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arXiv:2610.00274

Frame dependence of Kochen-Specker contextuality for relativistic spin systems

Jonte R. Hance, Ruben Campos Delgado

quant-phphysics.opticsSubmitted Sep 24, 2026• Announced Sep 25, 2026• 0 citations
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Abstract

We ask whether Kochen-Specker contextuality is Lorentz invariant. On the full Hilbert space, it is, since boosts act unitarily. Restricted to spin, it is not: momentum acts as an environment applying momentum-dependent Wigner rotations, leaving spin observables unsharp and breaking the operator identities behind state-independent contextuality. For spin $1/2$ one scalar governs the loss, the packet average $Δ$ of $\sin^2(θ_W/2)$ (where $θ_W$ is the Wigner angle); for any spin, a ray degrades at a rate set by its spin variance transverse to the boost. In the spin-only account, the Free Will Theorem's SPIN axiom fails for every $Δ>0$; the Yu-Oh set loses contextuality between $Δ\approx0.008$ and $0.016$, with state independence going first; the Peres-Mermin square loses contextuality at $Δ\approx0.059$ or $0.074$ by orientation; singlet CHSH loses nonlocality for $Δ\geq0.159$; and a GHZ-state OR gate loses Raussendorf's contextuality certificate at $Δ\approx0.206$. A $q$-plate implements the channel exactly on photon polarisation, with a retardation in place of the rapidity, so the nonlocality, GHZ and aimed-source results are testable without motion.

Comments
15+3 pages, 10 figures

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We ask whether Kochen-Specker contextuality is Lorentz invariant. On the full Hilbert space, it is, since boosts act unitarily. Restricted to spin, it is not: momentum acts as an environment applying momentum-dependent Wigner rotations, leaving spin observables unsharp and breaking the operator identities behind state-independent contextuality. For spin 1/21/21/2 one scalar governs the loss, the packet average ΔΔΔ of sin⁡2(θW/2)\sin^2(θ_W/2)sin2(θW​/2) (where θWθ_WθW​ is the Wigner angle); for any spin, a ray degrades at a rate set by its spin variance transverse to the boost. In the spin-only account, the Free Will Theorem's SPIN axiom fails for every Δ>0Δ>0Δ>0; the Yu-Oh set loses contextuality between Δ≈0.008Δ\approx0.008Δ≈0.008 and 0.0160.0160.016, with state independence going first; the Peres-Mermin square loses contextuality at Δ≈0.059Δ\approx0.059Δ≈0.059 or 0.0740.0740.074 by orientation; singlet CHSH loses nonlocality for Δ≥0.159Δ\geq0.159Δ≥0.159; and a GHZ-state OR gate loses Raussendorf's contextuality certificate at Δ≈0.206Δ\approx0.206Δ≈0.206. A qqq-plate implements the channel exactly on photon polarisation, with a retardation in place of the rapidity, so the nonlocality, GHZ and aimed-source results are testable without motion.
We show that the operational definition of contextuality introduced by Spekkens is, in general, not Lorentz invariant. Specifically, we consider an explicit example with particle states consisting of both spin and momentum, we apply a Lorentz transformation to obtain the states in a new inertial frame, and then trace out the momentum degrees of freedom in both frames. We find that, while an observer in the first inertial frame describes a contextual ontological model with respect to spin states and all possible spin measurements, an observer in the boosted frame describes a noncontextual model with respect to the transformed spin states and all transformed spin measurements. Hence, the Spekkens' notion of contextuality, when restricted to spin degress of freedom only, is a frame-dependent concept. We apply our results to predict a relativistic effect concerning the task of discriminating between two quantum states. We show that the probability of success for a moving observer exceeds that of an observer at rest.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the structural essence of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often remains unclear. This issue already looms large at the level of the Bell-Kochen-Specker theorem: while traditional proofs proceed by showing the nonexistence of valuations, the notion of state-independent contextuality in the marginal approach allows to prove contextuality from seemingly weaker assumptions. In the light of this, and at the absence of a unified mathematical framework for Kochen-Specker contextuality, the original algebraic approach has been widely abandoned, in favour of the study of contextual correlations. Here, we reinstate the algebraic perspective on contextuality. Concretely, by building on the novel concept of context connections, we reformulate the algebraic relations between observables originally postulated by Kochen and Specker, and we explicitly demonstrate their consistency with the notion of state-independent contextuality. In the present paper, we focus on the new conceptual ideas and discuss them in the concrete setting of spin-1 observables, specifically those in the example of [S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012)]; in a companion paper, we generalise these ideas, obtain a complete characterisation of Kochen-Specker contextuality and provide a detailed comparison with the related notions of contextuality in the marginal and graph-theoretic approach.
Contextuality provides one of the fundamental characterizations of quantum phenomena, and can be used as a resource in lots of quantum information processing. In this paper, we summarize and derive some equivalent noncontextual inequalities from different noncontextual models of the proofs for the Kochen-Specker theorem based on Greenberger-Horne-Zeilinger states. These noncontextual inequalities are equivalent up to some correlation items which hold both for noncontextual hidden variable theories and quantum mechanics. Therefore, using single-photon hyperentangled Greenberger-Horne-Zeilinger states encoded by spin, path, and orbital angular momentum, we experimentally verify several state-dependent noncontextual models of the proofs for the Kochen-Specker theorem by testing an extreme simplest Mermin-like inequality.
A Lorentz boost acts on the canonical spin of a massive particle through a momentum-dependent Wigner rotation. We show that, for one fixed observer boost, reducing over an uncertain momentum can strictly contract every pairwise spin-state trace distance without producing a channel that is degradable from the less contracted one. For spin 1/21/21/2, we first characterize the exact inversion-symmetric channel cone generated by a fixed Wigner angle and transverse momentum directions. Inside this cone lies the Pauli family Mα=diag⁡(1−α,1−α,1−2α)M_α=\operatorname{diag}(1-α,1-α,1-2α)Mα​=diag(1−α,1−α,1−2α), 0≤α<1/20\leqα<1/20≤α<1/2. For 0<α<β<1/20<α<β<1/20<α<β<1/2, all trace distances between distinct spin states are strictly smaller after MβM_βMβ​ than after MαM_αMα​, yet the unique linear post-processing factor has a negative normalized Choi eigenvalue. We solve the optimization over all physical converters exactly: 12inf⁡Λ∈CPTP∥Φβ−Λ∘Φα∥⋄=α(β−α)2−3α\frac{1}{2}\inf_{Λ\in\mathrm{CPTP}}\|Φ_β-Λ\circΦ_α\|_\diamond=\frac{α(β-α)}{2-3α}21​infΛ∈CPTP​∥Φβ​−Λ∘Φα​∥⋄​=2−3αα(β−α)​, whereas the reverse deficiency is β−αβ-αβ−α. Thus the identity dominates the family, while all positive-noise members are pairwise incomparable under CPTP post-processing. The ideal construction is realized as the narrow-packet limit of pure, normalizable five-component momentum states, and explicit perturbation and finite-shot tomography bounds certify an open set of examples. Separately, every nonidentity member fails embedding in a time-homogeneous Pauli-diagonal Lindblad semigroup. Hence ordering all unassisted spin distinguishabilities does not determine the quantum statistical post-processing order.
The Kochen-Specker theorem, Bell inequalities, and several other tests that were designed to rule out hidden-variable theories, assume the existence of observables having infinitely sharp eigenvalues. A paradigmatic example is spin-1/2. It is measured with a Stern-Gerlach array whose outputs are divided into two classes, spin-up and spin-down, in correspondence to the two spots observed on a detection screen. The spot's finite size is attributed to imperfections of the measuring device. This assumption turns the experimental output into a dichotomic, discrete one, thereby allowing the assignment of each spot to an infinitely sharp eigenvalue. Alternatively, one can assume that the spot's finite size stems from eigenvalues spanning a continuous range. Can we disprove such an assumption? Can we rule out hidden-variable theories that reproduce quantum predictions by assuming that, e.g., the electron's magnetic moment is not exactly the same for all electrons? We address these questions by focusing on the Peres-Mermin version of the Bell-Kochen-Specker theorem. It is shown that the assumption of unsharp eigenvalues precludes ruling out non-contextual hidden-variable theories and hence quantum contextuality does not arise. Analogous results hold for Bell-like inequalities. This represents a new loophole that spoils several fundamental tests of quantum mechanics and issues the challenge to close it.
At eleven parameters a frame is a body with a momentum. The quantum step of the theory makes that momentum an operator, and a frame body in a superposition of momenta is a quantum reference frame. This letter asks what using such a frame does to it, in a model with one assumption. An ideal meter comoving with the frame body reads the canonical spins of test bodies, and nothing else. Relative to the frame each spin is turned by the Wigner rotation of the composition law, by an angle that depends on the frame's own rapidity. So every spin the frame serves is a weak measurement of the frame's velocity. The quantum Fisher information per spin is the square of the derivative of the Wigner angle, in closed form. For a frame at rest it is tanh 2 (θt/2), θt the test body's rapidity, and it is bounded by 1/ cosh 2 θF . With polarised spins and the records kept, n spins sharpen the frame's rapidity, to the van Trees bound 1/(σ -2 + n F ) within one percent. With the records discarded, or with unpolarised spins, the frame's momentum coherence decays by the overlap of the two rotated spin states per spin. For a Gaussian frame its purity is 1/ 1 + N F σ 2 , equal to the purity of the spins relative to the frame. The origin of the frame is the conjugate reference. Each discarded spin kicks the frame's position by ±ℏφ ′ /2M , a random walk, and the kept records widen it by the Heisenberg bound. The velocity reference is kept or sharpened while the origin degrades, the two being the conjugate blocks of the eleven. The frame of an entangled pair with a sharp total momentum has nothing to learn and no origin to lose. The sixteen of a body being the operators of two qubits, a boost is a coupling that is not a gate, and its cost in a simulation is stated. The statements are checked numerically.