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Frame dependence of Spekkens' contextuality for relativistic spin systems

Ruben Campos Delgado

TL;DR

The paper investigates whether Spekkens' contextuality remains Lorentz invariant when quantum systems include both spin and momentum. By constructing explicit relativistic qubit models and applying Lorentz boosts, it shows that contextuality can be frame-dependent when momentum is traced out, though full spin–momentum treatment preserves contextuality under boosts; a spherical-symmetry condition can restore invariance for one-qubit states, while multi-qubit cases do not. It further analyzes state discrimination scenarios, finding that boosts generally hinder four-state discrimination but can offer a slight moving-observer advantage for discriminating between two ensembles. Overall, the work highlights the necessity of including momentum degrees of freedom to achieve frame-invariant contextuality and motivates a more relativistically consistent formulation of Spekkens' notions in quantum information tasks.

Abstract

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 non-contextual 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 novel 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.

Frame dependence of Spekkens' contextuality for relativistic spin systems

TL;DR

The paper investigates whether Spekkens' contextuality remains Lorentz invariant when quantum systems include both spin and momentum. By constructing explicit relativistic qubit models and applying Lorentz boosts, it shows that contextuality can be frame-dependent when momentum is traced out, though full spin–momentum treatment preserves contextuality under boosts; a spherical-symmetry condition can restore invariance for one-qubit states, while multi-qubit cases do not. It further analyzes state discrimination scenarios, finding that boosts generally hinder four-state discrimination but can offer a slight moving-observer advantage for discriminating between two ensembles. Overall, the work highlights the necessity of including momentum degrees of freedom to achieve frame-invariant contextuality and motivates a more relativistically consistent formulation of Spekkens' notions in quantum information tasks.

Abstract

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 non-contextual 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 novel 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.

Paper Structure

This paper contains 6 sections, 3 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relativistic qubits
  3. Spekkens contextuality for one-qubit states
  4. A relativistic effect in state discrimination
  5. Conclusions
  6. Acknowledgments

Key Result

theorem 1

Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{D}(\mathcal{H})$ be the space of states in $\mathcal{H}$. If the states $\{\rho_i\}_{i} \subseteq \mathcal{D}(\mathcal{H})$ are linearly independent, then they are Spekkens non-contextual.

Figures (2)

  • Figure 1: Eve's probability of success in discriminating between four states, as a function of the rapidity of the Lorentz boost. Data is shown for $\epsilon=0.1$, $m=1$, $\sigma_{\uparrow}=2$, $\sigma_{\downarrow}=4$, $\sigma_{+}=3$, $\sigma_{-}=6$.
  • Figure 2: Eve's probability of success in discriminating between two ensembles, as a function of the rapidity of the Lorentz boost. Data is shown for $\epsilon=0.1$, $m=1$, $\sigma_{\uparrow}=2$, $\sigma_{\downarrow}=4$, $\sigma_{+}=3$, $\sigma_{-}=6$.

Theorems & Definitions (4)

  • theorem 1
  • theorem 2
  • theorem 3
  • proof