Monogamy of Entanglement Bounds and Improved Approximation Algorithms for Qudit Hamiltonians

TL;DR

This paper establishes new bounds on entanglement monogamy for qudit Hamiltonians and introduces improved algorithms that approximate maximum energy more effectively than previous methods, especially for specific graph classes.

Contribution

It provides novel monogamy bounds for qudit Hamiltonians and develops simple matching-based algorithms with better approximation guarantees than prior approaches.

Findings

Proves monogamy bounds for two-local qudit Hamiltonians without one-local terms.

Shows a matching-based algorithm approximates maximum energy to at least 1/d for general graphs.

Achieves a 0.595 approximation ratio for d=2, surpassing previous results.

Abstract

We prove new monogamy of entanglement bounds for two-local qudit Hamiltonians of rank-one projectors without one-local terms. In particular, we certify the maximum energy in terms of the maximum matching of the underlying interaction graph via low-degree sum-of-squares proofs. Algorithmically, we show that a simple matching-based algorithm approximates the maximum energy to at least $1/d$ for general graphs and to at least $1/d + \Theta(1/D)$ for graphs with bounded degree, $D$. This outperforms random assignment, which, in expectation, achieves energy of only $1/d^2$ of the maximum energy for general graphs. Notably, on $D$-regular graphs with degree, $D \leq 5$, and for any local dimension, $d$, we show that this simple matching-based algorithm has an approximation guarantee of $1/2$. Lastly, when $d=2$, we present an algorithm achieving an approximation guarantee of $0.595$, beating that of [PT22, arXiv:2206.08342], which gave an approximation ratio of $1/2$.