  Many entanglement measures are first defined for pure states of a bipartite
Hilbert space, and then extended to mixed states via the convex roof extension.
In this article we alter the convex roof extension of an entanglement measure,
to produce a sequence of extensions that we call $f$-$d$ extensions, for $d \in
\mathbb{N}$, where $f:[0,1]\to [0, \infty)$ is a fixed continuous function
which vanishes only at zero. We prove that for any such function $f$, and any
continuous, faithful, non-negative function, (such as an entanglement measure),
$\mu$ on the set of pure states of a finite dimensional bipartite Hilbert
space, the collection of $f$-$d$ extensions of $\mu$ detects entanglement, i.e.
a mixed state $\rho$ on a finite dimensional bipartite Hilbert space is
separable, if and only if there exists $d \in \mathbb{N}$ such that the $f$-$d$
extension of $\mu$ applied to $\rho$ is equal to zero. We introduce a quantum
variational algorithm which aims to approximate the $f$-$d$ extensions of
entanglement measures defined on pure states. However, the algorithm does have
its drawbacks. We show that this algorithm exhibits barren plateaus when used
to approximate the family of $f$-$d$ extensions of the Tsallis entanglement
entropy for a certain function $f$ and unitary ansatz $U(\theta)$ of sufficient
depth. In practice, if additional information about the state is known, then
one needs to avoid using the suggested ansatz for long depth of circuits.
