Cost of Simulating Entanglement in Steering Scenarios

Yujie Zhang; Jiaxuan Zhang; Eric Chitambar

doi:10.22331/q-2025-10-31-1902

Cost of Simulating Entanglement in Steering Scenarios

Yujie Zhang¹, Jiaxuan Zhang¹, and Eric Chitambar²

¹Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
²Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

Published:	2025-10-31, volume 9, page 1902
Editor:	Aaron Goldberg
Eprint:	arXiv:2302.09060v4
Doi:	https://doi.org/10.22331/q-2025-10-31-1902
Citation:	Quantum 9, 1902 (2025).

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Abstract

Quantum entanglement is a fundamental feature of quantum mechanics, yet certain entangled states that are unsteerable can be classically simulated in steering scenarios, making them unable to exhibit quantum steering. Despite their significance, a systematic comparison of such entangled states has not been explored. In this work, we quantify the resource content of unsteerable quantum states in terms of the amount of shared randomness required to simulate the assemblages they generate in the steering scenario. We rigorously demonstrate that the simulation cost is unbounded even for certain unsteerable two-qubit states. Moreover, the simulation cost of entangled two-qubit states is always strictly larger than that for any separable state.
A significant portion of our results rests on the relationship between the simulation cost of two-qubit Werner states and that of noisy spin measurements. Using noisy spin measurements as our central example, we also investigate the minimum number of outcomes a parent measurement requires to simulate a given set of compatible measurements. Although certain continuous measurement families admit a finite-outcome parent measurement, we identify scenarios where the simulation cost is unbounded. Our results establish previously unknown lower bounds and upper bounds on the shared randomness simulation cost, supported by connections between the simulation cost of noisy spin measurements and various geometric inequalities, including ones from the zonotope approximation problem in Banach space theory.

Featured image: Certain entangled states can be reproduced by a local hidden state model. In this picture, the simulation cost is quantified by the cardinality of the hidden variable set Λ, representing how much shared randomness is required to mimic the quantum correlations.

Popular summary

How hard is it to mimic quantum entanglement with a classical model? We address this by quantifying the simulation cost in shared randomness. In contrast to the usual benchmark—how much classical communication is needed—we ask how much shared randomness suffices to reproduce the 'correlation' of entangled states that are nevertheless simulable (i.e., local or unsteerable). Tight bounds on this cost have been missing: existing tools give weak lower bounds and offer no route to upper bounds. We close this gap in the steering scenario by linking the problem to a family of geometric inequalities, including those arising from the zonotope approximation problem in Banach space theory. This connection reveals two key facts: the shared-randomness cost can be unbounded, and it has clear operational meaning—yielding sharp separations between separable and entangled states in the two-qubit case.

► BibTeX data

@article{Zhang2025costofsimulating,
  doi = {10.22331/q-2025-10-31-1902},
  url = {https://doi.org/10.22331/q-2025-10-31-1902},
  title = {Cost of {S}imulating {E}ntanglement in {S}teering {S}cenarios},
  author = {Zhang, Yujie and Zhang, Jiaxuan and Chitambar, Eric},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1902},
  month = oct,
  year = {2025}
}

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[3] Yujie Zhang and Eric Chitambar, "Exact Steering Bound for Two-Qubit Werner States", Physical Review Letters 132 25, 250201 (2024).

[4] Yujie Zhang, Jonah Spodek, David Schmid, Carter Reid, Liam J. Morrison, Thomas Jennewein, Kevin J. Resch, and Robert W. Spekkens, "A new paradigm for entanglement certification using noncontextuality inequalities", arXiv:2507.01122, (2025).

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