A computational test of quantum contextuality, and even simpler proofs of quantumness

TL;DR

This work shows that an arbitrary contextuality game can be compiled into an “operational test of contextuality” involving a single quantum device, by only making the assumption that the device is computationally bounded.

Abstract

Bell non-locality is a fundamental feature of quantum mechanics whereby measurements performed on "spatially separated" quantum systems can exhibit correlations that cannot be understood as revealing predetermined values. This is a special case of the more general phenomenon of "quantum contextuality", which says that such correlations can occur even when the measurements are not necessarily on separate quantum systems, but are merely "compatible" (i.e. commuting). Crucially, while any non-local game yields an experiment that demonstrates quantum advantage by leveraging the "spatial separation" of two or more devices (and in fact several such demonstrations have been conducted successfully in recent years), the same is not true for quantum contextuality: finding the contextuality analogue of such an experiment is arguably one of the central open questions in the foundations of quantum mechanics. In this work, we show that an arbitrary contextuality game can be compiled into an operational "test of contextuality" involving a single quantum device, by only making the assumption that the device is computationally bounded. Our work is inspired by the recent work of Kalai et al. (STOC '23) that converts any non-local game into a classical test of quantum advantage with a single device. The central idea in their work is to use cryptography to enforce spatial separation within subsystems of a single quantum device. Our work can be seen as using cryptography to enforce "temporal separation", i.e. to restrict communication between sequential measurements. Beyond contextuality, we employ our ideas to design a "proof of quantumness" that, to the best of our knowledge, is arguably even simpler than the ones proposed in the literature so far.

Figures (10)

• Figure 1: The compiler in this work compiles a much larger set of games compared to the one in KLVY22.
• Figure 2: A schematic comparing the non-local game compiler KLVY22 with our contextuality game compiler. The key idea in KLVY is to ask the first question of a non-local game under a homomorphic encryption and the second one in the clear, with the prover using two entangled subsystems (one that is encrypted, and one in the clear). In our compiler, the oblivious pad ($\mathsf{OPad}$) allows the prover to "re-encrypt" its post-measurement state, just before Round 2. Upon obtaining information about the "re-encryption" that took place, the verifier can then safely reveal the "overall" decryption key in Round 2, allowing the prover to proceed with the next measurement in the clear.
• Figure 3: A two-player non-local game. Alice and Bob get $(x,y)$ from the referee and respond with $(a,b)$. They cannot communicate once the game starts.
• Figure 4: An illustration of the optimal quantum strategy corresponding to the KCBS Game defined in \ref{['Exa:KCBSintro']}. Here the red vector denotes the quantum state $\ket{\psi}$ and the blue ones $\ket{v_q}$ correspond to projective measurements, $\Pi_q = \ket{v_q}\bra{v_q}$. Consecutively indexed blue vectors, i.e. $\ket{v_q},\ket{v_{q+1}}$, are orthogonal (indexing is periodic).
• Figure 5: The folklore memoryless interpretation of contextuality: a memoryless device (left) and a memoryless operational test, illustrated using the Peres-Mermin magic square (right).

Theorems & Definitions (71)

• Theorem 1: informal, simplified
• Theorem 2: Informal
• Example 1: Peres-Mermin (Magic Square) peres1990incompatiblemermin1990simple
• Example 2: 2-player non-local games
• Example 3: KCBS klyachko2008simple
• Remark 3: Criteria for being an Operational Test of Contextuality
• Theorem : KLVY22, informal
• Theorem : restatement of \ref{['Thm:infMain1']}
• Theorem 4: informal
• Definition 5: Quantum Homomorphic Encryption