arXiv subjects

Shi-Jiao Geng

Publications and source records attributed to Shi-Jiao Geng.

3 recordsLinked to original sources

arXiv2024

Geometry of Kirkwood-Dirac classical states: A case study based on discrete Fourier transform

The characterization of Kirkwood-Dirac (KD) classicality or non-classicality is very important in quantum information processing. In general, the set of KD classical states with respect to two bases is not a convex polytope[J. Math. Phys. \textbf{65} 072201 (2024)], which makes us interested in finding out in which circumnstances they do form a polytope. In this paper, we focus on the characterization of KD classicality of mixed states for the case where the transition matrix between two bases is a discrete Fourier transform (DFT) matrix in Hilbert space with dimensions $p^2$ and $pq$, respectively, where $p, q$ are prime. For the two particular cases we investigate, the sets of extremal points are finite, implying that the set of KD classical states we characterize forms a convex polytope. We show that for $p^2$ dimensional system, the set $\rm{KD}_{\mathcal{A},\mathcal{B}}^+$ is a convex hull of the set $\rm {pure}({\rm {KD}_{\mathcal{A},\mathcal{B}}^+})$ based on DFT, where $\rm{KD}_{\mathcal{A},\mathcal{B}}^+$ is the set of KD classical states with respect to two bases and $\rm {pure}({\rm {KD}_{\mathcal{A},\mathcal{B}}^+})$ is the set of all the rank-one projectors of KD classical pure states with respect to two bases. In $pq$ dimensional system, we believe that this result also holds. Unfortunately, we do not completely prove it, but some meaningful conclusions are obtained about the characterization of KD classicality.

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arXiv2023

Characterizing Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform

In this paper, we investigate the Kirkwood-Dirac nonclassicality and uncertainty diagram based on discrete Fourier transform (DFT) in a $d$ dimensional system. The uncertainty diagram of complete incompatibility bases $\mathcal {A},\mathcal {B}$ are characterized by De Bièvre [arXiv: 2207.07451]. We show that for the uncertainty diagram of the DFT matrix which is a transition matrix from basis $\mathcal {A}$ to basis $\mathcal {B}$, there is no ``hole" in the region of the $(n_{\mathcal {A}}, n_{\mathcal {B}})$-plane above and on the line $n_{\mathcal {A}}+n_{\mathcal {B}}\geq d+1$, whether the bases $\mathcal {A},\mathcal {B}$ are not complete incompatible bases or not. Then we present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation $n_{\mathcal {A}}(ψ)n_{\mathcal {B}}(ψ)\geq d$, where $n_{\mathcal {A}}(ψ)$ and $n_{\mathcal {B}}(ψ)$ count the number of nonvanishing coefficients in the basis $\mathcal {A}$ and $\mathcal {B}$ representations, respectively. That is, a state $|ψ\rangle$ is KD nonclassical if and only if $n_{\mathcal {A}}(ψ)n_{\mathcal {B}}(ψ)> d$, whenever $d$ is prime or not. That gives a positive answer to the conjecture in [Phys. Rev. Lett. \textbf{127}, 190404 (2021)].

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arXiv2018

Construnctions of LOCC indistinguishable set of generalized Bell states

In this paper, we mainly consider the local indistinguishability of the set of mutually orthogonal bipartite generalized Bell states (GBSs). We construct small sets of GBSs with cardinality smaller than $d$ which are not distinguished by one-way local operations and classical communication (1-LOCC) in $d\otimes d$. The constructions, based on linear system and Vandermonde matrix, is simple and effective. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs, and greatly improve previous results in [Zhang \emph{et al.}, Phys. Rev. A 91, 012329 (2015); Wang \emph{et al.}, Quantum Inf. Process. 15, 1661 (2016)]. The case that $d$ is odd of the results also shows that the set of 4 GBSs in $5\otimes 5$ in [Fan, Phys. Rev. A 75, 014305 (2007)] is indeed a 1-LOCC indistinguishable set which can not be distinguished by Fan's method.

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