16 research outputs found
A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces
Entanglement witnesses (EWs) constitute one of the most important
entanglement detectors in quantum systems. Nevertheless, their complete
characterization, in particular with respect to the notion of optimality, is
still missing, even in the decomposable case. Here we show that for any
qubit-qunit decomposable EW (DEW) W the three statements are equivalent: (i)
the set of product vectors obeying \bra{e,f}W\ket{e,f}=0 spans the
corresponding Hilbert space, (ii) W is optimal, (iii) W=Q^{\Gamma} with Q
denoting a positive operator supported on a completely entangled subspace (CES)
and \Gamma standing for the partial transposition. While, implications
and are known, here we prove that
(iii) implies (i). This is a consequence of a more general fact saying that
product vectors orthogonal to any CES in C^{2}\otimes C^{n} span after partial
conjugation the whole space. On the other hand, already in the case of
C^{3}\otimes C^{3} Hilbert space, there exist DEWs for which (iii) does not
imply (i). Consequently, either (i) does not imply (ii), or (ii) does not imply
(iii), and the above transparent characterization obeyed by qubit-qunit DEWs,
does not hold in general.Comment: 13 pages, proof of lemma 4 corrected, theorem 3 removed, some parts
improve
Bounds on the entanglement of two-qutrit systems from fixed marginals
We discuss the problem of characterizing upper bounds on entanglement in a bipartite quantum system when only the reduced density matrices (marginals) are known. In particular, starting from the known two-qubit case, we propose a family of candidates for maximally entangled mixed states with respect to fixed marginals for two qutrits. These states are extremal in the convex set of two-qutrit states with fixed marginals. Moreover, it is shown that they are always quasidistillable. As a by-product we prove that any maximally correlated state that is quasidistillable must be pure. Our observations for two qutrits are supported by numerical analysis
Constructing new optimal entanglement witnesses
We provide a new class of indecomposable entanglement witnesses. In 4 x 4
case it reproduces the well know Breuer-Hall witness. We prove that these new
witnesses are optimal and atomic, i.e. they are able to detect the "weakest"
quantum entanglement encoded into states with positive partial transposition
(PPT). Equivalently, we provide a new construction of indecomposable atomic
maps in the algebra of 2k x 2k complex matrices. It is shown that their
structural physical approximations give rise to entanglement breaking channels.
This result supports recent conjecture by Korbicz et. al.Comment: 9 page
Fidelity of dynamical maps
We introduce the concept of fidelity for dynamical maps in an open quantum system scenario. We derive an inequality linking this quantity to the distinguishability of the inducing environmental states. Our inequality imposes constraints on the allowed set of dynamical maps arising from the microscopic description of system plus environment. Remarkably, the inequality involves only the states of the environment and the dynamical map of the open system and, therefore, does not rely on the knowledge of either the microscopic interaction Hamiltonian or the environmental Hamiltonian characteristic parameters. We demonstrate the power of our result by applying it to two different scenarios: quantum programming and quantum probing. In the first case, we use it to derive bounds on the dimension of the processor for approximate programming of unitaries. In the second case we present an intriguing proof-of-principle demonstration of the ability to extract information on the environment via a quantum probe without any a priori assumption on the form of the system-environment coupling Hamiltonian
Spectral properties of entanglement witnesses
Entanglement witnesses are observables which when measured, detect
entanglement in a measured composed system. It is shown what kind of relations
between eigenvectors of an observable should be fulfilled, to allow an
observable to be an entanglement witness. Some restrictions on the signature of
entaglement witnesses, based on an algebraic-geometrical theorem will be given.
The set of entanglement witnesses is linearly isomorphic to the set of maps
between matrix algebras which are positive, but not completely positive. A
translation of the results to the language of positive maps is also given. The
properties of entanglement witnesses and positive maps express as special cases
of general theorems for -Schmidt witnesses and -positive maps. The
results are therefore presented in a general framework.Comment: published version, some proofs are more detailed, mistakes remove