Abstract
A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Pl\"ucker map well-known form twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\geq 3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side.
Highlights
Since the advent of holography [1,2] and the AdS/CFT correspondence [3] physicists have realized that in order to understand the properties of a physical system sometimes one can find clues for achieving this goal within the realm of another one of a wildly different kind
We identify the bulk as the finite geometric version of an object called the Brody-Hughston “quantum space-time” [31,32]
We show that in our finite geometric version, just like in the n 1⁄4 2 case, this real slice of the bulk for n ≥ 3 can be embedded into a hyperbolic quadric residing in a projective space PðVÞ
Summary
Since the advent of holography [1,2] and the AdS/CFT correspondence [3] physicists have realized that in order to understand the properties of a physical system sometimes one can find clues for achieving this goal within the realm of another one of a wildly different kind. We show that in our finite geometric version, just like in the n 1⁄4 2 case, this real slice of the bulk for n ≥ 3 can be embedded into a hyperbolic quadric residing in a projective space PðVÞ This time the underlying vector space V is a one of dimension 2n taken over the finite field GFð2Þ, equipped with a natural symplectic structure. We work out the simplest case of encoding two qubit observables (boundary observables) into a special set of three qubit ones (bulk observables) via error correction Being very simple this case serves as an excellent playing ground for showing our basic finite geometric structures at work. These detailed considerations aim at helping the reader to visualize our basic ideas, to be generalized later for an arbitrary number of qubits
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