Geometric measure of mixing of quantum state

Abstract

We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states. An explicit expression for the geometric measure is obtained. It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix. As an example, geometric measure of mixing for spin-1/2 states is calculated.

The controlled SWAP test for entanglement of mixed quantum states

“…The results of theorem 2 can be used to explore some relevant properties of quantum states. Here we give a representation of the degree of mixing of quantum states [22] based on theorem 2.…”

“…Let ρ in C dA ⊗ C dB be any bipartite state. We can obtain the following representation of degree of mixing [22] of quantum states after passing the controlled SWAP test in fig. 2:…”

“…Proof. According to [22], the linear entropy S L = tr(ρ − ρ 2 ) = 1 − tr(ρ 2 ) can be used as a measure of the degree of mixing of quantum state ρ, i.e., how close it is to a pure state. From theorem 2, we obtain tr(ρ 2 ) = 1 − 2(p(|01 ) + p(|10 )), then it is easy to get our result.…”

“…According to theorem 6, we can obtain the purity of quantum state tr(ρ 2 ) = 1 − 2 c z S is odd z∈{0,1} n p(z). Then according to [22], we can obtain the above result.…”