Maximal magic for two-qubit states

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>16</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>7</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≈</mml:mo> <mml:mn>0.827</mml:mn> </mml:mrow> </mml:math> , establishing a tighter bound than the prior <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≈</mml:mo> <mml:mn>0.916</mml:mn> </mml:mrow> </mml:math> . 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 ≠ 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> </mml:mrow> </mml:math> , as quantified by the concurrence; none is maximally entangled.