I have a rather limited understanding about error correcting codes. Therefore I was happy to learn that there is a conference in Stanford in which leading gurus of quantum gravity and quantum information sciences are talking about these topics. The first lecture that I listened was about a possible connection between holography and quantum error correcting codes. The lecturer was Preskill and the title of the talk was "Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence" (see this and this). A detailed representation can be found in the article of Preskill et al ).
The idea is that time= constant section of AdS, which is hyperbolic space allowing tessellations, can define tensor networks. So called perfect tensors are building bricks of the tensor networks providing representation for holography. There are three observations that put bells ringing and actually motivated this article.
- Perfect tensors define entanglement which TGD framework corresponds negentropic entanglement playing key role in TGD inspired theory of consciousness and of living matter.
- In TGD framework the hyperbolic tesselations are realized at hyperbolic spaces H3(a) defining light-cone proper time hyperboloids of M4 light-cone.
- TGD replaces AdS/CFT correspondence with strong form of holography.
One can criticize AdS/CFT based holography because it has Minkowski space only as a rather non-unique conformal boundary resulting from conformal compactification. Situation gets worse as one starts to modify AdS by populating it with blackholes. And even this is not enough: one can imagine anything inside blackhole interiors: wormholes connecting them to other blackholes, anything. Entire mythology of mystic creatures filling the white (or actually black) areas of the map. Post-modernistic sloppiness is the problem of recent day theoretical physics - everything goes - and this leads to inflationary story telling. Minimalism would be badly needed.
AdS/CFT is very probably mathematically correct. The question is whether the underlying conformal symmetry - certainly already huge - is large enough and whether its proper extension could allow to get rid of admittedly artificial features of AdS/CFT.
In TGD framework conformal symmetries are generalized thanks due to the metric 2-dimensionality of light-cone boundary and of light-like 3-surfaces in general. The resulting generalization of Kac-Moody group as super-symplectic group replaces finite-dimensional Lie group with infinite-dimensional group of symplectic transformations and leads to what I call strong form of holography in which AdS is replaced with 4-D space-time surface and Minkowski space with 2-D partonic 2-surfaces and their light-like orbits defining the boundary between Euclidian and Minkowskian space-time regions: this is very much like ordinary holography. Also imbedding space M4× CP2 fixed uniquely by twistorial considerations plays an important role in the holography.
AdS/CFT realization of holography is therefore not absolutely essential. Even better, its generalization to TGD involves no fictitious boundaries and is free of problems posed by closed time-like geodesics.
Perfect tensors and tensor networks realized in terms of magnetic body carrying negentropically entangled dark matter
Preskill et al suggest a representation of holography in terms of tensor networks associated with the tesselations of hyperbolic space and utilizing perfect tensors defining what I call negentropic entanglement. Also Minkowski space light-cone has hyperbolic space as proper time=constant section (light-cone proper time constant section in TGD) so that the model for the tensor network realization of holography cannot be distinguished from TGD variant, which does not need AdS at all.
The interpretational problem is that one obtains also states in which interior local states are non-trivial and are mapped by holography to boundary states are: holography in the standard sense should exclude these states. In TGD this problem disappears since the macroscopic surface is replaced with what I call wormhole throat (something different as GRT wormhole throat for which magnetic flux tube is TGD counterpart) can be also microscopic.
Physics of living matter as physics condensed dark matter at magnetic bodies?
A very attractive idea is that in living matter magnetic flux tube networks defining quantum computational networks provide realization of tensor networks realizing also holographic error correction mechanism: negentropic entanglement - perfect tensors - would be the key element! As I have proposed, these flux tube networks would define kind of central nervous system make it possible for living matter to experience consciously its biological body using magnetic body.
These networks would also give rise to the counterpart of condensed matter physics of dark matter at the level of magnetic body: the replacement of lattices based on subgroups of translation group with infinite number of tesselations means that this analog of condensed matter physics describes quantum complexity.
I am just a novice in the field of quantum error correction (and probably remain such) but from experience I know that the best manner to learn something new is to tell the story with your own words. Of course, I am not at all sure whether this story helps anyone to grasp the new ideas. In any case, if one have a new vision about physical world, the situation becomes considerably easier since creative elements enter to the story re-telling. How these new ideas could be realized in the Universe of TGD bringing in new features relating to the new views about space-time, quantum theory, and living matter and consciousness in relation to quantum physics.
For the details see the new chapter Holography and Quantum Error Correcting Codes: TGD View or the article with the same title
For a summary of earlier postings see Links to the latest progress in TGD.