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I Justification for introducting "Disentaglers" in MERA

  1. Jun 27, 2016 #1
    A class of quantum many-body states that can be efficiently simulated
    G. Vidal
    (Submitted on 12 Oct 2006)
    We introduce the multi-scale entanglement renormalization ansatz (MERA), an efficient representation of certain quantum many-body states on a D-dimensional lattice. Equivalent to a quantum circuit with logarithmic depth and distinctive causal structure, the MERA allows for an exact evaluation of local expectation values. It is also the structure underlying entanglement renormalization, a coarse-graining scheme for quantum systems on a lattice that is focused on preserving entanglement.

    http://arxiv.org/abs/quant-ph/0610099

    Just trying to follow some of the Condensed Matter and Area Laws Papers of late. Find myself re-reading Swingle's paper http://arxiv.org/abs/0905.1317 and the one above - which as I understand it first introduced the MERA process - which seems to hinge especially on these things called "disentanglers". I'm hoping someone can answer the following question without too much effort:

    Are disentanglers (and I guess isometries as well) just part of the ansatz (guess). In other words without speculating on what could cause them - they are key parts of a proposed theory which is Ansatz. Or is there a more extensive history to them conceptually. If MERA accurately models Quantum Space-Time Geometry where would one suggest the disentanglers come from? Not where literally, but what mechanism are they rooted in. What physical notion supports their existence and function. Is related to the negative curvature of Ads?

    Hope that makes a little sense.
     
  2. jcsd
  3. Jun 28, 2016 #2

    atyy

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    Science Advisor

    Disentanglers are part of the Ansatz. An Ansatz is just a physically motivated guess. You can think of them as removing short range entanglement, but leaving long range entanglement.

    The other way to think about it is that the MERA produces wave functions for systems whose entanglement entropy obey area laws. The disentanglers allow the Ansatz to have that property.
     
    Last edited: Jun 29, 2016
  4. Jun 29, 2016 #3
    Thanks.

    From Vidal:
    "We have defined a MERA for |Ψi in terms of a quantum circuit C that transforms a product state |0i ⊗N into |Ψi by means of Θ layers of unitary gates."

    "An alternative interpretation of the MERA can be obtained by considering the sequence of states {|Ψ0i, |Ψ1i, |Ψ2i, . . . }, which correspond to undoing the quantum evolution of C back in time. Notice that |Ψτ+1i is obtained from |Ψτ i by applying two layers of tensors in M. The first layer is made of disentanglers that transform |Ψτ i into a less entangled state |Ψ′ τ i. The second layer is made of isometries that combine pairs of nearest neighbor wires into single wires, turning the state |Ψ′ τ i of Nτ wires into the state |Ψτ+1i of Nτ /2 wires, where Nτ = 2Θ−τ . That is, M implements a class of real space coarse-graining transformations known as entanglement renormalization [7]."


    So the "unitary gates" represent "evolution of observables" or "measurements" or "un-observed collapse inducing interactions" (I'm trying to avoid the can of interpretation questions - while checking my association with same).

    And the "Isometries" or renormalization step is the "re-etangling" of the new unobserved state - the subsequent evolution of which must be a unitary disentangler.

    Am I following?
     
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