Short proofs of the Quantum Substate Theorem

TL;DR

This paper provides new, simpler, and shorter proofs of the Quantum Substate Theorem, which relates the observational divergence of quantum states to their proximity and substate relationships, with optimal dependence on divergence.

Contribution

The authors introduce novel proofs of the Quantum Substate Theorem that are both conceptually clearer and shorter, achieving optimal dependence on observational divergence.

Findings

Proofs are shorter and more elegant than previous ones.

The new proofs are conceptually simpler.

The dependence on observational divergence is optimal up to a constant.

Abstract

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.