Exploring entanglement in the context of quantum sensing

Abstract

Motivated by an interest in quantum sensing, we define carefully a degree of entanglement, starting with bipartite pure states and building up to a definition applicable to any mixed state on any tensor product of finite-dimensional vector spaces. For mixed states the degree of entanglement is defined in terms of a minimum over all possible decompositions of the mixed state into pure states. Using a variational analysis we show a property of minimizing decompositions. Combined with data about the given mixed state, this property determines the degrees of entanglement of a given mixed state. For pure or mixed states symmetric under permutation of particles, we show that no partial trace can increase the degree of entanglement. For selected less-than-maximally-entangled pure states, we quantify the degree of entanglement surviving a partial trace.