Witness High-Dimensional Quantum Steering via Majorization Lattice

Abstract

Quantum steering enables one party to influence another remote quantum state by local measurement. While steering is fundamental to many quantum information tasks, the existing detection methods in the literature are mainly constrained to either specific measurement scenario or low-dimensional systems. In this work, we propose a majorization lattice framework for steering detection, which is capable of exploring the steering in arbitrary dimension and measurement setting. Steering inequalities for two-qubit states, high-dimensional Werner states and isotropic states are obtained, which set even stringent bars than what has been reached yet. Notably, the known high-dimensional results turn out to be some kind of approximate limits of the new approach.

Figures (4)

• Figure 1: The steering thresholds of the isotropic states for different dimensions $d$, in case of a complete set of MUBs. The red and green lines represent the steering threshold derived from the approximate upper bounds $\bar{\Theta}_N$ and $\bar{\Gamma}_N$, respectively. The cyan line indicates the optimal steering threshold from $\bar{\Omega}_N$, while the magenta line represents the critical value of isotropic states derived from the LHS model werner89wiseman07almeida07. Note that, due to the existence problem of $6$-dimensional MUBs, the threshold for $d=6$ is calculated using only three MUBs.
• Figure 1: Infinite measurement settings. Measurements are performed for the whole hemisphere i.e., the all $\vec{b}_\mu$ with $\forall \theta_\mu\in[0,\pi/2],\phi_\mu\in[0,2\pi]$.
• Figure 2: The steering threshold value of the qutrit isotropic and Werner states for $N$ measurement settings. Here, we plot the steering threshold of qutrit isotropic states (a) and Werner states (b) obtained from the Cross-Entropy Method (CEM) for $N=2$ to $13$ and $N=2$ to $8$, respectively. The red points represent the critical values $5/12$ and $2/3$ of qutrit isotropic and Werner states from LHS model werner89wiseman07almeida07.
• Figure 3: Lorenz curves for $\rho(\lambda,\theta,\phi)$ with Schmidt rank $1(\theta=0,\phi=\pi/2)$, $2(\theta=\pi/4,\phi=\pi/2)$ and $3(\theta=\phi=\pi/3,\theta=\pi/4,\phi=\arctan\sqrt{2})$ for MUBs and non-MUBs cases. The black solid line represents the majorization bound $\vec{\omega}(\mathcal{B})$. The shadow region above the majorization bound curve indicates the steerable region detected by \ref{['eq:majo_steering_ineq_gene']}.

Theorems & Definitions (7)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• proof
• Lemma 4
• proof