741 research outputs found
Constraints on Multipartite Quantum Entropies
The von Neumann entropy plays a vital role in quantum information theory. As the Shannon entropydoes in classical information theory, the von Neumann entropy determines the capacities of quan-tum channels. Quantum entropies of composite quantum systems are important for future quantumnetwork communication their characterization is related to the so calledquantum marginal problem.Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropiesof multipartite quantum systems is the main object of interest. The problem of characterizing this setis not new – however, progress has been sparse, indicating that the problem may be considered hardand that new methods might be needed. Here, a variety of different and complementary aprroachesare taken.First, I look at global properties. It is known that the von Neumann entropy region – just likeits classical counterpart – forms aconvex cone. I describe the symmetries of this cone and highlightgeometric similarities and differences to the classical entropy cone.In a different approach, I utilize thelocalgeometric properties ofextremal raysof a cone. I showthat quantum states whose entropy lies on such an extremal ray of the quantum entropy cone have avery simple structure.As the set of all quantum states is very complicated, I look at a simple subset calledstabilizerstates. I improve on previously known results by showing that under a technical condition on the localdimension, entropies of stabilizer states respect an additional class of information inequalities that isvalid for random variables from linear codes.In a last approach I find a representation-theoretic formulation of the classical marginal problemsimplifying the comparison with its quantum mechanical counterpart. This novel correspondenceyields a simplified formulation of the group characterization of classical entropies (IEEE Trans. Inf.Theory, 48(7):1992–1995, 2002) in purely combinatorial terms
Entanglement, Purity, and Information Entropies in Continuous Variable Systems
Quantum entanglement of pure states of a bipartite system is defined as the
amount of local or marginal ({\em i.e.}referring to the subsystems) entropy.
For mixed states this identification vanishes, since the global loss of
information about the state makes it impossible to distinguish between quantum
and classical correlations. Here we show how the joint knowledge of the global
and marginal degrees of information of a quantum state, quantified by the
purities or in general by information entropies, provides an accurate
characterization of its entanglement. In particular, for Gaussian states of
continuous variable systems, we classify the entanglement of two--mode states
according to their degree of total and partial mixedness, comparing the
different roles played by the purity and the generalized entropies in
quantifying the mixedness and bounding the entanglement. We prove the existence
of strict upper and lower bounds on the entanglement and the existence of
extremally (maximally and minimally) entangled states at fixed global and
marginal degrees of information. This results allow for a powerful, operative
method to measure mixed-state entanglement without the full tomographic
reconstruction of the state. Finally, we briefly discuss the ongoing extension
of our analysis to the quantification of multipartite entanglement in highly
symmetric Gaussian states of arbitrary -mode partitions.Comment: 16 pages, 5 low-res figures, OSID style. Presented at the
International Conference ``Entanglement, Information and Noise'', Krzyzowa,
Poland, June 14--20, 200
Tensor Network Models of Unitary Black Hole Evaporation
We introduce a general class of toy models to study the quantum
information-theoretic properties of black hole radiation. The models are
governed by a set of isometries that specify how microstates of the black hole
at a given energy evolve to entangled states of a tensor product
black-hole/radiation Hilbert space. The final state of the black hole radiation
is conveniently summarized by a tensor network built from these isometries. We
introduce a set of quantities generalizing the Renyi entropies that provide a
complete set of bipartite/multipartite entanglement measures, and give a
general formula for the average of these over initial black hole states in
terms of the isometries defining the model. For models where the dimension of
the final tensor product radiation Hilbert space is the same as that of the
space of initial black hole microstates, the entanglement structure is
universal, independent of the choice of isometries. In the more general case,
we find that models which best capture the "information-free" property of black
hole horizons are those whose isometries are tensors corresponding to states of
tripartite systems with maximally mixed subsystems.Comment: 22 pages, 4 figure
Infinitely many constrained inequalities for the von Neumann entropy
We exhibit infinitely many new, constrained inequalities for the von Neumann
entropy, and show that they are independent of each other and the known
inequalities obeyed by the von Neumann entropy (basically strong
subadditivity). The new inequalities were proved originally by Makarychev et
al. [Commun. Inf. Syst., 2(2):147-166, 2002] for the Shannon entropy, using
properties of probability distributions. Our approach extends the proof of the
inequalities to the quantum domain, and includes their independence for the
quantum and also the classical cases.Comment: 11 page
Witnessing entanglement by proxy
Entanglement is a ubiquitous feature of low temperature systems and believed
to be highly relevant for the dynamics of condensed matter properties and
quantum computation even at higher temperatures. The experimental certification
of this paradigmatic quantum effect in macroscopic high temperature systems is
constrained by the limited access to the quantum state of the system. In this
paper we show how macroscopic observables beyond the energy of the system can
be exploited as proxy witnesses for entanglement detection. Using linear and
semi-definite relaxations we show that all previous approaches to this problem
can be outperformed by our proxies, i.e. entanglement can be certified at
higher temperatures without access to any local observable. For an efficient
computation of proxy witnesses one can resort to a generalized grand canonical
ensemble, enabling entanglement certification even in complex systems with
macroscopic particle numbers.Comment: 22 pages, 8 figure
Inequalities for the Ranks of Quantum States
We investigate relations between the ranks of marginals of multipartite
quantum states. These are the Schmidt ranks across all possible bipartitions
and constitute a natural quantification of multipartite entanglement
dimensionality. We show that there exist inequalities constraining the possible
distribution of ranks. This is analogous to the case of von Neumann entropy
(\alpha-R\'enyi entropy for \alpha=1), where nontrivial inequalities
constraining the distribution of entropies (such as e.g. strong subadditivity)
are known. It was also recently discovered that all other \alpha-R\'enyi
entropies for satisfy only one trivial linear
inequality (non-negativity) and the distribution of entropies for
is completely unconstrained beyond non-negativity. Our result
resolves an important open question by showing that also the case of \alpha=0
(logarithm of the rank) is restricted by nontrivial linear relations and thus
the cases of von Neumann entropy (i.e., \alpha=1) and 0-R\'enyi entropy are
exceptionally interesting measures of entanglement in the multipartite setting
Entanglement of Purification and Multiboundary Wormhole Geometries
We posit a geometrical description of the entanglement of purification for
subregions in a holographic CFT. The bulk description naturally generalizes the
two-party case and leads to interesting inequalities among multi-party
entanglements of purification that can be geometrically proven from the
conjecture. Further, we study the relationship between holographic
entanglements of purification in locally-AdS3 spacetimes and entanglement
entropies in multi-throated wormhole geometries constructed via quotienting by
isometries. In particular, we derive new holographic inequalities for
geometries that are locally AdS3 relating entanglements of purification for
subregions and entanglement entropies in the wormhole geometries.Comment: 23 pages, 12 figures; v2 added references; v3 fixed inequality
direction in Eq.(2), expanded discussion - reflects published versio
- …