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Protection of quantum steering ellipsoids in non-Markovian environments

Wen-Jie Zhang, Jun-Hong An

TL;DR

The paper demonstrates that bound-state formation in a bipartite quantum system coupled to local non-Markovian environments can protect the quantum steering ellipsoid (QSE) geometry and associated steering properties. By solving an exact non-Markovian dynamics with Ohmic-like spectra, the authors show that bound states on both sides preserve nontrivial QSEs and enable two-way EPR steering, while one-sided bound states yield one-way steering, and absent bound states cause complete QSE collapse. The mechanism offers a reservoir-engineering approach to tailor steering resources, potentially enhancing decoherence resilience for quantum communication and information-processing tasks. The results bridge geometric QSE analysis with dynamical control of quantum correlations in open systems, highlighting practical strategies for steering-based technologies.

Abstract

The quantum steering ellipsoid (QSE) provides a geometric representation, within the Bloch picture, of all possible states to which one qubit can be steered through measurements performed on another correlated qubit. However, in most realistic settings, quantum systems are inevitably coupled to their surrounding environment, resulting in decoherence and the consequent degradation of the QSE. Here, by investigating how local dissipative environments coupled separately to each qubit affect the steering properties geometrized by the QSE within an exact non-Markovian framework, we find that the geometry of each party's QSE is closely tied to whether a bound state forms in the energy spectrum of the total qubit-environment system. We systematically examine the characteristics of QSEs under three distinct scenarios: two-sided bound states, one-sided bound states, and no bound state, revealing a diverse range of steering types. Our work establishes quantum reservoir engineering as a tunable strategy for protecting and controlling quantum steering in open systems, offering a practical pathway toward robust steering-based quantum technologies.

Protection of quantum steering ellipsoids in non-Markovian environments

TL;DR

The paper demonstrates that bound-state formation in a bipartite quantum system coupled to local non-Markovian environments can protect the quantum steering ellipsoid (QSE) geometry and associated steering properties. By solving an exact non-Markovian dynamics with Ohmic-like spectra, the authors show that bound states on both sides preserve nontrivial QSEs and enable two-way EPR steering, while one-sided bound states yield one-way steering, and absent bound states cause complete QSE collapse. The mechanism offers a reservoir-engineering approach to tailor steering resources, potentially enhancing decoherence resilience for quantum communication and information-processing tasks. The results bridge geometric QSE analysis with dynamical control of quantum correlations in open systems, highlighting practical strategies for steering-based technologies.

Abstract

The quantum steering ellipsoid (QSE) provides a geometric representation, within the Bloch picture, of all possible states to which one qubit can be steered through measurements performed on another correlated qubit. However, in most realistic settings, quantum systems are inevitably coupled to their surrounding environment, resulting in decoherence and the consequent degradation of the QSE. Here, by investigating how local dissipative environments coupled separately to each qubit affect the steering properties geometrized by the QSE within an exact non-Markovian framework, we find that the geometry of each party's QSE is closely tied to whether a bound state forms in the energy spectrum of the total qubit-environment system. We systematically examine the characteristics of QSEs under three distinct scenarios: two-sided bound states, one-sided bound states, and no bound state, revealing a diverse range of steering types. Our work establishes quantum reservoir engineering as a tunable strategy for protecting and controlling quantum steering in open systems, offering a practical pathway toward robust steering-based quantum technologies.
Paper Structure (6 sections, 15 equations, 4 figures)

This paper contains 6 sections, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. QSE and steering Classification
  3. QSE in noisy environments
  4. numerical results
  5. conclusion
  6. Explicit expressions for $h_A$ and $h_B$

Figures (4)

  • Figure 1: (a) Energy spectrum of each qubit-environment subsystem. (b) Evolution of the QSEs $\mathcal{E}_A$ and $\mathcal{E}_B$ with $p=0.9$ and $\theta=\pi/8$. The solid surfaces depict the numerical evolution, with the analytical steady-state solution overlaid as a golden wireframe on the final ellipsoid. The green tetrahedron represents any tetrahedron inscribed in the Bloch sphere. It cannot fully enclose the steady-state steering ellipsoid. In all plots, we use $\omega_c=20\omega_0$ and $\eta_A=\eta_B=0.06$.
  • Figure 2: (a) Evolution of the concurrence $C$ and the EPR steering witnesses $\Delta S_{AB}$ and $\Delta S_{BA}$ corresponding to the parameter set in Fig. \ref{['FIG1']}(a). Green dashed lines show the analytical results for $\Delta S_{AB}$ and $\Delta S_{BA}$ evaluated from Eqs. \ref{['e10']}. Blue and orange solid curves are the corresponding numerical solutions. The red dashed line denotes the numerical evolution of the $C$. (b) Same as (a) but with $\theta=\pi/12$.
  • Figure 3: (a) Evolution of the QSEs $\mathcal{E}_A$ and $\mathcal{E}_B$ with a bound state on Alice's side ($\eta_A=0.06$) and no bound state on Bob's side ($\eta_B=0.03$). (b) The reversed scenario with the bound state on Bob's side ($\eta_B=0.06$) and no bound state on Alice's side ($\eta_A=0.03$). The solid surfaces depict the numerical evolution, with the analytical steady-state solution overlaid as a golden wireframe on the final QSE. (c) Evolution of the $C$, $\Delta S_{AB}$ and $\Delta S_{BA}$ for the bound-state cases shown in (a). Green dashed lines show the analytical results for $\Delta S_{AB}$ and $\Delta S_{BA}$; blue and orange solid curves are the corresponding numerical solutions. The red dashed line denotes the numerical evolution of the $C$. In all plots, we use $\omega_c=20\omega_0$.
  • Figure 4: Same as Fig \ref{['FIG3']}(a) but with $\eta_A=\eta_B=0.03$, showing the evolution of the QSEs $\mathcal{E}_A$ and $\mathcal{E}_B$ with no bound state on either side.