Short proofs of the Quantum Substate Theorem
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.
Cite
@article{arxiv.1103.6067,
title = {Short proofs of the Quantum Substate Theorem},
author = {Rahul Jain and Ashwin Nayak},
journal= {arXiv preprint arXiv:1103.6067},
year = {2012}
}
Comments
11 pages. Rewritten; included new references; presented the results in terms of smooth relative min-entropy; stronger results; included converse and proof using SDP duality