Uncertainty principle, which was first introduced by Werner Heisenberg
in 1927, forms a fundamental component of quantum mechanics.
A graceful aspect of quantum mechanics is that the uncertainty
relations between incompatible observables allow for succinct quan-
titative formulations of this revolutionary idea: it is impossible to
simultaneously measure two complementary variables of a particle in
precision. In particular, information theory offers two basic ways to
express the Heisenberg’s principle: variance-based uncertainty relations
and entropic uncertainty relations.
We first investigate the uncertainty relations based on the sum of
variances and derive a family of weighted uncertainty relations to
provide an optimal lower bound for all situations. Our work indicates
that it seems unreasonable to assume a priori that incompatible
observables have equal contribution to the variance-based sum form
uncertainty relations. We also study the role of mutually exclusive
physical states in the recent work and generalize the variance-based
uncertainty relations to mutually exclusive uncertainty relations.
Next, we develop a new kind of entanglement detection criteria within
the framework of marjorization theory and its matrix representation.
By virtue of majorization uncertainty bounds, we are able to construct
the entanglement criteria which have advantage over the scalar detect-
ing algorithms as they are often stronger and tighter.
Furthermore, we explore various expression of entropic uncertainty
relations, including sum of Shannon entropies, majorization uncer-
tainty relations and uncertainty relations in presence of quantum
memory. For entropic uncertainty relations without quantum side
information, we provide several tighter bounds for multi-measurements,
with some of them also valid for Rényi and Tsallis entropies besides
the Shannon entropy. We employ majorization theory and actions
of the symmetric group to obtain an admixture bound for entropic
uncertainty relations with multi-measurements. Comparisons among
existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical
side information. For modern uncertainty relations, those who allowed
for non-trivial amount of quantum side information, their bounds
have been strengthened by our recent result for both two and multi-
measurements.
Finally, we propose an approach which can extend all uncertainty
relations on Shannon entropies to allow for quantum side information
and discuss the applications of our entropic framework. Combined with
our uniform entanglement frames, it is possible to detect entanglement
via entropic uncertainty relations even if there is no quantum side in-
formation. With the rising of quantum information theory, uncertainty
relations have been established as important tools for a wide range of
applications, such as quantum cryptography, quantum key distribution,
entanglement detection, quantum metrology, quantum speed limit and
so on. It is thus necessary to focus on the study of uncertainty relations.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:15-qucosa-219674 |
| Date | 06 March 2017 |
| Creators | Xiao, Yunlong |
| Contributors | Institute of Mathematics, The Faculty of Mathematics and Computer Science, Prof. Dr. Jürgen Jost, Prof. Dr. Gilad Gour |
| Publisher | Universitätsbibliothek Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
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