Lower bounds on the violation of the monogamy inequality for quantum correlation measures

Abstract

In multiparty quantum systems, the monogamy inequality proposes an upper bound on the distribution of bipartite quantum correlation between a single party and each of the remaining parties in the system, in terms of the amount of quantum correlation shared by that party with the rest of the system taken as a whole. However, it is well-known that not all quantum correlation measures universally satisfy the monogamy inequality. In this work, we aim at determining the non-trivial value by which the monogamy inequality can be violated by a quantum correlation measure. Using an information-theoretic complementarity relation between the normalized purity and quantum correlation in any given multiparty state, we obtain a non-trivial lower bound on the negative monogamy score for the quantum correlation measure. In particular, for the three-qubit states the lower bound is equal to the negative von Neumann entropy of the single qubit reduced density matrix. We analytically examine the tightness of the derived lower bound for certain n-qubit quantum states. Further, we report numerical results of the same for monogamy violating correlation measures using Haar uniformly generated three-qubit states.

Forbidden regimes in the distribution of bipartite quantum correlations due to multiparty entanglement

“…Significantly, this allows us to readily obtain the upper bound of monogamy score, even for quantum correlation measures, such as distillable entanglement and entanglement cost, where the monogamy score is either intractable or not computable. Along with a recent result giving a lower bound [32] on the monogamy score, the present work provides an estimate of the constraints of monogamy in a quantum system. This letter is organized as follows.…”

Monogamy of Quantum Correlations - A Review

“…The proof of the relation (57) requires that Q(ρ xy ) satisfies the conditions Q(ρ xy ) ≤ S(ρ x ). However, it is independently satisfied by several important quantum correlation measures [75,172].…”

“…If we now consider a non-monogamous normalized bipartite quantum corelation measure, Q, which could, for example, be normalized quantum discord or normalized quantum work deficit, we can obtain a useful lower bound on the monogamy score in terms of purity. For an N -qudit state, using (8) and (57), one obtains the relation [172],…”

Lectures on General Quantum Correlations and their Applications

“…If we now consider a non-monogamous normalized bipartite quantum corelation measure, Q, which could, for example, be normalized quantum discord or normalized quantum work deficit, we can obtain a useful lower bound on the monogamy score in terms of purity. For an N -qudit state, using (8) and (57), one obtains the relation [172],…”