Short Proofs of the Quantum Substate Theorem

The Quantum Substate Theorem due to Jain (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 divergenc...

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Bibliographic Details
Published in:IEEE transactions on information theory 2012-06, Vol.58 (6), p.3664-3669
Main Authors: Jain, R., Nayak, A.
Format: Article
Language:English
Subjects:
Divergence
Educational institutions
Entropy
Hilbert space
Information theory
Observational divergence
Optimization
Proving
Quantum computing
quantum information theory
Quantum physics
relative entropy
Relativistic quantum mechanics
smooth relative min-entropy
substate theorem
Theorems
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Summary:The Quantum Substate Theorem due to Jain (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 ρ, σ is small, then there is a quantum state ρ ' close to ρ in trace distance, such that ρ ' when scaled down by a small factor becomes a substate of σ. 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.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2012.2184522