constituent points
are associated with product states carrying Mutual Unbiased Bases (MUB) labels.
This representation is shown to be convenient for the study of the Mean King
Problem and a variant thereof, termed Tracking the King which proves to be a
novel quantum communication channel. The main topics, notions used are reviewed
in an attempt to have the paper self contained.Comment: 8. arXiv admin note: substantial text overlap with arXiv:1206.3884,
arXiv:1206.035
Tensor network states and geometry
Tensor network states are used to approximate ground states of local
Hamiltonians on a lattice in D spatial dimensions. Different types of tensor
network states can be seen to generate different geometries. Matrix product
states (MPS) in D=1 dimensions, as well as projected entangled pair states
(PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the
lattice model; in contrast, the multi-scale entanglement renormalization ansatz
(MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on
homogeneous tensor networks, where all the tensors in the network are copies of
the same tensor, and argue that certain structural properties of the resulting
many-body states are preconditioned by the geometry of the tensor network and
are therefore largely independent of the choice of variational parameters.
Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for
D=1 systems is seen to be determined by the structure of geodesics in the
physical and holographic geometries, respectively; whereas the asymptotic
scaling of entanglement entropy is seen to always obey a simple boundary law --
that is, again in the relevant geometry. This geometrical interpretation offers
a simple and unifying framework to understand the structural properties of, and
helps clarify the relation between, different tensor network states. In
addition, it has recently motivated the branching MERA, a generalization of the
MERA capable of reproducing violations of the entropic boundary law in D>1
dimensions.Comment: 18 pages, 18 figure
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