109 research outputs found
Short proofs of the Quantum Substate Theorem
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
We consider the problem of whether the canonical and microcanonical ensembles
are locally equivalent for short-ranged quantum Hamiltonians of spins
arranged on a -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
spins. The proof rests on a variant of the Berry--Esseen
theorem for quantum lattice systems and ideas from quantum information theory
Exponential Decay of Correlations Implies Area Law
We prove that a finite correlation length, i.e. exponential decay of
correlations, implies an area law for the entanglement entropy of quantum
states defined on a line. The entropy bound is exponential in the correlation
length of the state, thus reproducing as a particular case Hastings proof of an
area law for groundstates of 1D gapped Hamiltonians.
As a consequence, we show that 1D quantum states with exponential decay of
correlations have an efficient classical approximate description as a matrix
product state of polynomial bond dimension, thus giving an equivalence between
injective matrix product states and states with a finite correlation length.
The result can be seen as a rigorous justification, in one dimension, of the
intuition that states with exponential decay of correlations, usually
associated with non-critical phases of matter, are simple to describe. It also
has implications for quantum computing: It shows that unless a pure state
quantum computation involves states with long-range correlations, decaying at
most algebraically with the distance, it can be efficiently simulated
classically.
The proof relies on several previous tools from quantum information theory -
including entanglement distillation protocols achieving the hashing bound,
properties of single-shot smooth entropies, and the quantum substate theorem -
and also on some newly developed ones. In particular we derive a new bound on
correlations established by local random measurements, and we give a
generalization to the max-entropy of a result of Hastings concerning the
saturation of mutual information in multiparticle systems. The proof can also
be interpreted as providing a limitation on the phenomenon of data hiding in
quantum states.Comment: 35 pages, 6 figures; v2 minor corrections; v3 published versio
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