where ρ is the density matrix for a system describing an ensemble of particles. Based on the previous example it is clear that the entropy functional is zero if and only if ρ is a pure state. The property of Markovianity is numerically explicit, and can be further defined through the entropy function and quantum information theory. A Markovian process defines each future state in time based solely on the previous time step, and this corresponds to the entropy function being concave such that ρ → S(ρ), and S ( i λ i ρ i ) ≥ i λ i S(ρ i ), for non-vanishing λ i . Furthermore, when the system is composite, there is a subadditivity condition: H = H (1) ⊗ H (2) where S(ρ) ≤ S(ρ (1) ) + S(ρ (2) ). Consequently, during a Markovian evolution, the system is monotonically losing information to the environment, as each successive state evolves independently of its history towards equilibrium . Monotonic loss of information for quantum states is defined by decreasing distinguishability of different quantum systems with time . In contrast, non-Markovianity corresponds to any situation which invalidates the Markovian condition of monotonic information loss. With reference to the previous example, this could be considered as a mixed state where information about the system in a historical state imprinted onto the environment returns through a quantum channel to influence the future dynamics.