Differentiators and the geometry of polynomials

Quantum coherence, present whenever a quantum system exists in a superposition of multiple classically distinct states, marks one of the fundamental departures from classical physics. Quantum coherence has recently been investigated rigorously within a resource-theoretic formalism. However, the finer-grained notion of multilevel coherence, which explicitly takes into account the number of superposed classical states, has remained relatively unexplored. A comprehensive analysis of multilevel coherence, which acts as the single-party analogue to multipartite entanglement, is essential for understanding natural quantum processes as well as for gauging the performance of quantum technologies. Here, we develop the theoretical and experimental groundwork for characterizing and quantifying multilevel coherence. We prove that nontrivial levels of purity are required for multilevel coherence, as there is a ball of states around the maximally mixed state that do not exhibit multilevel coherence in any basis. We provide a simple, necessary, and sufficient analytical criterion to verify the presence of multilevel coherence, which leads to a complete classification of multilevel coherence for three-level systems. We present the robustness of multilevel coherence, a bona fide quantifier, which we show to be numerically computable via semidefinite programming and experimentally accessible via multilevel coherence witnesses, which we introduce and characterize. We further verify and lower bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, which is implemented experimentally with four-level quantum probes in a photonic setup. Our results contribute to understanding the operational relevance of genuine multilevel coherence, also by demonstrating the key role it plays in enhanced phase discrimination-a primitive for quantum communication and metrology-and suggest new ways to reliably and effectively test the quantum behavior of physical systems.

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Certification and Quantification of Multilevel Quantum Coherence

In the
last few years there has been considerable research
on numerical methods for differential algebraic equations (DAEs)
 
$f(x', x, t) = 0$
 where 
$f_{x'}$
 is identically singular.
The index provides one measure of the singularity of a DAE. Most of
the numerical analysis literature on DAEs to date has dealt with
DAEs with indices no larger than three. Even in this case,
the systems were often assumed to have a special structure.
Recently a numerical method was proposed that could, in principle,
be used to integrate general unstructured higher index solvable DAEs.
However, that method did not preserve constraints.
This paper will discuss a modification of that approach which can
be used to design constraint preserving integrators for general
nonlinear higher index DAEs.

Constraint preserving integrators for general nonlinear higher index DAEs

An informal personal survey of some major outstanding questions in matrix theory is given These are grouped into several categories and illustrate the faci that modern work in matrix theory relies predominantly on neither linear nor algebraic techniques Occasional references are given Category 1 involves simultaneous conditions on sets of parameters associated with a matrix; category 2 includes restricted similarity questions, and category 3 is represented by the D-stability problem In category 4 are questions associated with the geometry of major classes of matrices and category 5 is that of determinant optimization and inequality problems Finallycategory 6 recognizes an area of rapidly growing interest, that of combinatorial aspects of matrix problems

Some outstanding problems in the theory of matrices

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n−1)x(n−1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.

Herbert A. Robinson

Papers

Eigenvalue inequalities for principal submatrices