Maximal Magic for Two-qubit States
Qiaofeng Liu, Ian Low, Zhewei Yin
TL;DR
We address identifying states with maximal magic under the Stabilizer Rényi Entropy for two-qubit systems. Using a second-order SRE, we derive a tighter bound $M_2 \le \ln(16/7)$ than the prior $\ln(5/2)$ and numerically locate 480 two-qubit states saturating this bound; these states are WH MUB fiducials generated by Weyl–Heisenberg group orbits. The study reveals that WH MUB fiducials attain maximal non-stabilizerness in the absence of WH-SICs for two qubits and likely for general $n$-qubit systems with $n \neq 1,3$, with an explicit entanglement pattern: the concurrence of maximal magic states is restricted to $\tfrac{1}{2}$ or $\tfrac{1}{\sqrt{2}}$, never maximal. The results strengthen the link between magic, MUBs, and group-covariant constructions, and they point to WH MUBs as central objects for maximal magic in composite systems.
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 Rényi 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.