Cost of Simulating Entanglement in Steering Scenario

TL;DR

This paper investigates the classical simulation costs of unsteerable entangled quantum states, revealing unbounded resource requirements for certain states and establishing bounds using geometric inequalities.

Contribution

It provides a systematic analysis of the resource cost for simulating unsteerable quantum states and links this to geometric inequalities, advancing understanding of quantum steering simulation.

Findings

Simulation cost is unbounded for some unsteerable two-qubit states.

Entangled two-qubit states require more shared randomness than separable states for simulation.

Connections between simulation costs and geometric inequalities are established.

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.