Monogamy of entanglement without inequalities

TL;DR

This paper introduces a new inequality-free definition of monogamous entanglement measures, linking it to traditional monogamy relations and demonstrating its applicability to quantum states like Markov states and G-concurrence.

Contribution

It proposes a novel, equality-based 'disentangling condition' for monogamous entanglement measures, expanding understanding beyond traditional inequalities.

Findings

Quantum Markov states satisfy the disentangling condition for all entanglement monotones.

Entanglement monotones with convex roof extensions are monogamous if monogamous on pure states.

For states satisfying the disentangling condition, entanglement of formation equals entanglement of assistance.

Abstract

We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.