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Peres-type Criterion of Einstein-Podolsky-Rosen Steering for Two Qubits

Yu-Xuan Zhang, Jing-Ling Chen

TL;DR

The paper addresses the lack of a simple spectral criterion for Einstein-Podolsky-Rosen (EPR) steering by constructing a Peres-type criterion based on eigenvalue invariants of the partially transposed state $\rho^{T_B}$. The authors introduce a symmetric form $\mathcal{S}= (1+\mu) \Lambda_4 (1+\tau \Lambda_2)+ \Lambda_3^2$ with $\Lambda_1=1$, and show it is necessary and sufficient for steerability for the two-qubit Werner state, all pure states, and all rank-2 states; higher-rank cases align with known steering inequalities. They derive relations among the eigenvalue sums $\Lambda_k$ (e.g., $\Lambda_1=1$, $\Lambda_2=\nu_1\nu_2$, $\Lambda_3=-(\nu_1-\nu_2)\Omega$, $\Lambda_4=\Gamma-\Omega^2$) and prove key non-positivity properties (e.g., $\Gamma\le 0$ and $\Gamma-\Omega^2$ is tied to concurrence). The results provide a unified spectral framework linking entanglement detection, steering, and Bell nonlocality via partial transposition, with promising implications for theory and experiments. Future work includes optimizing $\mu$, extending to higher dimensions or multipartite settings, and pursuing experimental verification.

Abstract

Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.

Peres-type Criterion of Einstein-Podolsky-Rosen Steering for Two Qubits

TL;DR

The paper addresses the lack of a simple spectral criterion for Einstein-Podolsky-Rosen (EPR) steering by constructing a Peres-type criterion based on eigenvalue invariants of the partially transposed state . The authors introduce a symmetric form with , and show it is necessary and sufficient for steerability for the two-qubit Werner state, all pure states, and all rank-2 states; higher-rank cases align with known steering inequalities. They derive relations among the eigenvalue sums (e.g., , , , ) and prove key non-positivity properties (e.g., and is tied to concurrence). The results provide a unified spectral framework linking entanglement detection, steering, and Bell nonlocality via partial transposition, with promising implications for theory and experiments. Future work includes optimizing , extending to higher dimensions or multipartite settings, and pursuing experimental verification.

Abstract

Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.
Paper Structure (7 sections, 107 equations, 2 figures)

This paper contains 7 sections, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Brief Review of Peres's Criterion
  3. Brief Review of an Alternative Peres-type Criterion
  4. Construction of a New Peres-type Criterion
  5. The Proof for All Two-Qubit Rank-2 States
  6. Examples for Rank-3 and Rank-4 states
  7. Conclusion and Outlook

Figures (2)

  • Figure 1: $\mathcal{S}$ versus $V$ for the fixed value $\alpha=\pi/6$. Here $\mu \approx 0$. For $\alpha=\pi/6$, the criterion (\ref{['eq:marek']}) fails to detect the steerability of $\rho_{GW}$, however, the criterion (\ref{['eq:crite']}) can detect the region of $V\in (0.5844, 1)$.
  • Figure 2: $\mathcal{S}$ versus $\alpha$ for $V=3/[2(1+2\sin^2\alpha)]$. Here $\mu \approx 0$. For $V=3/[2(1+2\sin^2\alpha)]$, the criterion (\ref{['eq:marek']}) fails to detect the steerability of $\rho_{GW}$, however, the criterion (\ref{['eq:crite']}) can still detect the steerability.

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6