  The Quantum Substate Theorem due to Jain, Radhakrishnan, and Sen (2002) gives
us a powerful operational interpretation of relative entropy, in fact, of the
observational divergence of two quantum states, a quantity that is related to
their relative entropy. Informally, the theorem states that if the
observational divergence between two quantum states rho, sigma is small, then
there is a quantum state rho' close to rho in trace distance, such that rho'
when scaled down by a small factor becomes a substate of sigma. We present new
proofs of this theorem. The resulting statement is optimal up to a constant
factor in its dependence on observational divergence. In addition, the proofs
are both conceptually simpler and significantly shorter than the earlier proof.
