Short proofs of the Quantum Substate Theorem

Understand

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.