53 research outputs found
Entanglement distillation from Greenberger-Horne-Zeilinger shares
We study the problem of converting a product of Greenberger-Horne-Zeilinger
(GHZ) states shared by subsets of several parties in an arbitrary way into GHZ
states shared by every party. Our result is that if SLOCC transformations are
allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of
the initial state. This generalizes a result by Strassen on the asymptotic
subrank of the matrix multiplication tensor.Comment: 8 pages, v2: minor correction
Distillation of Greenberger-Horne-Zeilinger states by combinatorial methods
We prove a lower bound on the rate of Greenberger-Horne-Zeilinger states
distillable from pure multipartite states by local operations and classical
communication (LOCC). Our proof is based on a modification of a combinatorial
argument used in the fast matrix multiplication algorithm of Coppersmith and
Winograd. Previous use of methods from algebraic complexity in quantum
information theory concerned transformations with stochastic local operations
and classical operation (SLOCC), resulting in an asymptotically vanishing
success probability. In contrast, our new protocol works with asymptotically
vanishing error.Comment: 26 pages, 2 figures; v2: updated to match published versio
Local unitary invariants for multipartite quantum systems
A method is presented to obtain local unitary invariants for multipartite
quantum systems consisting of fermions or distinguishable particles. The
invariants are organized into infinite families, in particular, the
generalization to higher dimensional single particle Hilbert spaces is
straightforward. Many well-known invariants and their generalizations are also
included.Comment: 13 page
The asymptotic spectrum of LOCC transformations
We study exact, non-deterministic conversion of multipartite pure quantum
states into one-another via local operations and classical communication (LOCC)
and asymptotic entanglement transformation under such channels. In particular,
we consider the maximal number of copies of any given target state that can be
extracted exactly from many copies of any given initial state as a function of
the exponential decay in success probability, known as the converese error
exponent. We give a formula for the optimal rate presented as an infimum over
the asymptotic spectrum of LOCC conversion. A full understanding of exact
asymptotic extraction rates between pure states in the converse regime thus
depends on a full understanding of this spectrum. We present a characterisation
of spectral points and use it to describe the spectrum in the bipartite case.
This leads to a full description of the spectrum and thus an explicit formula
for the asymptotic extraction rate between pure bipartite states, given a
converse error exponent. This extends the result on entanglement concentration
in [Hayashi et al, 2003], where the target state is fixed as the Bell state. In
the limit of vanishing converse error exponent the rate formula provides an
upper bound on the exact asymptotic extraction rate between two states, when
the probability of success goes to 1. In the bipartite case we prove that this
bound holds with equality.Comment: v1: 21 pages v2: 21 pages, Minor corrections v3: 17 pages, Minor
corrections, new reference added, parts of Section 5 and the Appendix
removed, the omitted material can be found in an extended form in
arXiv:1808.0515
The asymptotic induced matching number of hypergraphs: balanced binary strings
We compute the asymptotic induced matching number of the -partite
-uniform hypergraphs whose edges are the -bit strings of Hamming weight
, for any large enough even number . Our lower bound relies on the
higher-order extension of the well-known Coppersmith-Winograd method from
algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam.
Our result is motivated by the study of the power of this method as well as of
the power of the Strassen support functionals (which provide upper bounds on
the asymptotic induced matching number), and the connections to questions in
tensor theory, quantum information theory and theoretical computer science.
Phrased in the language of tensors, as a direct consequence of our result, we
determine the asymptotic subrank of any tensor with support given by the
aforementioned hypergraphs. In the context of quantum information theory, our
result amounts to an asymptotically optimal -party stochastic local
operations and classical communication (slocc) protocol for the problem of
distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement
Asymptotic tensor rank of graph tensors: beyond matrix multiplication
We present an upper bound on the exponent of the asymptotic behaviour of the
tensor rank of a family of tensors defined by the complete graph on
vertices. For , we show that the exponent per edge is at most 0.77,
outperforming the best known upper bound on the exponent per edge for matrix
multiplication (), which is approximately 0.79. We raise the question
whether for some the exponent per edge can be below , i.e. can
outperform matrix multiplication even if the matrix multiplication exponent
equals 2. In order to obtain our results, we generalise to higher order tensors
a result by Strassen on the asymptotic subrank of tight tensors and a result by
Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our
results have applications in entanglement theory and communication complexity
The Role of Topology in Quantum Tomography
We investigate quantum tomography in scenarios where prior information
restricts the state space to a smooth manifold of lower dimensionality. By
considering stability we provide a general framework that relates the topology
of the manifold to the minimal number of binary measurement settings that is
necessary to discriminate any two states on the manifold. We apply these
findings to cases where the subset of states under consideration is given by
states with bounded rank, fixed spectrum, given unitary symmetry or taken from
a unitary orbit. For all these cases we provide both upper and lower bounds on
the minimal number of binary measurement settings necessary to discriminate any
two states of these subsets
- …