5 research outputs found
On asymptotic continuity of functions of quantum states
A useful kind of continuity of quantum states functions in asymptotic regime
is so-called asymptotic continuity. In this paper we provide general tools for
checking if a function possesses this property. First we prove equivalence of
asymptotic continuity with so-called it robustness under admixture. This allows
us to show that relative entropy distance from a convex set including maximally
mixed state is asymptotically continuous. Subsequently, we consider it arrowing
- a way of building a new function out of a given one. The procedure originates
from constructions of intrinsic information and entanglement of formation. We
show that arrowing preserves asymptotic continuity for a class of functions
(so-called subextensive ones). The result is illustrated by means of several
examples.Comment: Minor corrections, version submitted for publicatio
A Generalization of Quantum Stein's Lemma
We present a generalization of quantum Stein's Lemma to the situation in
which the alternative hypothesis is formed by a family of states, which can
moreover be non-i.i.d.. We consider sets of states which satisfy a few natural
properties, the most important being the closedness under permutations of the
copies. We then determine the error rate function in a very similar fashion to
quantum Stein's Lemma, in terms of the quantum relative entropy.
Our result has two applications to entanglement theory. First it gives an
operational meaning to an entanglement measure known as regularized relative
entropy of entanglement. Second, it shows that this measure is faithful, being
strictly positive on every entangled state. This implies, in particular, that
whenever a multipartite state can be asymptotically converted into another
entangled state by local operations and classical communication, the rate of
conversion must be non-zero. Therefore, the operational definition of
multipartite entanglement is equivalent to its mathematical definition.Comment: 30 pages. (see posting by M. Piani arXiv:0904.2705 for a different
proof of the strict positiveness of the regularized relative entropy of
entanglement on every entangled state). published version
Faithful Squashed Entanglement
Squashed entanglement is a measure for the entanglement of bipartite quantum
states. In this paper we present a lower bound for squashed entanglement in
terms of a distance to the set of separable states. This implies that squashed
entanglement is faithful, that is, strictly positive if and only if the state
is entangled. We derive the bound on squashed entanglement from a bound on
quantum conditional mutual information, which is used to define squashed
entanglement and corresponds to the amount by which strong subadditivity of von
Neumann entropy fails to be saturated. Our result therefore sheds light on the
structure of states that almost satisfy strong subadditivity with equality. The
proof is based on two recent results from quantum information theory: the
operational interpretation of the quantum mutual information as the optimal
rate for state redistribution and the interpretation of the regularised
relative entropy of entanglement as an error exponent in hypothesis testing.
The distance to the set of separable states is measured by the one-way LOCC
norm, an operationally-motivated norm giving the optimal probability of
distinguishing two bipartite quantum states, each shared by two parties, using
any protocol formed by local quantum operations and one-directional classical
communication between the parties. A similar result for the Frobenius or
Euclidean norm follows immediately. The result has two applications in
complexity theory. The first is a quasipolynomial-time algorithm solving the
weak membership problem for the set of separable states in one-way LOCC or
Euclidean norm. The second concerns quantum Merlin-Arthur games. Here we show
that multiple provers are not more powerful than a single prover when the
verifier is restricted to one-way LOCC operations thereby providing a new
characterisation of the complexity class QMA.Comment: 24 pages, 1 figure, 1 table. Due to an error in the published
version, claims have been weakened from the LOCC norm to the one-way LOCC
nor