Operationally classical simulation of quantum states

A classical state-preparation device cannot generate states in relative superposition. We introduce classical models in which devices that are individually unable to generate states with relative superposition can be stochastically coordinated to simulate sets of quantum states. These models have natural operational interpretation in prepare-and-measure scenarios and they can account for many non-commuting quantum state sets. We develop systematic methods both for classically simulating quantum sets and for showing that no such simulation exists, thereby certifying quantum coherence. In particular, we determine the exact noise rates required to classically simulate the entire state space of quantum theory. We also reveal connections between the operational classicality of sets and the well-known fundamental concepts of joint measurability and Einstein-Podolsky-Rosen steering. Here, we present an avenue to understand how and to what extent quantum states defy generic models based on classical devices, which also has relevant implications for quantum information applications.

💡 Research Summary

The paper introduces a novel operational framework for classically simulating sets of quantum states that are not mutually commuting. The authors start by defining a “classical state‑preparation device” as one that can only emit states diagonal in a single, fixed basis – i.e., all its output states commute. While a single such device can only reproduce commuting quantum states, the authors show that a collection of many such devices, each possibly associated with a different basis, can be coordinated by a shared random variable λ. By sampling λ according to a probability density q(λ) and then, for each desired quantum state ρₓ, selecting an output from the corresponding device with a conditional probability p(z|x,λ), any target state can be expressed as

 ρₓ = ∫ dλ q(λ) τₓ,λ,

where each τₓ,λ is a commuting set of states (diagonal in the basis associated with λ). This definition (Equation 2 in the paper) constitutes a “classical model” for the set {ρₓ}. The key requirement is that λ be independent of the input label x, mirroring the freedom‑of‑choice assumption used in Bell‑type hidden‑variable models.

The authors then explore two complementary problems: (i) given a quantum set, construct a classical model; (ii) certify that no such model exists. For (i) they develop analytical constructions and numerical algorithms based on linear and semidefinite programming. A central result is the exact noise threshold needed to render the entire d‑dimensional quantum state space classically simulable. They prove that adding isotropic white noise with visibility v ≤ 1/(d+1) (for qubits, v ≤ 1/√2) suffices to make every quantum state a convex combination of commuting states drawn from suitably chosen devices. This extends earlier work that only considered pure states or commuting sets.

For (ii) they derive a hierarchy of SDP‑based witnesses that test the impossibility of a decomposition of the form (2). These witnesses are stronger than simple commutation tests and can be applied to experimental data with realistic imperfections. The authors demonstrate that the failure of a classical model in their sense is a necessary (though not sufficient) condition for observing quantum advantages in semi‑device‑independent prepare‑and‑measure scenarios.

A substantial portion of the paper connects the operational model to the prepare‑and‑measure paradigm. If Alice and Bob share the random variable λ, Bob knows the basis in which Alice’s state τₓ,λ is diagonal, independently of Alice’s input x. Consequently, any quantum correlation p(b|x,y)=Tr