[Paper Review] Equivalence of contextuality and Wigner function negativity in continuous-variable quantum optics

Jonas Haferkamp, Juan Bermejo-Vega|arXiv (Cornell University)|Dec 29, 2021

Quantum Mechanics and Applications4 citations

TL;DR

This paper establishes a rigorous equivalence between contextuality and Wigner function negativity in continuous-variable quantum optics, proving that non-contextual hidden variable models for Gaussian quantum systems correspond exactly to non-negative Wigner representations. The key result is that both contextuality and Wigner negativity are equivalent notions of non-classicality and necessary resources for quantum computational speed-up in Gaussian optics.

ABSTRACT

One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigner function description. Conversely, it was known that the Wigner representation provides a non-negative non-contextual description of Gaussian quantum optics. It follows that contextuality and Wigner negativity are equivalent notions of non-classicality and equivalent resources for this quantum subtheory. In particular, both contextuality and Wigner negativity are necessary for a computational speed-up of quantum Gaussian optics. At the technical level, our result holds true for any subfamily of Gaussian measurements that include homodyne measurements, i.e., measurements of standard quadrature observables.

Motivation & Objective

• To resolve the foundational question of identifying what distinguishes quantum systems from classical ones in continuous-variable quantum optics.
• To clarify the relationship between two central notions of non-classicality: contextuality (via non-contextual hidden variable models) and Wigner function negativity.
• To establish that in Gaussian quantum optics, non-contextuality is equivalent to the existence of a non-negative Wigner representation.
• To demonstrate that both contextuality and Wigner negativity are necessary resources for quantum computational speed-up in continuous-variable systems.
• To extend the equivalence from discrete-variable systems (e.g., odd-dimensional qudits) to the continuous-variable regime, including full Gaussian operations and transformations.

Proposed method

• Adapting the proof technique from discrete-variable systems (Delfosse et al., 2015) to the continuous-variable setting, focusing on integrability and measure-theoretic properties of phase-space functions.
• Using Spekkens’ generalized framework for contextuality (2005) to define non-contextual hidden variable models for Gaussian quantum optics, including state preparations, measurements, and transformations.
• Proving that any non-contextual model for Gaussian systems with homodyne measurements must admit a non-negative Wigner function representation (Theorem 5).
• Demonstrating the uniqueness of the Wigner representation as the non-contextual hidden variable model for the full Gaussian subtheory.
• Leveraging the composition rule for Wigner functions of quantum channels ($W_{ ho_2 ho_1} = W_{ ho_2} W_{ ho_1}$) to extend the equivalence to sequential and dynamical contextuality.
• Comparing results with independent work (Booth et al., 2021) using sheaf-theoretic contextuality frameworks, confirming consistency in the continuous-variable extension.

Experimental results

Research questions

• RQ1 Is contextuality in continuous-variable quantum optics equivalent to Wigner function negativity for Gaussian states and measurements?
• RQ2 Can a non-contextual hidden variable model for Gaussian quantum optics be fully characterized by a non-negative Wigner function representation?
• RQ3 Does the equivalence between contextuality and Wigner negativity extend to full Gaussian operations, including quantum channels and transformations?
• RQ4 What is the role of Wigner function negativity and contextuality in enabling quantum computational speed-up in continuous-variable quantum computing?
• RQ5 How does the equivalence in the continuous-variable regime compare to known results in discrete-variable systems, particularly for odd-dimensional qudits?

Key findings

• The paper proves that non-contextuality with respect to homodyne measurements in Gaussian quantum optics is equivalent to the non-negativity of the Wigner function (Theorem 5).
• Any non-contextual hidden variable model for Gaussian quantum optics admits a unique non-negative Wigner function representation, establishing the Wigner function as the canonical non-contextual model.
• Conversely, the existence of a non-negative Wigner function implies that the system admits a non-contextual hidden variable description, confirming the bidirectional equivalence.
• The equivalence extends to the full Gaussian subtheory, including all Gaussian states, measurements, and completely positive trace-preserving (CPTP) maps, due to the well-behaved composition rule of Wigner functions.
• Contextuality and Wigner function negativity are both necessary for quantum computational speed-up in continuous-variable quantum optics, as both rule out efficient classical simulation.
• The result implies that the Wigner function is the unique non-contextual hidden variable model for the entire Bosonic Gaussian subtheory, resolving a foundational question in continuous-variable quantum foundations.