Characterizing the geometry of the Kirkwood-Dirac positive states

Langrenez, Christopher; Arvidsson-Shukur, David R. M.; De Bievre, Stephan

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1063/5.0164672

Title :

Characterizing the geometry of the Kirkwood-Dirac positive states

Author(s) :

Langrenez, Christopher [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Université de Lille
Systèmes de particules et systèmes dynamiques [Paradyse]
Arvidsson-Shukur, David R. M. [Auteur]
Hitachi Cambridge Laboratory [University of Cambridge]
De Bievre, Stephan [Auteur]

Systèmes de particules et systèmes dynamiques [Paradyse]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Université de Lille

Journal title :

Journal of Mathematical Physics

Publisher :

American Institute of Physics (AIP)

Publication date :

2024-07-10

ISSN :

0022-2488

HAL domain(s) :

Physique [physics]
Mathématiques [math]

English abstract : [en]

The Kirkwood-Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables $A$ and $B$. KD distributions behave similarly to classical joint probability distributions ...
Show more >The Kirkwood-Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables $A$ and $B$. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we characterize how the full convex set of states with positive KD distributions depends on the eigenbases of $A$ and $B$. In particular, we identify three regimes where convex combinations of the eigenprojectors of $A$ and $B$ constitute the only KD-positive states: $(i)$ any system in dimension $2$; $(ii)$ an open and dense set of bases in dimension $3$; and $(iii)$ the discrete-Fourier-transform bases in prime dimension. Finally, we investigate if there can exist mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We show that for some choices of observables $A$ and $B$ this phenomenon does indeed occur. We explicitly construct such states for a spin-$1$ system.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Comment :

35 pages, 2 figures

Collections :