Maximal Magic for Two-qubit States

TL;DR

This paper investigates two-qubit states with maximal quantum magic using Stabilizer Rénnyi Entropy, establishing a tighter bound, identifying states that saturate it, and exploring their entanglement properties and relation to mutually unbiased bases.

Contribution

It provides a new tighter bound for maximal magic in two-qubit states, identifies states that achieve this bound, and explores their connection to MUBs and entanglement.

Findings

Maximal second order SRE is approximately 0.827, tighter than previous bounds.

480 states saturate the new bound, related to MUBs generated by Weyl-Heisenberg group.

Maximal magic states have restricted entanglement, with concurrence values 1/2 and 1/√2.

Abstract

Magic is a quantum resource essential for universal quantum computation and represents the deviation of quantum states from those that can be simulated efficiently using classical algorithms. Using the Stabilizer R\'enyi Entropy (SRE), we investigate two-qubit states with maximal magic, which are most distinct from classical simulability, and provide strong numerical evidence that the maximal second order SRE is $\ln (16/7)\approx 0.827$, establishing a tighter bound than the prior $\ln (5/2)\approx 0.916$. We identify 480 states saturating the new bound, which turn out to be the fiducial states for the mutually unbiased bases (MUBs) generated by the orbits of the Weyl-Heisenberg (WH) group, and conjecture that WH-MUBs are the maximal magic states for $n$-qubit, when $n\neq 1$ and 3. We also reveal a striking interplay between magic and entanglement: the entanglement of maximal magic states is restricted to two possible values, $1/2$ and $1/\sqrt{2}$, as quantified by the concurrence; none is maximally entangled.