  Quantities invariant under local unitary transformations are of natural
interest in the study of entanglement. This paper deduces and studies a
particularly simple quantity that is constructed from a combination of two
standard permutations of the density matrix, namely realignment and partial
transpose. This bipartite quantity, denoted here as $R_{12}$, vanishes on large
classes of separable states including classical-quantum correlated states,
while being maximum for only maximally entangled states. It is shown to be
naturally related to the 3-tangle in three qubit states via their two-qubit
reduced density matrices. Upper and lower bounds on concurrence and negativity
of two-qubit density matrices for all ranks are given in terms of $R_{12}$.
Ansatz states satisfying these bounds are given and verified using various
numerical methods. In rank-2 case it is shown that the states satisfying the
lower bound on $R_{12}$ {\it vs} concurrence define a class of three qubit
states that maximizes the tripartite entanglement (the 3-tangle) given an
amount of entanglement between a pair of them. The measure $R_{12}$ is
conjectured, via numerical sampling, to be always larger than the concurrence
and negativity. In particular this is shown to be true for the physically
interesting case of $X$ states.
