Reassessing the boundary between classical and nonclassical for individual quantum processes

TL;DR

The paper introduces a unified notion of classical explainability for individual quantum processes based on generalized noncontextuality, enabling a consistent classical–nonclassical boundary across states, measurements, channels, and more complex processes. Central to the approach are dual processes and factorizing duals, which reduce the verification of classicality to frame representations over operator spaces, yielding necessary and sufficient conditions for a wide range of process types via structure theorems. It shows that all entangled states, incompatible measurements, non-entanglement-breaking channels, and steerable assemblages are nonclassical, while also admitting nonclassical subsets within their classical counterparts (e.g., certain separable states and compatible measurements). A key result is that nonclassicality of arbitrary processes can be inferred from the nonclassicality of associated multipartite states via the Choi isomorphism, flag-convexification, and dephasing arguments, providing a powerful bridge between process-level and state-level analyses. The framework is designed to be robust to noise and applicable to generalized probabilistic theories, with future work outlined on quantification, composition, and resource-theoretic treatments of nonclassical resources.

Abstract

There is a received wisdom about where to draw the boundary between classical and nonclassical for various types of quantum processes. For multipartite states, it is the divide between separable and entangled; for channels, the divide between entanglement-breaking and not; for sets of measurements, the divide between compatible and incompatible; for assemblages, the divide between steerable and unsteerable. However, these choices have not been motivated by any unified notion of what it means to be classically explainable. One well-motivated notion of classical explainability is the one based on generalized noncontextuality: a set of circuits is classically explainable if a generalized-noncontextual ontological model can realize the statistics they generate. In this work, we show that this notion can be leveraged to define a classical-nonclassical divide for individual quantum processes of arbitrary type. We begin the task of characterizing where the classical-nonclassical divide lies according to this proposal for a variety of different types of processes. In particular, we show that all of the following are judged to be nonclassical: every entangled state, every set of incompatible measurements, every non-entanglement-breaking channel, and every steerable assemblage. Our proposal differs from the received wisdom, however, insofar as it also judges certain subsets of the complementary classes to be nonclassical, including certain separable states, compatible sets of measurements, entanglement-breaking channels, and unsteerable assemblages. Finally, we prove structure theorems characterizing the classical-nonclassical divide based on whether a process admits of a specific type of frame representation.

Figures (29)

• Figure 1: Schematic showing the main results of the paper. Yellow: in each section, we study the class of quantum circuits built from the type of target process of interest together with some dual processes (Definition \ref{['def:dualprocess']}) that close the circuit; the conditions for the resulting statistics to be taken to be classically explainable is realizability by a noncontextual ontological model (Definition \ref{['defn:classical']}). Blue: a process is then defined to be classical if the resulting statistics of the set of circuits obtained by contracting it with all possible dual processes are classically explainable (Definition \ref{['def:og_process']} and Definition \ref{['maindefn']}); the classical-nonclassical divide for processes of each type is then characterized by corresponding structure theorems. Green: traditional notions of nonclassicality are recovered as special cases, i.e., sufficient conditions for the proposed notion of nonclassicality.
• Figure 2: Examples of processes with a single quantum output $A$ (top) and with a single quantum input $A$ (bottom) on a single quantum system. Throughout this paper, quantum systems are represented by double wires, whereas classical systems are represented by single wires.
• Figure 3: A prepare-transform-measure circuit (left) and an ontological representation of it (right). The circuit is classically explainable if and only if there exists a linear and diagram-preserving ontological representation taking each quantum system to a classical random variable and each quantum process to a substochastic matrix. The map must, moreover, respect the circuit's topology and reproduce the quantum predictions.
• Figure 4: In the prepare-transform-measure circuit, one can define different composite processes. (a) Boxing the multi-channel and the multi-state defines a new multi-state on $\mathcal{H}^B$, $\tilde{\rho}_{\cdot|zx}^B\coloneqq\mathcal{E}_{\cdot|z}^{B|A}(\rho_{\cdot|x}^A)$; (b) Boxing the multi-channel and the measurement defines a multi-measurement on $\mathcal{H}^A$, $\tilde{M}_{b|z}^A\coloneqq[\mathcal{E}^{B|A}]^{\dagger }_{\cdot|z}(M_{b}^{B})$; (c) More subtly, boxing the (composite) multi-state $\{\tilde{\rho}_{\cdot|zx}^B\}_{zx}$ with the (composite) multi-measurement $\{\{\tilde{M}_{b|z}^A\}_b\}_z$ defines a measure-and-prepare multi-instrument from $\mathcal{H}^A$ to $\mathcal{H}^B$ with $\tilde{\mathcal{E}}^{B|A}_{b"|z'x'z"}(\cdot)\coloneqq\tilde{\rho}_{\cdot|z'x'}^B\textrm{Tr}[\tilde{M}_{b"|z"}^A(\cdot)]$.
• Figure 5: Left: Probing schemes (in blue) that can generate statistics that are not classically explainable independently of the identity of the given process of interest (in yellow). In (a), the process of interest is a multi-source. In (b), the process of interest is a multi-measurement. Right: Treating these same circuits in a manner that coarse-grains over all processes involving systems that are not connected directly to the process of interest leads to probing schemes (in blue) such that whether the statistics can be explained classically now always depends nontrivially on the identity of the process of interest. In (a), the effective process is simply an effect $\bar{M}_A$; in (b), the effective process is simply a state $\bar{\rho}_A$.

Theorems & Definitions (85)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• Definition $3'$
• Definition 5
• Theorem 2
• Remark
• Corollary 1