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Cost of Simulating Entanglement in Steering Scenario

Yujie Zhang, Jiaxuan Zhang, Eric Chitambar

TL;DR

The paper introduces the classical simulation cost $\gamma(\rho_{AB})$ for steering scenarios and proves that this cost can be unbounded for some unsteerable entangled states, while entangled two-qubit states have strictly larger costs than separable ones. It builds a tight bridge between steering, measurement incompatibility, and zonotope geometry by examining Werner states and noisy spin measurements, deriving both lower and upper bounds on $\gamma$ and its planar variant $\gamma^p$. Key results show $\gamma(\rho_{AB})>4$ iff the state is entangled for two-qubit systems, with explicit thresholds and asymptotic rates such as $\gamma(\omega_r)\ge c(\tfrac{1}{2}-r)^{-2/5}$ and $\gamma^p(\omega_r)\ge c'(\tfrac{2}{\pi}-r)^{-1/2}$ near steering-compatibility boundaries. The work leverages zonotope-approximation theorems to connect abstract resource costs with concrete geometric structures, yielding both lower and upper bounds (including planar bounds) and revealing a separability-entanglement boundary encoded in simulation cost. These insights offer a resource-based, semi-device-independent lens on entanglement and nonlocality, with practical implications for entanglement verification under limited shared randomness and potential extensions to higher-dimensional systems and LHV scenarios.

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.

Cost of Simulating Entanglement in Steering Scenario

TL;DR

The paper introduces the classical simulation cost for steering scenarios and proves that this cost can be unbounded for some unsteerable entangled states, while entangled two-qubit states have strictly larger costs than separable ones. It builds a tight bridge between steering, measurement incompatibility, and zonotope geometry by examining Werner states and noisy spin measurements, deriving both lower and upper bounds on and its planar variant . Key results show iff the state is entangled for two-qubit systems, with explicit thresholds and asymptotic rates such as and near steering-compatibility boundaries. The work leverages zonotope-approximation theorems to connect abstract resource costs with concrete geometric structures, yielding both lower and upper bounds (including planar bounds) and revealing a separability-entanglement boundary encoded in simulation cost. These insights offer a resource-based, semi-device-independent lens on entanglement and nonlocality, with practical implications for entanglement verification under limited shared randomness and potential extensions to higher-dimensional systems and LHV scenarios.

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.
Paper Structure (18 sections, 31 theorems, 131 equations, 3 figures, 5 tables)

This paper contains 18 sections, 31 theorems, 131 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Simulation Cost of Werner States and Noisy Spin Measurements
  4. Compatibility Radius and Its Geometric Characterization
  5. Lower Bound on Simulation Cost
  6. Upper bound on the simulation cost
  7. Qubit planar measurements
  8. General qubit measurements
  9. Simulating Cost of General Unsteerable States
  10. Geometry of compatible region
  11. Proof of proposition \ref{['Prop:Main']}, and Corollary \ref{['Cor: Main']}
  12. Proof of Theorem \ref{['thm:SC lowerbound']}, Theorem \ref{['thm: zonotope approx']} and connection to Zonotope approximation
  13. Criterion for compatible radius $R(\{ \Pi_{\lambda}\})$
  14. Example of $R(\{ \Pi_{\lambda}\})$ for symmetric POVMs, and compatible models
  15. Simulation cost of entangled state and compatible measurements
  16. ...and 3 more sections

Key Result

Lemma 1

$\mathcal{P}_{r}$ can be simulated with an $n$-element POVM if and only if the simulation cost of Werner states $\omega_r$ under projective measurements equals $n$. $\gamma_{\rm PM}(\omega_r) = N(r)$.

Figures (3)

  • Figure 1: The bipartite state $\rho_{AB}$ induces a c-to-cq channel represented by the assemblage $\mathcal{A}=\{\sigma_{a|x}\}$.
  • Figure 2: Schematic showing the connections between different quantities in this paper: (a) Left: Relation between simulation cost of Werner states $\gamma(\omega_r)$ and simulation cost (compatible complexity) $N(r)$ for the set of all noisy spin measurements ; (b) Right: Compatible radius for noisy spin measurements $R(n)$ and its derivatives, i.e., various lower and upper bounds; (c) The left and right parts are connected by the relation in the middle box (see Eq. \ref{['Eq:compatibility-radius-complexity']}); (d) $\rightarrow$(dashed): Upper bound from left to right (in special cases).
  • Figure 3: Compatible region for Platonic solid with $n=4,6,8,12$. From left to right: Octahedron, Cube, Rhombic dodecahedron and Rhombic triacontahedron .

Theorems & Definitions (58)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Proposition 1
  • Conjecture 1
  • Proposition 2
  • Definition 3
  • Proposition 3
  • proof
  • Corollary 1
  • ...and 48 more