Monogamy of entanglement of maximal dimension

TL;DR

The paper addresses how entanglement can be shared in higher-dimensional tripartite systems by formulating a CKW-style monogamy inequality using the G-concurrence measure. The authors develop an upper-bound framework for G-concurrence on mixed states and prove a monogamy relation $G^d_{12}+G^d_{13} \le G^d_{1(23)}$ for $d>2$, with a detailed $d=3$ example. They illustrate the results using states such as a tripartite qutrit χ, a generalized W-class state, and a qudit GHZ state, showing saturation or strictness of the bound in different cases. This work extends monogamy constraints beyond qubits, recovers CKW for qubits, and provides a pathway toward tighter bounds and multipartite generalizations in higher dimensions.

Abstract

In the present paper, a trade off of sharing of entanglement between subsystems of a higher dimensional quantum state is derived. It is presented in terms of an inequality which is analogous to the Coffman-Kundu-Wootters inequality that succinctly describes monogamy of entanglement in $\mathcal{C}^2\otimes \mathcal{C}^2\otimes \mathcal{C}^2$ dimensional pure state. To derive the monogamy inequality in $\mathcal{C}^d\otimes \mathcal{C}^d\otimes \mathcal{C}^d$ dimension, G-concurrence measure of entanglement is considered as a measure of entanglement of maximal dimension. The approach of the present paper incidentally points towards a rigorous framework which enables us to obtain an upper bound of G-concurrence of a bipartite qudit mixed state. Obtained upper bound of G-concurrence is then shown to satisfy a monogamy relation.