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arXiv:2605.09558

Classical Limit: Dissipation of Spekkens' Generalised Contextuality under Decoherence

Enrico Bozzetto, Jonte R. Hance

quant-phphysics.hist-phSubmitted May 10, 2026• Announced May 12, 2026• 0 citations• Revised Sep 22, 2026 (v3)
v1 · May 10, 2026v2 · May 13, 2026v3 · Sep 22, 2026
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Abstract

When does a decohering quantum system become classical? Generalised noncontextuality gives one criterion, for scenarios rich enough in preparations: a system is classical once its statistics admit an underlying model in which operationally indistinguishable procedures are physically identical. Where it crosses depends not on the state and channel alone but on which preparations, transformations and measurements the experiment offers, and if every measurement is allowed there is no crossing. For any pairwise unbiased bases in any dimension, given preparations holding their eigenstates and surrounding the maximally mixed state, the question reduces to a finite linear program; for a qudit of odd prime dimension $d$ restricted to the stabiliser subtheory, with one depolarised magic state, we solve it in closed form. With preparations and measurements alone the qutrit strange state is classical at zero noise; with the Clifford gates added, each channel represented once, it crosses at $3/5$; and with a random gate represented, as Spekkens' treatment asks, by the matching mixture of stochastic maps, at $3/4$. A Clifford orbit of assignments supports such a representation only if it has $d^2$ members, leaving, in every odd prime dimension, only Gross's discrete Wigner function: the crossing sits where it stops being negative.

Comments
24+11 pages, 2 figures. Erroneous KD/Wigner claims fixed, three subtheory cases differentiated, numerics added

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When does a decohering quantum system become classical? Generalised noncontextuality gives one criterion, for scenarios rich enough in preparations: a system is classical once its statistics admit an underlying model in which operationally indistinguishable procedures are physically identical. Where it crosses depends not on the state and channel alone but on which preparations, transformations and measurements the experiment offers, and if every measurement is allowed there is no crossing. For any pairwise unbiased bases in any dimension, given preparations holding their eigenstates and surrounding the maximally mixed state, the question reduces to a finite linear program; for a qudit of odd prime dimension ddd restricted to the stabiliser subtheory, with one depolarised magic state, we solve it in closed form. With preparations and measurements alone the qutrit strange state is classical at zero noise; with the Clifford gates added, each channel represented once, it crosses at 3/53/53/5; and with a random gate represented, as Spekkens' treatment asks, by the matching mixture of stochastic maps, at 3/43/43/4. A Clifford orbit of assignments supports such a representation only if it has d2d^2d2 members, leaving, in every odd prime dimension, only Gross's discrete Wigner function: the crossing sits where it stops being negative.
The emergence of classicality is fundamentally driven by the interaction between a quantum system and its environment. Foundational open-system approaches, notably the Caldeira-Leggett model, successfully captured how these interactions lead to macroscopic effects like quantum dissipation and decoherence. However, these approaches often leave the precise definitions of classicality and quantumness ambiguous. In quantum information theory, this boundary is a heavily scrutinized question, and Kochen-Specker contextuality emerges as a hallmark of nonclassicality. It is therefore natural to investigate whether decoherence can actually suppress this property. Taking this path creates an apparent conundrum, once there exist two distinct manifestations of quantum contextuality: state-dependent and state-independent ones. While state-dependent contextuality naturally vanishes under state degradation, state-independent contextuality could persist for any quantum state, since it shows up even for the maximally mixed state! In this paper, we resolve this apparent paradox by analyzing sequential measurement implementations of the paradigmatic Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) and Peres-Mermin prepare-and-measure scenarios under the influence of depolarizing channels. By introducing noise both prior to and in between measurements, and by analyzing the resulting sequential correlators in both the Schrödinger and Heisenberg pictures, we show how open-system dynamics suppress the correlations required to witness contextuality, leading to classicalization.
Abstract The emergence of classicality is fundamentally driven by the interaction between a quantum system and its environment. Foundational open-system approaches, notably the Caldeira-Leggett model, successfully captured how these interactions lead to macroscopic effects like quantum dissipation and decoherence. However, these approaches often leave the precise definitions of classicality and quantumness ambiguous. In quantum information theory, this boundary is a heavily scrutinized question, and Kochen-Specker contextuality emerges as a hallmark of nonclassicality. It is therefore natural to investigate whether decoherence can actually suppress this property. Taking this path creates an apparent conundrum, once there exist two distinct manifestations of quantum contextuality: state-dependent and state-independent ones. While state-dependent contextuality naturally vanishes under state degradation, state-independent contextuality could persist for any quantum state, since it shows up even for the maximally mixed state! In this paper, we resolve this apparent paradox by analyzing sequential measurement implementations of the paradigmatic Klyachko, Can, Binicioğlu, and Shumovsky (KCBS) and Peres-Mermin prepare-and-measure scenarios under the influence of depolarizing channels. By introducing depolarizing noise both prior to and in between measurements, and by analyzing the resulting sequential correlators in both the Schrödinger and Heisenberg pictures, we show how open-system dynamics suppress the correlations required to witness contextuality, leading to classicalization.
Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we show how, for a certain class of quantum systems, the quantum description of an experiment can be decomposed into classical descriptions that are epistemically restricted, though in a different sense than in Spekkens' work. For each aspect of the experiment—the preparation, the transformations, and the measurement—the epistemic restriction limits the form of the probability distribution an imagined classical observer may use. There are also global constraints that the whole collection of classical descriptions must satisfy. Each classical description generates its own prediction regarding the outcome of the experiment. One recovers the quantum prediction via a simple but highly nonclassical rule: the "nonrandom part" of the predicted quantum probabilities is obtained by summing the nonrandom parts of the classically predicted probabilities. By "nonrandom part" we mean the deviation from complete randomness, that is, from what one would expect upon measuring the fully mixed state.
We show that, under certain conditions, generalised contextuality, in the sense introduced by Spekkens, cannot be created from the vacuum of a quantum field. Initial noncontextual qudit states do not become contextual just by interacting with a quantum field. This is verified in two scenarios. First, in a perturbative approach where the time evolution operator is expanded in powers of the coupling constant. Second, in the limit of zero energy gap in the spectrum of the free detector Hamiltonian. Our results are at the opposite side of what has been recently obtained in the literature regarding the harvesting of Kochen-Specker contextuality. Our results demonstrate that the existing notions of contextuality can be independent of one another and that contextuality harvesting depends on which precise notion is used.
Spekkens has introduced an epistemically restricted classical theory of discrete systems, based on discrete phase space. The theory manifests a number of quantum-like properties but cannot fully imitate quantum theory because it is noncontextual. In this paper we start with a theory resembling Spekkens's model: we imagine a classical observer describing an experiment in terms of discrete phase space, and for each aspect of the experiment -- the state preparation, the transformations, and the measurement -- there is an epistemic restriction limiting the form of the probability distribution the classical observer may use. We show that for a certain class of systems, one can obtain the correct quantum mechanical prediction for the experiment by combining the (typically weaker) predictions of all the conceivable classical observers. The rule for effecting this combination is highly nonclassical: the ``nonrandom part of the quantum prediction is obtained by summing the ``nonrandom parts of the classical predictions.
Coherent errors leave stabiliser-code-encoded magic states in superpositions of syndrome sectors. We ask how much magic these coherent errors add, and whether syndrome extraction depletes logical magic. Using Robustness of Magic, we show that, for a unitary error U on a correctable region which supports no stabiliser, the encoded state has the same robustness as the logical state plus one copy of the Choi state of U , in every stabiliser code encoding. Syndrome extraction removes this copy, returning the logical robustness. The removed magic is positive for every non-Clifford error: adding a non-stabiliser state to any state multiplies one plus the robustness by a (tightly- bounded) factor above one. The magic capacity of two-qubit gates Clifford-equivalent to diagonal or single-qubit gates is equal to their Choi-state robustness. We give a linear-algebraic test for when a Clifford error makes syndrome extraction measure a logical operator, and give magic lifetimes of stored single-qubit magic states under Pauli noise. Choosing the plane of the stored state extends these lifetimes about 1.5x on rotated surface codes under low code-capacity depolarising noise. We identify states and channels in odd-prime dimension for which Wigner negativity and robustness show equal lifetimes.