Stabilizer Renyi entropy as a measure of magic

Short answer: The stabilizer Rényi entropy is a measure of nonstabilizerness, commonly called "magic", defined from the Rényi entropy of the probability distribution given by the squared expectation values of Pauli strings in a pure state. Its 2022 authors say it is easily computable because it needs no minimization, and a 2023 Quantum paper studies it further and finds it is not a monotone under some stabilizer protocols. These are theory papers about a quantity, not experiments.

Definition

The 2022 Physical Review Letters paper introduces the measure for pure states. It takes the Rényi entropy of the probability distribution given by the square of the expectation value of Pauli strings in the state. The authors say it is a good measure of nonstabilizerness from the point of view of resource theory, and show bounds with other known measures. They say it has the advantage of being easily computable because it does not need a minimization procedure.

Monotonicity

The 2023 Quantum paper compares stabilizer entropies with known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. By explicit examples, it shows that for Rényi index n between 0 and 2, the stabilizer entropies are not monotones with respect to stabilizer protocols that include computational-basis measurements, even for pure states, while the question remains open for n of 2 or more. It also shows that for any Rényi index they do not satisfy a strong monotonicity condition with respect to computational-basis measurements. So the 2022 claim that it is a good resource measure sits alongside these limits in later work.

Chaos and measurement

The 2022 authors present a protocol for measuring it experimentally by randomized measurements. They also show that nonstabilizerness is intimately connected to out-of-time-order correlation functions and that maximal levels of nonstabilizerness are necessary for quantum chaos.

Numerical methods

The 2023 paper studies the stabilizer entropies in different classes of many-body states and compares them with other measures, either proving or giving numerical evidence for inequalities. It discusses exact or efficient tensor-network methods for matrix-product states with large numbers of qubits. In addition to previously developed exact methods for the Rényi entropies, it puts forward a scheme based on perfect matrix-product-state sampling that computes the von Neumann stabilizer entropy efficiently for large bond dimensions.

Limits

This is a theory quantity with proposed measurement and numerical methods. The papers do not report a hardware measurement in the abstracts used here. The monotonicity results mean it should not be treated as a drop-in replacement for every other magic measure.

FAQ

What does it measure?

Nonstabilizerness, or magic, of a pure quantum state.

Why is it easy to compute?

The 2022 authors say it needs no minimization procedure.

Is it a monotone?

The 2023 paper shows it is not under some stabilizer protocols with computational-basis measurements for n between 0 and 2, and leaves n of 2 or more open for that case.

Can it be measured?

The 2022 authors present a protocol using randomized measurements.

How does it relate to chaos?

They show that maximal levels of nonstabilizerness are necessary for quantum chaos.