Polynomial Monogamy Relations for Entanglement Negativity

The notion of nonclassical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations—the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under local operations and classical communication, and derive a necessary and sufficient inequality which, for the first time, is a nonlinear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.

Figure 1

Achievable qubit concurrence.

Figure 2

Achievable qubit negativity.

Figure 3

Achieved $D=3$ negativity of states in Eq. (15). Red indicates the (nonbounding) part of the polynomial surface for $d<1/\sqrt{D}$.