Exploring degrees of entanglement

In spite of a long history, the quantification of entanglement still calls for exploration. What matters about entanglement depends on the situation, and so presumably do the numbers suitable for its quantification. Regardless of situational complications, a necessary first step is to make available for calculation some quantitative measure of entanglement. Here we define a geometric degree of entanglement, distinct from earlier definitions, but in the case of bipartite pure states related to that proposed by Shimony (Ann N Y Acad Sci 755:675–679, 1995). The definition offered here applies also to multipartite mixed states, and a variational method simplifies the calculation. We analyze especially states that are invariant under permutation of particles, states that we call bosonic. Of interest to quantum sensing, for bosonic states, we show that no partial trace can increase a degree of entanglement. For some sample cases we quantify the degree of entanglement surviving a partial trace. As a function of the degree of entanglement of a bosonic 3-qubit pure state, we show the range of degree of entanglement for the 2-qubit reduced density matrix obtained from it by a partial trace. Then we calculate an upper bound on the degree of entanglement of the mixed state obtained as a partial trace over one qubit of a 4-qubit bosonic state. As a reminder of the situational dependence of the advantage of entanglement, we review the way in which entanglement combines with scattering theory in the example of light-based radar.