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Quantum-Kaniadakis entropy as a measure of quantum correlations through implicit bounds

Narayan S Iyer, Shraddha Sharma

TL;DR

This work investigates how the negative values of the quantum Kaniadakis conditional entropy $S^{\\mathcal{K}}_{\\alpha}(A|B)$ relate to the fully entangled fraction $FEF(\\rho_{AB})$ as a predictor of a state's usefulness for teleportation and its steerability. By deriving explicit expressions for $S^{\\mathcal{K}}_{\\alpha}(A|B)$ and $FEF$ for key states (two-qubit Werner and Weyl states, and their $d\\otimes d$ generalizations including isotropic and Werner states), the authors establish implicit bounds on $FEF$ that must hold when $S^{\\mathcal{K}}_{\\alpha}(A|B)<0$, and identify exception regions where these bounds are inconclusive. They further connect this entropic framework to $k$-copy steering, showing that negative $S^{\\mathcal{K}}_{\\alpha}$ implies $k$-copy steerability for isotropic states under projective measurements for certain $(d,\\alpha, F)$ regimes, thereby providing a practical route to assess steerability via $FEF$. The results extend the entropic-characterization toolkit to higher dimensions and link generalized quantum entropy to operational tasks in quantum information processing, offering new criteria to certify non-usefulness for teleportation and to certify steerability via FEF-based bounds.

Abstract

In the present article, we examine the relationship of negative conditional quantum Kaniadakis entropy ($α-$CQKE) with the fully entangled fraction (FEF) which is a substantial yardstick for quantum information processing protocols including teleportation, and quantum steerability, executed over four vital quantum states with maximally mixed marginals, the 2-qubit Werner state, the 2-qubit Weyl state, the 2-qudit Werner state and the isotropic state. We initiate our analysis in 2$\otimes$2 systems where we derive implicit bounds on FEF when the $α-$CQKE takes negative values, i.e. when $α-$CQKE $\in$ $R^{-}$ for 2-qubit Werner state. Consequently, we derive the sufficient implicit bounds for a definitive claim on the non-usefulness of Werner state for quantum teleportation provided its visibility parameter succeeds to elude a critical region, the exception region 1, where the situation becomes inconclusive. Subsequently, we replicate the same for the 2-qubit Weyl state with some constraints augmented by an analogous exception region 2 and the correlation tensor matrix elements. Furthermore, we extend our investigation to d $\otimes$ d states, commencing our analysis with the Isotropic state. We derive implicit bounds on FEF of the Isotropic state and the 2-qudit Werner state resembling the ones in the 2$\otimes$2 analysis. Additionally, we utilize the convoluted relationship between the FEF and quantum steerability to formulate propositions linking negative $α-$CQKE to the k-copy steerability of isotropic states for projective measurements, thereby reducing the intricacy of the study of k-copy steerability directly via FEF. In the appendix section of the article, we provide corroborative calculations and supplementary materials to the theorems presented in the main sections.

Quantum-Kaniadakis entropy as a measure of quantum correlations through implicit bounds

TL;DR

This work investigates how the negative values of the quantum Kaniadakis conditional entropy relate to the fully entangled fraction as a predictor of a state's usefulness for teleportation and its steerability. By deriving explicit expressions for and for key states (two-qubit Werner and Weyl states, and their generalizations including isotropic and Werner states), the authors establish implicit bounds on that must hold when , and identify exception regions where these bounds are inconclusive. They further connect this entropic framework to -copy steering, showing that negative implies -copy steerability for isotropic states under projective measurements for certain regimes, thereby providing a practical route to assess steerability via . The results extend the entropic-characterization toolkit to higher dimensions and link generalized quantum entropy to operational tasks in quantum information processing, offering new criteria to certify non-usefulness for teleportation and to certify steerability via FEF-based bounds.

Abstract

In the present article, we examine the relationship of negative conditional quantum Kaniadakis entropy (CQKE) with the fully entangled fraction (FEF) which is a substantial yardstick for quantum information processing protocols including teleportation, and quantum steerability, executed over four vital quantum states with maximally mixed marginals, the 2-qubit Werner state, the 2-qubit Weyl state, the 2-qudit Werner state and the isotropic state. We initiate our analysis in 22 systems where we derive implicit bounds on FEF when the CQKE takes negative values, i.e. when CQKE for 2-qubit Werner state. Consequently, we derive the sufficient implicit bounds for a definitive claim on the non-usefulness of Werner state for quantum teleportation provided its visibility parameter succeeds to elude a critical region, the exception region 1, where the situation becomes inconclusive. Subsequently, we replicate the same for the 2-qubit Weyl state with some constraints augmented by an analogous exception region 2 and the correlation tensor matrix elements. Furthermore, we extend our investigation to d d states, commencing our analysis with the Isotropic state. We derive implicit bounds on FEF of the Isotropic state and the 2-qudit Werner state resembling the ones in the 22 analysis. Additionally, we utilize the convoluted relationship between the FEF and quantum steerability to formulate propositions linking negative CQKE to the k-copy steerability of isotropic states for projective measurements, thereby reducing the intricacy of the study of k-copy steerability directly via FEF. In the appendix section of the article, we provide corroborative calculations and supplementary materials to the theorems presented in the main sections.

Paper Structure

This paper contains 36 sections, 8 theorems, 86 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation And Preliminaries
  3. Entanglement, Separability and Bloch-Fano Decomposition
  4. Quantum Entropies
  5. Kaniadakis statistics and Quantum Kaniadakis entropy
  6. Quantum (Kaniadakis) Mutual Information
  7. FEF
  8. Quantum Nonlocality: Bell Nonlocality and Quantum Steering 27
  9. k-copy nonlocality
  10. k-copy steering
  11. $\alpha$-CQKE and FEF For Two-Qubit States
  12. Werner states
  13. Exception Region 1
  14. Weyl state
  15. Exception Region(s) 2
  16. ...and 21 more sections

Key Result

Theorem 1

If the $\alpha$-CQKE of a two-qubit Werner state is negative, then the FEF must satisfy the implicit bounds induced by the inequality where $\delta = min \{\delta_{1},\delta_{2}\}$, $\delta_{i}$ are eigenvalue of $\rho_{2}^{wer}$, the multiplicity of eigenvalue $\delta_2$ is three and $\hat{K}_{\alpha} (x) = x^{1-\alpha} - x^{1+\alpha}$.

Figures (5)

  • Figure 1: $(a)$ Plot of $\hat{K}_{\alpha}(x)$. The plots $(b)$ and $(c)$ correspond to the cases of $\hat{g}(\alpha)> 1/4$ (case 1, section \ref{['constr_excep_reg_1']}) and $\hat{g}(\alpha)<1/4$ (case 2, \ref{['constr_excep_reg_1']}) to facilitate the computations regarding the construction of exception region 1 detailed in section \ref{['constr_excep_reg_1']}.
  • Figure 2: Plots of ${S_{\alpha}^{\mathcal{K}}(A|B)}_{\rho_2^{iso}}$ vs $FEF(\rho_2^{iso})$ as a supplementary material for proposition \ref{['prop_7']} in section \ref{['sec_5']}. We consider the $\alpha$ values $0^{+},~0.1,~0.3,~0.5,~0.75$ in order to portray that $FEF(\rho_2^{iso})$ is lower bounded by 0.81 regardless of value of $\alpha \in (0,1).$
  • Figure 3: Plots of ${S_{\alpha}^{\mathcal{K}}(A|B)}_{\rho_6^{iso}}$ vs $FEF(\rho_6^{iso})$ as a supplementary material for proposition \ref{['prop_8']} observation 1 in section \ref{['sec_5']}. We consider the $\alpha$ values $0^{+},~0.1,~0.3,~0.5,~0.75$ in order to portray that $FEF(\rho_6^{iso})$ is lower bounded by 0.66 regardless of value of $\alpha \in (0,1).$
  • Figure 4: Plots of ${S_{\alpha}^{\mathcal{K}}(A|B)}_{\rho_d^{iso}}$ vs $FEF(\rho_d^{iso})$ as a supplementary material for proposition \ref{['prop_8']} observation 2 in section \ref{['sec_5']}. We consider 2 values of $\alpha = 0.1,~0.5$ and 3 values of $d=6,~7,~8$ for our analysis.
  • Figure 5: Plot of ${S_{0^+}^{\mathcal{K}}(A|B)}_{\rho_{\infty}^{iso}}$ vs $FEF(\rho_{\infty}^{iso})$ as a supplementary material for proposition \ref{['prop_8']} observation 3 in section \ref{['sec_5']}. It clearly highlights the fact that $\lim_{(d,\alpha)\rightarrow(\alpha,0^{+})}S_{\alpha}^{\mathcal{K}}(\rho_d^{iso}) = 0.506 \approx 0.51$.

Theorems & Definitions (19)

  • Definition 1: von-Neumann Entropy
  • Definition 2: Conditional von-Neumann Entropy
  • Definition 3: Quantum Kaniadakis $\alpha-$entropy
  • Definition 4: Conditional quantum Kaniadakis $\alpha-$entropy
  • Definition 5: Quantum Kaniadakis mutual information
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 9 more