A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems

TL;DR

The paper investigates the geometry of the stabilizer polytope in few-qubit systems by quantifying non-stabilizerness with the trace-distance measure $\mathcal{M}(\rho)$ to the stabilizer set. It develops analytical expressions linking facet violations to $\mathcal{M}$ for low-dimensional systems, constructs Bell-like witnesses, and studies the distribution and noise resilience of non-stabilizerness under Haar and Hilbert–Schmidt random state generation. By comparing $\mathcal{M}$ with Robustness of Magic and stabilizer Rényi entropies, and by analyzing entanglement classifications, the work uncovers tight connections and trade-offs between non-stabilizerness and entanglement in 1–3 qubit systems, including concentration results near stabilizer vertices. The results offer geometric and operational insights into stabilizer resources, with potential implications for near-term quantum devices, device-dependent Bell witnesses, and future extensions to higher dimensions.

Abstract

Non-stabilizerness is a fundamental resource for quantum computational advantage, differentiating classically simulable circuits from those capable of universal quantum computation. Recently, non-stabilizerness has been shown to be relevant for a few qubit systems. In this work, we investigate the geometry of the stabilizer polytope in few-qubit quantum systems, using the trace distance to the stabilizer set to quantify non-stabilizerness. By randomly sampling quantum states, we analyze the distribution of non-stabilizerness for both pure and mixed states and compare the trace distance with other non-stabilizerness measures, as well as entanglement. Additionally, we give an analytical expression for the introduced quantifier, classify Bell-like inequalities corresponding to the facets of the stabilizer polytope, and establish a general concentration result connecting non-stabilizerness and entanglement via Fannes' inequality. Our findings provide new insights into the geometric structure of non-stabilizerness and its role in small-scale quantum systems, offering a deeper understanding of the interplay between quantum resources

Figures (11)

• Figure 1: The $P^{\text{STAB}}_{2,1}$ polytope inscribed in the Bloch sphere.
• Figure 2: A heat-map of the NTD of the Bloch sphere surface.
• Figure 3: NTD histograms of $360{,}000$ randomly sampled pure (blue) and mixed (orange) states for (a) 1-qubit, (b) 2-qubit, and (c) 3-qubit systems.
• Figure 4: Tight lower bound of $\mathcal{M}(\rho)$ restricted to a value $I^3_i(\rho)$ for $i=1,2$. In this figure, we show how the numerical approach allows to get an analytical expression for the SDP [see Eq. (\ref{['eq: Magic_TD']})]. The dots are the numerical result of the SDP by restricting the states to satisfy a certain value of the witnesses $I^3$ given by Eq. (\ref{['eq: Ineq1_qtrits']}) (black dots) and Eq. (\ref{['eq: Ineq2_qtrits']}) (green dots). We show the corresponding fitting in the legend, giving the analytical expression for the trace distance in terms of $I^3$.
• Figure 5: Detection capability of the facets of the 2-qubit polytope. The plot shows the percentage of states that violate each facet out of $10^6$ states sampled randomly from the Hilbert-Schmidt ensemble.

Theorems & Definitions (10)

• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Lemma 5.1: Fannes-Audenaert inequality
• Theorem 5.2
• Corollary 5.3
• proof
• Lemma B.1
• proof
• proof