Basis-independent stabilizerness and maximally noisy magic states

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.

Figures (8)

• Figure 1: CNC sets cannot contain observables forming a Mermin square-style proof of contextuality. (a) A subset of an $m=1$ CNC set, and (b) a subset of an $m=2$ CNC set on two qubits (see Theorem \ref{['Theorem:CNCClassification']}). In both cases, the Mermin square proof is avoided.
• Figure 2: The single-qubit Bloch ball is shown in gray containing the stabilizer octahedron in red and the set of single-qubit absolutely stabilizer states in blue. In the single-qubit case, the set of absolutely stabilizer states is a Hilbert-Schmidt ball inscribed in the stabilizer octahedron.
• Figure 3: Cross section of the set of $2$-qubit states parametrized by $\rho(x,y)=\frac{1}{4}I+x(ZZ+XX-YY)+y(Z_1+Z_2)$. The set of physical density matrices is shown in gray, $\mathrm{STAB}_0$ in red, and $\mathrm{ASTAB}_0$ in blue.
• Figure 4: Left: Ternary plot of the spectra of single-qutrit absolutely stabilizer states (blue) and absolutely Wigner-positive states (red). Right: the associated Weyl chamber (i.e. the subset of ordered spectra) with special points of intersection displayed.
• Figure 5: Left: Ternary plot of the spectra of single-qutrit absolutely stabilizer states (blue), absolutely Wigner-positive states (red), together with the inradius (yellow) common to both. Yellow dots denote absolutely stabilizer states on the boundary of both $\mathrm{ASTAB}$ and its inradius. Right: the associated Weyl chamber with the spectra of these special points displayed together with the Hilbert-Schmidt radius $r(\mathrm{ASTAB})=\frac{1}{\sqrt{24}}$.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: RaussendorfZurel2020; § IV
• Corollary 1
• Theorem 2: ZurelHeimendahl2024b; Theorem 1
• Definition 5
• Lemma 1
• proof : Proof of Lemma \ref{['Lemma:STABHRepresentation']}