The trace distance can be seen as a direct quantum generalization of the total variation distance between probability distributions. Given a pair of probability distributions P , Q , their total variation distance is δ δ ( P , Q ) = 1 2 ‖ ‖ P − − Q ‖ ‖ 1 = 1 2 ∑ ∑ k | P k − − Q k | . Attempting to directly apply this definition to quantum states raises the problem that quantum states can result in different probability distributions depending on how they are measured. A natural choice is then to consider the total variation distance between the classical probability distribution obtained measuring the two states, maximized over the possible choices of measurement, which results precisely in the trace distance between the quantum states. More explicitly, this is the quantity max Π Π 1 2 ∑ ∑ i | Tr ⁡ ⁡ ( Π Π i ρ ρ ) − − Tr ⁡ ⁡ ( Π Π i σ σ ) | , with the maximization performed with respect to all possible POVMs { Π Π i } i .