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One pure steered state implies Einstein-Podolsky-Rosen steering for two-qubit entangled states

  • Qiu-Cheng Song
  • Joonwoo Bae
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Abstract

In this work, we show that a two-qubit entangled state admitting at least one pure steered state is Einstein-Podolsky-Rosen (EPR) steerable from Alice to Bob. Pure steered states signifies that the quantum steering ellipsoid of Bob is tangent to his Bloch sphere at least at a single point. Furthermore, we prove that for a two-qubit entangled state, Bob's quantum steering ellipsoid is tangent to his Bloch sphere at exactly N points, for N∈ { 0,1,2,∞}, if and only if Alice's quantum steering ellipsoid is tangent to her Bloch sphere at exactly N points. For any two-qubit entangled state, therefore, if one party can steer the other to at least one pure state, the state is two-way EPR steerable. We also present several illuminating examples of two-qubit entangled states such that the EPR steering can be verified in terms of pure steered states. Our result addresses the Gisin theorem in a EPR steering scenario: at least a single pure steered state implies two-way steering.

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Cite

BibTeXRIS
BibTeX
@misc{song2026pure,
  title = {{One pure steered state implies Einstein-Podolsky-Rosen steering for two-qubit entangled states}},
  author = {Qiu-Cheng Song and Joonwoo Bae},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.18243},
  eprint = {2605.18243},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.18243},
}
RIS
TY  - UNPB
TI  - One pure steered state implies Einstein-Podolsky-Rosen steering for two-qubit entangled states
AU  - Qiu-Cheng Song
AU  - Joonwoo Bae
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2605.18243
UR  - https://arxiv.org/abs/2605.18243
AB  - In this work, we show that a two-qubit entangled state admitting at least one pure steered state is Einstein-Podolsky-Rosen (EPR) steerable from Alice to Bob. Pure steered states signifies that the quantum steering ellipsoid of Bob is tangent to his Bloch sphere at least at a single point. Furthermore, we prove that for a two-qubit entangled state, Bob's quantum steering ellipsoid is tangent to his Bloch sphere at exactly N points, for N∈ { 0,1,2,∞}, if and only if Alice's quantum steering ellipsoid is tangent to her Bloch sphere at exactly N points. For any two-qubit entangled state, therefore, if one party can steer the other to at least one pure state, the state is two-way EPR steerable. We also present several illuminating examples of two-qubit entangled states such that the EPR steering can be verified in terms of pure steered states. Our result addresses the Gisin theorem in a EPR steering scenario: at least a single pure steered state implies two-way steering.
ER  -