The interplay between quantum entanglement, coherence, and convex optimization

In this thesis, we strive to advance the knowledge of relations between convex optimization
and the quantum phenomena entanglement and coherence. The main research
areas we explore are rank-constrained semidefinite programming, the quantum
pure-state marginal problem and the existence of AME states as well as quantum
codes, entanglement detection, and the certification of quantum memories with coherence.

First, we start with real and complex rank-constraint semidefinite optimization problems
and rephrase them as an optimization over separable two-copy states. This reformulation
allows to approach the problem through a hierarchy of efficiently solvable
semidefinite programs that provide better and better certified bounds. We apply the
new technique to various problems in quantum information theory and beyond, such
as the optimization over pure states or unitary channels and the well-known maximum
cut problem. Furthermore, we describe an inherent symmetry in our formulation that
significantly improves the performance.

Second, we consider the application of our method to the quantum pure-state marginal
problem. In particular, we prove that the existence of n-partite absolutely maximally
entangled states with local dimension d is equivalent to the bipartite separability of a
certain state of 2n particles, and we compute that state explicitly. This application is a
striking example of how symmetries can simplify semidefinite programs and we use
them to compute high orders of our hierarchy despite the rapidly increasing dimension.
Moreover, we rewrite the existence problem of quantum error-correcting codes
as a marginal problem making our method also applicable to this area of research.

Third, since entanglement is not only a theoretically interesting phenomenon, but also
a vital resource for quantum information protocols, we investigate entanglement detection
in practical experiments. We examine scrambled data, a scenario in which the
mapping between outcomes and their respective probabilities is lost. Furthermore, we
use the joint numerical range of observables to find measurements that allow entanglement
detection even when the confidence region due to statistical and systematic
errors is large.

Finally, we introduce a quality measure for quantum memories that quantifies the
performance based on the memory’s ability to preserve coherence. Remarkably, this
measure also distinguishes entanglement-breaking channels from genuine quantum
memories. For the case of single-qubit channels, we find various theoretical bounds
and a simple measurement scheme to approximate our performance measure.