Tsirelson's bound and supersymmetric entangled states

TL;DR

This paper explores supersymmetric extensions of qubits, called superqubits, and demonstrates that certain superqubit states can surpass Tsirelson's bound in nonlocal games depending on how probabilities are extracted from Grassmann amplitudes.

Contribution

It introduces superqubits as a supersymmetric extension of qubits and shows they can violate Tsirelson's bound under specific probability mappings.

Findings

Superqubits can reach Tsirelson's bound with certain probability maps.

One probability extraction method allows surpassing Tsirelson's bound.

Negative transition probabilities can be induced through basis changes.

Abstract

A superqubit, belonging to a $(2|1)$-dimensional super-Hilbert space, constitutes the minimal supersymmetric extension of the conventional qubit. In order to see whether superqubits are more nonlocal than ordinary qubits, we construct a class of two-superqubit entangled states as a nonlocal resource in the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the result depends on how we extract real probabilities and we examine three choices of map: (1) DeWitt (2) Trigonometric (3) Modified Rogers. In cases (1) and (2) the winning probability reaches the Tsirelson bound $p_{win}=\cos^2{\pi/8}\simeq0.8536$ of standard quantum mechanics. Case (3) crosses Tsirelson's bound with $p_{win}\simeq0.9265$. Although all states used in the game involve probabilities lying between 0 and 1, case (3) permits other changes of basis inducing negative transition probabilities.