Operationally classical simulation of quantum states

TL;DR

This work introduces an operational framework for classical simulations of quantum states by coordinating multiple no-superposition state-preparation devices. It develops both analytical and numerical methods to determine when a quantum set admits a classical model and quantifies the minimal isotropic noise required to simulate the entire quantum state space, revealing how classicality scales with dimension. The authors connect their notion of classicality to joint measurability and EPR steering, and offer practical witnesses and optimization tools to certify absolute coherence or certify classicality in prepare-and-measure scenarios. The results illuminate how and to what extent quantum states defy classical device-based models, with implications for quantum information tasks, cryptography, and high-dimensional quantum technologies.

Abstract

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.

Figures (4)

• Figure 1: Classical models for quantum sets. For a given set of quantum states $\{\rho_x\}_x$, we ask if it be modelled only using classical devices. a) Many independent classical state-preparation devices $\mathcal{P}_1,\mathcal{P}_2,\ldots$ are called stochastically via a probability density function $q(\lambda)$. b) The classicality of $\mathcal{P}_\lambda$ meanse that its emitted states $\{\tau_{x,\lambda}\}_x$ commute, i.e. there exists a basis $\{\ket{e^\lambda_k}\}_k$ in which all $\{\tau_{x,\lambda}\}_x$ are diagonal. This is illustrated for a three-dimensional example.
• Figure 2: Classical models in prepare-and-measure scenarios. $\lambda$ is distributed between Alice and Bob. The existence of a classical model for Alice's preparations allows Bob to know the basis in which her quantum states are diagonal, independently of her input $x$. Then, any quantum correlation statistics $p(b|x,y)$ can be simulated sending a classical message limited to a $d$-valued alphabet.
• Figure 3: Quantum vs classical sets. The sets of quantum states that admit a classical model with complexity $r\in\{1,\ldots,d\}$ are convex and form a nested structure, leading up to the full collection of classical sets $\mathcal{S}$. Commuting sets $\mathcal{C}_r$ in dimension $r$ are represented as the red boundaries in the figure. The white region represents quantum sets that cannot be classically simulated, while the black boundary denotes the space of non-commuting pure states sets.
• Figure 4: Prepare and measure scenario. A set of measurements $\lbrace M_{b|y} \rbrace$, where $y$ denotes the measurement choice and $b$ the outcome, is performed on the set $\lbrace \rho_x \rbrace_x$. The outcome statistics is used to test for whether the set defies classical simulation.

Theorems & Definitions (9)

• Definition 1: Classical models
• Definition 2: Classical simulation complexity
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• Lemma 1