Postulates for measures of genuine multipartite correlations

A lot of research has been done on multipartite correlations, but the problem of satisfactorily defining genuine multipartite correlations—those not trivially reducible to lower partite correlations—remains unsolved. In this paper we propose three reasonable postulates which each measure or indicator of genuine multipartite correlations (or genuine multipartite entanglement) should satisfy. We also introduce the concept of degree of correlations, which gives partial characterization of multipartite correlations. Then, we show that covariance does not satisfy two postulates and hence it cannot be used as an indicator of genuine multipartite correlations. Finally, we propose a candidate for a measure of genuine multipartite correlations based on the work that can be drawn from a local heat bath by means of a multipartite state.

Figure 1

Illustration of Observation 1. (a) Four parties share a joint state $\rho$. (b1) The third and the fourth party add ancillas in a state which is product with the original system and the other ancilla. After this step the parties share the state $\rho \otimes {\rho }_{A3}\otimes {\rho }_{A4}$. (b2) The third and the fourth party perform local operations on their original qubits and ancillas. After this step the parties share the state ${\text{id}}_{O1}\otimes {\text{id}}_{O2}\otimes {\Lambda }_{O3A3}^1\otimes {\Lambda }_{O4A4}^2(\rho \otimes {\rho }_{A3}\otimes {\rho }_{A4})$, where ${\text{id}}_i$ denotes the identity map acting on qubit $i$ and ${\Lambda }_{\mathit{ij}}^a$ denotes any completely positive map acting on qubits $i$ and $j$. (c) The third and the fourth party send ancillas to the fifth and sixth party, respectively. If the initial state has degree of correlations less than $n$ then the final state cannot have degree of correlations greater than or equal to $n+2$.

Figure 2

Probability of distillation ($q$) versus fidelity ($F$). From bottom to top: 3-qubit state, 5-qubit state, 7-qubit state, 9-qubit state, 49-qubit state, and 499-qubit state.

Figure 3

Covariance ($C$) versus fidelity ($F$) for a 3-qubit state (thin line) and a 9-qubit state (thick line).