Nathaniel Johnston

Robustness of coherence: an operational and observable measure of quantum coherence

Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.

Robustness of asymmetry and coherence of quantum states

Quantum states may exhibit asymmetry with respect to the action of a given group. Such an asymmetry of states can be considered as a resource in applications such as quantum metrology, and it is a concept that encompasses quantum coherence as a special case. We introduce explicitly and study the robustness of asymmetry, a quantifier of asymmetry of states that we prove to have many attractive properties, including efficient numerical computability via semidefinite programming, and an operational interpretation in a channel discrimination context. We also introduce the notion of asymmetry witnesses, whose measurement in a laboratory detects the presence of asymmetry. We prove that properly constrained asymmetry witnesses provide lower bounds to the robustness of asymmetry, which is shown to be a directly measurable quantity itself. We then focus our attention on coherence witnesses and the robustness of coherence, for which we prove a number of additional results; these include an analysis of its specific relevance in phase discrimination and quantum metrology, an analytical calculation of its value for a relevant class of quantum states, and tight bounds that relate it to another previously defined coherence monotone.

A family of norms with applications in quantum information theory II

We consider the problem of computing the family of operator norms recently introducedin [1]. We develop a family of semidefinite programs that can be used to exactly computethem in small dimensions and bound them in general. Some theoretical consequencesfollow from the duality theory of semidefinite programming, including a new constructiveproof that for all r there are non-positive partial transpose Werner states that are r-undistillable. Several examples are considered via a MATLAB implementation of thesemidefinite program, including the case of Werner states and randomly generated statesvia the Bures measure, and approximate distributions of the norms are provided. Weextend these norms to arbitrary convex mapping cones and explore their implicationswith positive partial transpose states.

Minimal and maximal operator spaces and operator systems in entanglement theory

Abstract We examine k -minimal and k -maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k -minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k -positive linear maps and bound entanglement. Similarly, we investigate the k -super minimal and k -super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k -block positive operators and (unnormalized) states with Schmidt number no greater than k , respectively. We characterize a class of norms on the k -super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

Characterizing operations preserving separability measures via linear preserver problems

We use classical results from the theory of linear preserver problems to characterize operators that send the set of pure states with Schmidt rank no greater than k back into itself, extending known results characterizing operators that send separable pure states to separable pure states. We also provide a new proof of an analogous statement in the multipartite setting. We use these results to develop a bipartite version of a classical result about the structure of maps that preserve rank-1 operators and then characterize the isometries for two families of norms that have recently been studied in quantum information theory. We see, in particular, that for k ≥ 2 the operator norms induced by states with Schmidt rank k are invariant only under local unitaries, the swap operator and the transpose map. However, in the k = 1 case there is an additional isometry: the partial transpose map.

Uniqueness of quantum states compatible with given measurement results

We discuss the uniqueness of quantum states compatible with given measurement results for a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it was known that for a

Separability from spectrum for qubit-qudit states

d

The multiplicative domain in quantum error correction

-dimensional Hilbert space, there exists a set of

Limitations on Separable Measurements by Convex Optimization

4d\ensuremath{-}5

The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

observables that uniquely determines any pure state. We show that for case (2),