Reducing space-time to binary information

We present a new description of discrete space-time in 1+1 dimensions in terms of a set of elementary geometrical units that represent its independent classical degrees of freedom. This is achieved by means of a binary encoding that is ergodic in the class of space-time manifolds respecting coordinate invariance of general relativity. Space-time fluctuations can be represented in a classical lattice gas model whose Boltzmann weights are constructed with the discretized form of the Einstein–Hilbert action. Within this framework, it is possible to compute basic quantities such as the Ricci curvature tensor and the Einstein equations, and to evaluate the path integral of discrete gravity. The description as a lattice gas model also provides a novel way of quantization and, at the same time, to quantum simulation of fluctuating space-time.

identification of fundamental degrees of freedom non-trivial. Solving this problem corresponds to fixing the gauge of the coordinate invariance symmetry of GR at the discrete level.
Here we address this question and present a binary description of space-time in 1+1 dimensions. To do so, we borrow the discretization method of causal dynamical triangulation (CDT) [1]. Following this route leads to a formulation of space-times in terms of a statistical mechanical model, which we identify with a lattice gas model [2]. Fluctuations of space-time are thereby represented by different states of the lattice gas, and the Boltzmann factor for this statistical model is given by the discretized action of GR [3].
The central idea of this work is to replace the dynamical geometry of space-time with a fixed physical lattice with binary degrees of freedom at its vertices. This construction allows us to digitalize the geometrical information content of space-times. More precisely, we show how to construct a foliated triangulation T from a bit array λ by using 'forks', and conversely how to construct a bit array λ from a foliated triangulation T . For the latter construction, one first maps T to its dual T * , T * to an integer string S, and S to λ, as indicated in the following commuting diagram (the details of which will be explained below). Triangulation In addition, we will translate the Pachner moves to operations on the integer string encoding. (The Pachner moves are transformations in the set of triangulations which are ergodic, that is, any two triangulations are related by a finite sequence of Pachner moves [5]. See also [6,7] for related work.) Finally, we will use our binary encoding to formulate meaningful quantities of discrete gravity in 1+1 dimensions in the natural language of information processing.
In order to establish these results, let us first recall some basic properties of CDT [1]. In CDT the continuous manifold of space-time is approximated by a piecewise linear manifold [8], where the edge lengths of the simplices are assumed to be constant (space-like edges have length l 2 edge and time-like edges −αl 2 edge , we shall henceforth assume α = −1, which is one of the points in the 'Euclidean sector' that gives rise to an interesting new phase [4]). Additionally, only manifolds that obey a global proper discrete time are considered, i.e. manifolds with a discrete global time foliation. On the simplicial manifold one can define curvature, an action, and other quantities [8].
We shall here focus on the case of 1+1 dimensions, in which the configuration space T t is formed by all foliated triangulations T with t discrete proper-time steps (see figure 1). Additionally, we restrict ourselves to simplicial manifolds with a fixed topology, such as S 1 × S 1 (periodic boundary conditions (pbc) in time and space), or [0, 1] × S 1 (open boundary conditions in time and pbc in space). We consider only connected simplicial manifolds; this implies, in particular, that there is at least one simplex per spatial slice. The topology of the simplicial manifold is characterized by the Euler characteristic χ := N 0 − N 1 + N 2 , where N 0 , N 1 , N 2 is the number of vertices, edges and faces of T , respectively. Note that in 2 dimensions χ is related to the curvature via the Gauss-Bonnet theorem.
In this setting, the Pachner moves consist of two operations (called Rule 1 and Rule 2; see figure 1). The curvature R i is evaluated at every vertex i: where c i is the coordination number (i.e. number of adjacent vertices) of i. The discretized Einstein-Hilbert action takes the form foliated triangulation corresponding to a flat geometry, since the coordination number of every vertex is 6, excluding the boundaries (see equation (2)). Any other foliated triangulation, such as (b), can be obtained by applying a sequence of the Pachner moves Rule 1 and Rule 2.
where γ and κ are coupling constants related to Newton's constant G N and the effective cosmological constant in the continuum, respectively [1]. The quantization is carried out by means of a path integral formulation of the action (3), which (for a specific topology such as [0, 1] × S 1 ) takes the form Here C(T ) is the order of the automorphism group of T , and is the remnant of coordinate invariance of GR in the discrete theory. Considerable progress has been made over the last few years-in terms of analytic in 1 + 1 [9] and numerical studies in 2 + 1 [10] and 3 + 1 [4] dimensions-to investigate the continuum limit of CDT. Interesting connections with other continuum quantum gravity approaches have also been established [11][12][13]. A connection between foliated triangulations in terms of random walks was established in [14]. Finally, let us mention that other discrete models of quantum gravity have been proposed, e.g. [15,16]. Although we have motivated above that we may interpret our discretizations of space-time as real at a fundamental level, we also mention that some investigations have found that discrete models of space-time tend to suffer from non-adiabaticities of the evolution of fields that live on the space-time, which can lead to more particle production than what seems to be allowed by cosmological observational constraints [17].

From bit arrays to triangulations
We now show how a bit array λ encodes a foliated triangulation T , as indicated in diagram (1). The key observation is that every foliated triangulation T is built entirely (except for the boundaries) out of certain building blocks that we call 'forks'. The idea is the following: while, obviously, the building block of a general triangulation is the triangle, the basic unit of a foliated triangulation is a pair of triangles that share a space-like edge. We identify the fork with this unit; more precisely, each fork consists of one 'center', three legs and two faces (see little diagram on bottom of figure 2(b)). Thus, we can describe a foliated triangulation by 'comparing' it to a reference lattice and specifying what forks are present and what forks are absent. This renders a description which is in spirit similar to that of a lattice gas model, where the description of a fluid, with molecules absent or present, is mapped to the description of a magnet with two-level spins on a fixed lattice [2].
Let us be more precise. Consider a 2D square lattice where a binary variable λ nm is associated with every vertex (n, m) of the lattice, with 1 n N and 1 m M (see figure 2(a)). This forms a bit array λ. To transform this bit array to a foliated triangulation, we put a fork on every site (n, m) it λ nm = 1, and no fork if λ nm = 0. This collection of forks defines a graph T = (V, E ) in a natural way: each center of a fork defines a vertex v ∈ V , and its legs become edges e ∈ E of T . The space-like edge and the time-like edge pointing upwards connect to the first vertex to the left, and the time-like edge pointing downwards connects to the first vertex directly below or to the left (see figure 2). This can be formally described as follows. Let p be a rank 3 tensor with components Then the vertex at site (n, m) is adjacent to the vertices at sites (n + 1, a),

From triangulations to bit arrays
We now show how to map a foliated triangulation to a bit array. The recipe consists of six steps. First, construct the dual graph of T , denoted T * , whose basic building block is the d-fork  (1)). This string representation has certain symmetries. Two integer strings correspond to the same dual triangulation if they are related by a sequence of the following operations: (i) commutations of contiguous integers if the integers differ by at least two, i.e.
Notice that (ii) and (iii) are only applicable for pbc on the spatial slices, while (i) is a degeneracy introduced by the integer string encoding.
Fourth, define a string S flat := (S N , S N , . . . , S N ), consisting of complete integer-sequences, S N := (1, 2, . . . , N). Apply operation (i) to arrange any S in successive integer-sequences, by allowing for incomplete sequences, e.g. (1, 3, 4, 6) (see figure 3(d)). Fifth, define the extended string S E by adding zeros to S wherever there is a missing element, e.g. (1, 0, 3, 0, 0, 6) (see figure 3(e)). Generally, where A := (N, S 1 , S 2 , . . . , S F , 1), and is a vector with components i : Sixth, map S E to λ by arranging the entries of S in a two-dimensional array of size N × M (going from bottom to top and left to right), and then replacing each positive entry by a 1. Formally, where

Lattice size in binary encoding
The minimum lattice size necessary to encode all triangulations T up to F forks distributed on N spatial slices is given by N ×M, whereM = F − N + 1. To show this, let us temporarily put aside the requirement that there is at least one fork on every spatial slice, equivalent to the requirement that S contains every element in {1, 2, . . . , N} at least once. It is easy to see that S = {X, X, . . . , X}, where all elements X ∈ {1, 2, . . . , N} are the same, maximizes where is a vector of length F + 1, such that F E = F + (N − 1)F = F N andM = F. Consequently, by reintroducing the constraint to have at least one fork per spatial slice, the string S that will result in the largestM, has as many elements of the same kind, that are F − N + 1 d-forks on a single spatial slice X. It is best to study this case using the dual brick picture, where it is straightforward to see that d-forks, X ± n, for n 1, can be moved freely in both directions by using operations (i)-(iii) (see figure 4). Therefore any of the N − 1 d-forks can be absorbed in a sequence that contains already one X, and thus the number of sequences necessary to encode all possible triangulations with F forks distributed on N spatial slices is F − N + 1. According to the mapping, see equation (7) in the main text, every sequence in S E corresponds to one row in the corresponding λ.
Note that by construction the integer encoding represents a coordinate-free encoding of triangulations. Consequently, operations (i)-(iii) introduce an equivalence class on the space of all integer strings, and there exists a straightforward algorithm to single out its representatives.  1, 1, 2, 3). See appendix for a more thorough discussion.

Applications of binary and integer encoding
We now rephrase various quantities of CDT in binary/integer language. Last but not least, we compare the degeneracies in the configuration space in the binary with the integer encoding.

Binary equivalent of Ricci scalar
Consider first the Ricci scalar (equation (2)). The coordination number of a vertex at site (n, m) is 3 (because each fork has three edges) plus additional nm edges which depend on the surrounding bit array; explicitly: where F (m ) := m (mod M), and d is the index of the first non-zero entry to the left of m given by 0 < p(n, m + 1, d) < p(n, m + 1, m ) ∀m = d (see Equations (5), (6)). The Ricci scalar R nm is then given by The action (equation (3)) is also easily computed in the binary encoding. To compute the Euler characteristic, note that every fork is associated to one vertex, three edges and two faces, hence it does not contribute to the value of χ . Thus only forks relative to the boundary (that is, those which are placed at the boundary, either in the space or time dimension) contribute to χ . For instance, for a torus (topology S 1 × S 1 ), χ = 0. The volume is then given by twice the number of forks, N 2 = 2F. This, together with C(T ), which can be evaluated exactly for 1+1 dimensional triangulations [18], suffices to evaluate the discretized path integral (equation (4)).
Finally, the Pachner moves can also be expressed in the integer encoding. This allows us to show that the Pachner moves in 1+1 dimensions are ergodic, i.e they generate all simplicial triangulations.

Integer equivalent of Pachner moves
The Pachner moves are certain operations that when applied on a triangulation T generate other triangulations T . They are claimed to be ergodic, i.e. for any pair of triangulations, there always exists a sequence of Pachner moves that one can apply on one triangulation that yield the other. For foliated triangulations in 1+1 dimensions there are only two such moves (called Rule 1 and Rule 2, see figure 1 of the main text). Here we define the Pachner moves in the integer string description, that is, transformations on the integer strings which correspond to applying the Pachner move on the corresponding triangulation, as the following diagram illustrates The action of Rule 1 (R1) is easily understood in the dual triangulation, where it simply exchanges the order of two neighboring consecutive d-forks. In the integer description it corresponds to the following operation where |S i − S i+1 | = 1. Rule 2 (R2) has an equally simple equivalent, duplicating a d-fork and placing it right next to it. It is thus clear that R1 and R2 connect different equivalence classes in the string encoding. This demonstrates the ergodicity of the Pachner moves in 1+1 dimensions: starting from the minimal string S = (1, 2, 3, . . . , N − 2, N − 1, N), successive application of R1 and R2 generate all possible strings, and hence all possible triangulations. Figure 5 shows the action of the 'integer Pachner moves' on triangulations mentioned in the main text.

Degeneracies of the various encodings
The binary and the integer encoding differ in the size of their configuration spaces. To show this, let us for a moment relax the constraint of only considering connected triangulations to get a rough estimate of the size of the configuration space. For F N we can simplify the minimum lattice size required to encode all triangulations to N × F. Thus the number of triangulations in the binary encoding is growing ∝ (2 N ) F , while the size of the integer encoding grows ∝ N F . Since both encodings are ergodic (in the sense of containing all triangulations with F forks distributed over N spatial slices at least once) the binary encoding is less efficient than the integer encoding. The integer encoding has the additional advantage that the degeneracies of the configuration space are related to algebraic symmetry operations acting on the strings (see above).
This can be demonstrated with the help of the following example. Consider the integer string S = (1, 2, 3, 1, 2, 3, 1, 2, 3), representing a triangulation with coordination number 6 everywhere except at the boundaries. Even for this trivial example the mapping to a binary encoding is not unique, if the lattice size is larger than 3 × 3. In figure 6 we have depicted six seemingly different encodings of S.  . We show explicitly how to apply successively R1 and R2 on T to obtain T in both, the simplicial (a) and the integer string (b) encoding. In order to apply R1 it is sometimes necessary to use the pairwise commutation relations (PC), i.e.
Notice that the integer strings S and S correspond to the triangulations T and T of figure 1 in the main text, respectively. The differences in the configuration space, if not taken into account when taking the continuum limit, may lead to a different structure of the phases. We are currently investigating this issue further.

Conclusion and outlook
We have presented a binary encoding of discrete space-time in 1+1 dimensions. To this end, we have borrowed the discretization method of CDT and shown how to compute various quantities of interest of this theory in the binary description. Our results enable us to express a classical theory of discrete space-time as a lattice gas model. This approach has several potential applications. To begin with, it provides a natural framework for quantization. Forks, the elementary geometrical unit, constitute independent classical degrees of freedom or 'normal modes' of discrete space-time. In the quantized theory, the presence or absence of a fork will be interpreted as elementary excitations of a quantum lattice gas. These excitations will give rise to quantum fluctuations and, more generally, to quantum states representing superpositions of different space-times. Despite using a time foliation, our approach to quantizing spacetime is conceptually different from CDT quantum gravity, which is based on a path integral formulation.
Discretization of space-time was introduced as a regularization tool to compute the path integral of general relativity. If one considers discrete space-time as real, then forks could be seen as fundamental constituents of space-time, namely as basic geometric structures that connect events. A quantum formulation of this theory would naturally introduce quantum correlations in space-time.
As for the extension of our formalism to higher dimensions, like 2+1 or 3+1, we point out that the fork formalism that we have worked out in 1+1 dimensions, needs additional work and insight in order to apply it to higher dimensions. This can be seen from several facts, in particular, in higher dimensions we have additional geometrical constraints besides the Euler characteristic and we also have additional Pachner moves. Therefore, this important line of research calls for more elaboration and future work.
Finally, from an experimental perspective, by implementing quantum lattice gas models in the way we have introduced them, one could realize quantum simulations of fluctuating space-time. Such quantum simulators, analogue or digital, could be used to simulate models of quantum gravity in modern quantum optics laboratories, e.g. using ultra-cold atoms in optical lattices or ion-traps [19].

Appendix. Representatives of equivalence classes in all three encodings
To evaluate the path integral of CDT in any dimensions analytically, one has to first single out triangulations which are representatives of their equivalence classes. We demonstrate how this procedure can be done in integer string encoding. First, note that to encode all triangulations of size V = 2F distributed over N spatial slices, one has to generate all strings containing at least N forks and at most F of them containing each element of {1 . . . N} at least once. We then apply symmetry operations (i)-(iii) as illustrated in the following example.
Consider all triangulations containing up to five forks distributed over three spatial slices. We require at least one fork on every spatial slice; the smallest (in terms of 2-volume) triangulations contain three forks. The procedure to find the representatives for the equivalent classes is the following: (1) generate all possible integer strings of lengths 3, 4 and 5, with elements in S i ∈ {1, 2, 3}; (2) consider only strings that contain every element {1, 2, 3} at least once (so that only connected triangulations are taken into account); (3) apply symmetry operations to single out unique triangulations/integer strings: if two integer strings are related by a sequence of the symmetry operations (i) and (ii) they are in the same equivalence class. One representative of each equivalence class give all unique histories/triangulations. It is best to demonstrate this at hand of an example. What are the representatives of triangulations containing four forks distributed over three spatial slices? From basic combinatorics we are left with nine valid integer string encodings: Altogether we can for example single out the following representatives: S 1 , S 4 , S 5 , and S 7 .
Next we present the binary and the integer description for this set of triangulations explicitly. For the binary description we need to consider arrays of size N ×F − N + 1 = 3 × 3 (see section 2 in the main text).

A.1. Representatives for three forks distributed over three spatial slices
There is only one way to distribute three forks on three spatial slices, such that