Bounding Entanglement Entropy with Clifford Double Cosets

TL;DR

This paper introduces contracted graphs representing classes of quantum states with the same entropy, providing bounds on entropy diversity under Clifford circuits and exploring implications for holography.

Contribution

It develops a novel double coset formalism and contracted graph framework to analyze entropy evolution in Clifford orbits, extending previous reachability graph methods.

Findings

Derived an upper bound on entropy vectors generated by Clifford circuits.

Analyzed contracted graphs for stabilizer, W, and Dicke states.

Connected graph diameter to entropic diversity constraints.

Abstract

Following on our previous work arXiv:2204.07593 and arXiv:2306.01043 studying the orbits of quantum states under Clifford circuits via `reachability graphs', we introduce `contracted graphs' whose vertices represent classes of quantum states with the same entropy vector. These contracted graphs represent the double cosets of the Clifford group, where the left cosets are built from the stabilizer subgroup of the starting state and the right cosets are built from the entropy-preserving operators. We study contracted graphs for stabilizer states, as well as W states and Dicke states, discussing how the diameter of a state's contracted graph constrains the `entropic diversity' of its $2$-qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any $n$-qubit Clifford circuit, for any quantum state. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.