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X-states of a qubit pair of double classicality

Arsen Khvedelidze, Dimitar Mladenov, Astghik Torosyan

TL;DR

The work defines doubly classical states as the intersection $\mathfrak{C^{(++)}}_N = \mathfrak{S}_N \cap \mathfrak{C}^{(+)}_N$, combining separability and positive Wigner functions within a Stratonovich-Weyl phase-space framework to identify potential free resources in a quantum-resource theory. It specializes to X-states of a two-qubit system, deriving explicit separability criteria via PPT for the spectrum $\boldsymbol{r}$ and Euler angles, and isolates absolutely separable X-states through two cone inequalities on eigenvalues. The paper then analyzes Wigner-function positivity by introducing a WF-positivity polytope determined by the SW kernel spectra; it details the moduli spaces of SW kernels for quatrits and for 2-qubits, including explicit eigenvalue expressions and constraints. Together, these results reveal a nonempty region of double classicality beyond the WF-ball and clarify the geometric structure (cones, polytopes, moduli spaces) governing the intersection of classicality and separability, with implications for finite-dimensional quantum resource theories.

Abstract

A special class of states of 2-qubits which are simultaneously separable and have positive semidefinite Wigner functions is described.

X-states of a qubit pair of double classicality

TL;DR

The work defines doubly classical states as the intersection , combining separability and positive Wigner functions within a Stratonovich-Weyl phase-space framework to identify potential free resources in a quantum-resource theory. It specializes to X-states of a two-qubit system, deriving explicit separability criteria via PPT for the spectrum and Euler angles, and isolates absolutely separable X-states through two cone inequalities on eigenvalues. The paper then analyzes Wigner-function positivity by introducing a WF-positivity polytope determined by the SW kernel spectra; it details the moduli spaces of SW kernels for quatrits and for 2-qubits, including explicit eigenvalue expressions and constraints. Together, these results reveal a nonempty region of double classicality beyond the WF-ball and clarify the geometric structure (cones, polytopes, moduli spaces) governing the intersection of classicality and separability, with implications for finite-dimensional quantum resource theories.

Abstract

A special class of states of 2-qubits which are simultaneously separable and have positive semidefinite Wigner functions is described.

Paper Structure

This paper contains 16 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Two aspects of nonclassicality in quantum systems.
  3. Classifying states of N--dimensional quantum system
  4. Classifying states into separable vs. entangled sets.
  5. Classifying states according to the sign of Wigner functions.
  6. Double classicality states.
  7. Separable and absolute separable X-states of 2-qubits
  8. Separable states $\mathfrak{S}_X$.
  9. Absolutely separable states.
  10. WF positivity for quatrit and 2-qubits in X-states
  11. WF positivity polytope.
  12. SW kernels of X--states.
  13. The moduli space of quatrit.
  14. SW kernel of 2 qubits.
  15. Concluding remarks
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Left: Tetrahedron $ABCD$--- the simplex of partially ordered eigenvalues satisfying $1>r_1>r_2>0,\, 1>r_3>r_4>0$, while $ABC'D'$ --- the fundamental simplex with $1 \geq r_1 \geq r_2 \geq r_3 \geq r_4 \geq 0\,$. Right: Intersection of absolutely separable states with the fundamental simplex.
  • Figure 2: Left: Moduli space of quatrit vs. pair of qubits; Right: Typical pattern of intersection of a qubit pair Wigner function positivity supporting hyperplane plane (\ref{['eq:WFPosHyper']}) with the fundamental simplex, ($\pi_1=0.94, \pi_2=0.93, \pi_3=0.51$).