The trace distance is defined as half of the trace norm of the difference of the matrices: T ( ρ ρ , σ σ ) := 1 2 ‖ ‖ ρ ρ − − σ σ ‖ ‖ 1 = 1 2 T r [ ( ρ ρ − − σ σ ) † † ( ρ ρ − − σ σ ) ] , where ‖ ‖ A ‖ ‖ 1 ≡ ≡ Tr ⁡ ⁡ [ A † † A ] is the trace norm of A , and A is the unique positive semidefinite B such that B 2 = A (which is always defined for positive semidefinite A ). This can be thought of as the matrix obtained from A taking the algebraic square roots of its eigenvalues. For the trace distance, we more specifically have an expression of the form | C | ≡ ≡ C † † C = C 2 where C = ρ ρ − − σ σ is Hermitian. This quantity equals the sum of the singular values of C , which being C Hermitian, equals the sum of the absolute values of its eigenvalues. More explicitly, T ( ρ ρ , σ σ ) = 1 2 Tr ⁡ ⁡ | ρ ρ − − σ σ | = 1 2 ∑ ∑ i = 1 r | λ λ i | , where λ λ i ∈ ∈ R is the i -th eigenvalue of ρ ρ − − σ σ , and r is its rank.