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arXiv:2204.13649 · Annalen der Physik

Monogamy of Entanglement of Maximal Dimension

Sumit Nandi

Published Sep 16, 2025• 0 citations
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Abstract

Abstract In the present study, 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 dimensional pure state. To derive the monogamy inequality in dimension, G‐concurrence measure of entanglement is considered as a measure of entanglement of maximal dimension. The approach of the present study incidentally points toward a rigorous framework which enables 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.

DOI
10.1002/andp.202500216

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Abstract In the present study, 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 dimensional pure state. To derive the monogamy inequality in dimension, G‐concurrence measure of entanglement is considered as a measure of entanglement of maximal dimension. The approach of the present study incidentally points toward a rigorous framework which enables 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.
We provide a detailed proof of an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that though the bound does not allow one to detect bound entanglement, it is tight for some mixed states and can detect most of the free entanglement. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is, in general, not true for higher-dimensional quantum states. Inducing from the lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary pure quantum states.
Bipartite quantum entanglement for qutrits and higher-dimensional objects is considered. We analyze the possibility of violation of monogamy inequality, introduced by Coffman, Kundu, and Wootters, for some systems composed of such objects. An explicit counterexample with a three-qutrit totally antisymmetric state is presented. Since three-tangle has been confirmed to be a natural measure of entanglement for qubit systems, our result shows that the three-tangle is no longer a legitimate measure of entanglement for states with three qutrits or higher-dimensional objects.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [V. Coffman et al., Phys. Rev. A 61, 052306 (2000)], and its generalization [T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)] hold on general nnn-qubit states including mixed ones. In this Brief Report, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on nnn-qubit pure states.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.