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.
Abstract We investigate the entanglement of a bipartite partly non-orthogonal state of the form | ϕ ⟩ = u | ξ ⟩ 1 ⊗ | 1 ⟩ 2 + v | ξ ⟩ 1 ⊗ | 0 ⟩ 2 + w | ξ ′ ⟩ 1 ⊗ | 1 ⟩ 2 + z | ξ ′ ⟩ 1 ⊗ | 0 ⟩ 2 through 1 2 -spin coherent states by adopting the concurrence as a measure to quantify the degree of entanglement of the two-qubit system. After, we study the amount of entanglement of mixed states represented by a statistical mixture of partly non-orthogonal bipartite pure states. We also express the lower and upper bounds of the squared concurrence of mixed states in terms of the amplitudes of spin coherent states.