Exact evaluation of the stabilizer Rényi entropy of a Greenberger–Horne–Zeilinger state

Abstract

The stabilizer Rényi entropy (SRE) is a key metric for nonstabilizerness, or quantum magic, that measures the extent to which a quantum state deviates from the stabilizer framework. In this paper we present a calculation of the vanishing SRE of a Greenberger–Horne–Zeilinger (GHZ) stabilizer state from its matrix product state representation. We focus here on a demonstrative and exact proof that applies for all chain lengths. We find that the only Pauli strings with non-zero expectation values are binary strings that consist of either (a) identity and Pauli-Z operators, or (b) Pauli-X and Pauli-Y operators, that have an even number of Pauli-Z and Pauli-Y operators. These non-zero expectation values have unit magnitude, and an exact count of these strings yields an SRE that is identically zero. This result provides a practical example of an SRE calculation for a matrix product state with pedagogical value because of its tractability.