277 research outputs found

    Short proofs of the Quantum Substate Theorem

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    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.Comment: 11 pages. Rewritten; included new references; presented the results in terms of smooth relative min-entropy; stronger results; included converse and proof using SDP dualit

    Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems

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    We consider the problem of whether the canonical and microcanonical ensembles are locally equivalent for short-ranged quantum Hamiltonians of NN spins arranged on a dd-dimensional lattices. For any temperature for which the system has a finite correlation length, we prove that the canonical and microcanonical state are approximately equal on regions containing up to O(N1/(d+1))O(N^{1/(d+1)}) spins. The proof rests on a variant of the Berry--Esseen theorem for quantum lattice systems and ideas from quantum information theory
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