Basis-independent stabilizerness and maximally noisy magic states

TL;DR

This paper characterizes absolutely stabilizer states and absolutely Wigner-positive states across prime qudit dimensions, revealing geometric structures and purity bounds, and introduces a unitarily-invariant notion of bound magic.

Contribution

It provides a comprehensive geometric and spectral characterization of absolutely stabilizer and Wigner-positive states for all prime qudit dimensions, including new purity bounds.

Findings

The set of absolutely stabilizer states forms a specific geometric shape depending on dimension.

Absolutely Wigner-positive states include some states not stabilizer, indicating a form of bound magic.

Purity bounds are established for these state sets, informing their quantum resource properties.

Abstract

Absolutely stabilizer states are those that remain convex mixtures of stabilizer states after conjugation by any unitary. Here we give a characterization of such states for multiple qudits of all prime dimensions by introducing a polytope of their allowed spectra. We illustrate this through the examples of one qubit, two qubits, and one qutrit. In particular, the set of absolutely stabilizer states for a single qubit is a ball inscribed in the stabilizer octahedron, but for higher dimensions the geometry is more complicated. For odd-prime-dimensional qudits, we also give a complete characterization of absolutely Wigner-positive states, i.e., states whose Wigner function remains nonnegative after conjugation by any unitary. In so doing, we show there are absolutely Wigner-positive states that are not absolutely stabilizer, which can be seen as a unitarily-invariant version of bound magic. We then study the radii of the largest balls contained in the sets of absolutely stabilizer states and absolutely Wigner-positive states. These radii respectively tell us the lowest possible purity of nonstabilizer and Wigner-negative states. Conversely, we also find the radius of the smallest ball containing the set of absolutely Wigner-positive states, giving a tight purity-based necessary condition thereof.