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On their own, however, these framework based on the algebraic coding theory. This assumption are insufficient for reconstructing quantum theory provides a general model of communication and theory, as demonstrated by C∗-algebraic approaches [17]. deals mathematically with errors or ambiguity. Second, Quantum theory has emerged from the reconstruction how is quantum theory different from all other unambigu- program, not only as a description of individual systems ous descriptions? As emphasized above, it must obey a with continuous state spaces, but also requiring an extra condition of continuity and a bound on bipartite correla- axiom about how such systems compose [18, 19]. This tions. second insight must be complemented with a quantitative The use of coding theory is enabled by the definition bound on the amount of correlations given by the Bell in- of observer in information-theoretic terms introduced in equalities [20] and explored in postquantum models [21– Section III. It involves a limit on the complexity of 23]. There exists a fundamental fact about nature: the strings, which (to use a common-language expression) amount of correlations between distant subsystems is lim- the observer can ‘store and handle’. Strings contain all ited by a non-classical bound, e.g., the Tsirelson bound descriptions of states allowed by quantum theory but for bipartite correlations [24]. In our view, all mathe- also much more: they may not refer to systems or be matical alternatives to the Hilbert space formalism must interpretable in terms of preparations or measurements. strive to predict its empirical value. Using the work of Manin and Marcolli, we show that Another avenue leading to the question about the symbolic dynamics on such strings leads to an emergent mathematics of quantum theory begins with the prob- continuous model in the critical regime (Section IV). Re- lem of observer. Quantum theory says nothing about stricting this model to a subfamily of ‘quantum’ binary its physical composition. It only describes the observer’s codes describing ‘bipartite systems’ (Section V), we find information, which must be somehow registered. Hugh strong evidence of an upper bound on bipartite correla- Everett argued that observers are characterized by their tions equal to 2.82537. The difference between this num- memory, i.e., “parts... whose states are in correspon- ber and the Tsirelson bound 2√2 can be tested. The dence with past experience” [25]. It seems reasonable Hilbert space formalism, then, emerges from this math- to assume that differently constituted observers with the ematical approach as an effective description of a fun- same memory size will have equal capacity to register damental discrete theory of ‘quantum’ languages in the measurement results. This is because quantum mechan- critical regime, somewhat similarly to the description of ics uses abstract mathematics: it deals with observers phase transitions by the effective Landau theory. possessing information about systems, assuming noth- ing about material counterparts of these notions. Can this level of abstraction be used to describe observers? II. CODES We find a somewhat puzzling answer to this question in the work of Niels Bohr. Over time Bohr “became more Communication is based on encoding messages that are and more convinced of the need of a symbolization if transmitted in suitable codes using an alphabet shared one wants to express the latest results of physics” [26]. between communicating parties. An alphabet is a finite He had already written extensively on the problem of set A of cardinality q 2. Acodeisa subset C An con- objective description, but he only connected it in 1958 sisting of some of the≥ words of length n 1. A⊂ language with the choice of mathematical formalism for quantum is an ensemble of codes of different lengths≥ using the same theory: “The use of mathematical symbols secures the alphabet. As an example, take the alphabet A = Fq, the unambiguity of definition required for objective descrip- n finite field of q elements. Linear subspaces of Fq give tion” [27]. What exactly Bohr meant remains unclear. It rise to codes called linear. Linearity provides such codes is unlikely that his point was that quantum theory must with extra structure. Another example is given by bi- rely on the Hilbert space, by then a standard tool. Had nary codes of length n based on a two-letter alphabet, it been so, Bohr could have named the Hilbert space say, 0, 1 . Strings of zeros and ones of arbitrary length explicitly, yet he only vaguely referred to “mathemati- belong{ to} a language formed by binary codes with differ- cal symbols”. In the same text Bohr also rejected the ent values of n. Schr¨odinger wave function as a candidate mathematical In full generality, nothing can be stipulated about mes- tool. It is conceivable that Bohr’s view was that such sage semantics, material support of the encoding and de- mathematical symbols remained to be found. If so, their coding operations or their practical efficiency. One can discovery would purportedly guarantee the unambiguity observe, however, that decoding a message is less prone of communication and secure the objectivity of descrip- to error if the number of words in the code is small. On tion. It would then account to a “common-language” the other hand, reducing the number of code words re- basis of physical theory, in line with Bohr’s well-known quires the words to be longer. The number insistence on the role of classical concepts [28]. Our attempt to clarify the meaning of Bohr’s state- log #C R = q (1) ment leads to two questions. First, is there a mathemat- n ical framework that includes both ambiguous and unam- biguous descriptions? In Section II we introduce such a is called the (transmission) rate of code C. 3

One can associate a fractal to any code in the following Mathematical analysis proceeds, however, independently way [29, 30]. Define a rarified interval (0, 1) = [0, 1] of the choice of Zurek’s or our interpretation. For a set q \ m/qn m,n Z . Points x = (x ,...,x ) (0, 1)n can of strings that are code words of code C with rate R, the { | ∈ } 1 n ∈ q be identified with ( n) matrices whose k-th column lower Kolmogorov complexity satisfies [30]: ∞× is the q-ary decomposition of xk. For a code C define n sup κ(x)= R. (3) SC (0, 1) as the set of all matrices with rows in C. q x∈SˆC It is⊂ a Sierpinski fractal and its Hausdorff dimension is n R. The closure of SC inside the cube [0, 1] includes the For all words x SR in a language formed by codes ˆ ∈ rational points with q-ary digits. This new fractal SC is Cr, the lower Kolmogorov complexity is bounded by n a metric space in the induced topology from [0, 1] . Now κ(x) R. Hence the closure SˆR of the fractal SR is kr ≤ consider a family of codes Cr of #Cr = q words of a metric space that describes the handling of words of length nr, with rates: bounded Kolmogorov complexity. It is a ‘minimal’ geo- metric structure corresponding to the notion of observer. k r R. (2) nr ր IV. CRITICAL LANGUAGE DYNAMICS They define a fractal SR = Sr SCr of Hausdorff dimen- sion dim (S )= R. H R A change in observer’s information can be modeled via dynamical evolution on the fractal set SR. In quan- III. BOUNDED COMPLEXITY tum theory, new information enters when a projective or a POVM measurement produce a new string in ob- server’s memory. Taking inspiration from Manin and Any observer’s memory is limited in size. While their Marcolli [30], we represent this process as a statistical me- material constitution may be radically different, different chanical system evolving on the set of all possible strings observers with the same memory size should demonstrate in codes Cr. A change in observer’s information corre- similar performance in handling information. This intu- sponds to a change in the ‘occupation numbers’ λ of ition serves as a motivation for the following information- a words a Sr Cr. The evolution of λa is described via theoretic definition of observer. Hamiltonian∈ dynamics on the Fock space: Definition III.1. An observer is a subset of strings of bounded complexity, i.e., strings compressible below a Hstatǫa1...am = (λa1 + ... + λam )ǫa1...am , (4) certain threshold. with the Keane ‘ergodicity’ condition: This limit can be viewed as the length of observer’s −Rλa memory. If a string has high complexity, it cannot de- X e =1, (5) scribe an observer with a memory smaller that the mini- a∈∪rCr mal length required to store it; but it remains admissible where vectors ǫ 1 belong to the Fock space represen- for an observer with a larger memory. a ...am tation l2(W ( C )) of the set W of all words in the codes Definition III.1 requires a notion of string complex- Sr r C . To be precise, the Fock space is a representation of ity independent of the observer’s material organization. r the algebra defined below in (7); at this stage it is jus- Kolmogorov complexity is a suitable candidate. It has tified by the completeness of W , which by construction already been used in fundamental physics, e.g. by Zurek includes all possible observer information states. If ob- who argued that physical entropy should be defined as server’s information remains within W , then the Keane a sum of Shannon entropy H and algorithmic random- condition gives a meaning to the weights λ as normal- ness of available information [31–33]. The latter was to a ized logarithms of inverse probabilities that a is stored in be understood as information contained in a ‘binary im- observer’s memory. This evolution has a partition func- age’ of the state of the system, defined as Kolmogorov tion: complexity K of the shortest program able to generate it. When the state of the system is known sufficiently 1 Z(β)= . (6) well, K supplies the main contribution to entropy. Zurek −βλa 1 P e argued that this algorithmic component of physical en- − a∈∪rCr tropy can be made observer-independent by discretizing the system on a family of grids that are concisely de- Manin and Marcolli show that at the critical temper- scribable by a universal computer. To extend Zurek’s ature (equivalently, string complexity) β = R, the be- understanding of an observer who ‘interprets’ a string as haviour of this system is given by a KMS state on an containing information about the state of the system, we algebra respecting unitarity [30]. This algebra is built ˆ take such strings to be information-theoretic primitives. out of the geometric object, namely the fractal SC, as

In our approach, strings do not necessarily have an inter- follows. Consider characteristic functions χSˆC (w), where pretation as states of a system: they define the observer. w = w ...w runs over finite words composed of w C 1 m i ∈ 4 and SˆC (w) denotes the subset of infinite words x SˆC number of entangled degrees of freedom. Their informa- that begin with w. These functions can be identified∈ with tional content is represented by strings of identical com- the range projections plexity. For example, measurements in a CHSH-type ex- periment produce binary strings of results for a choice of ∗ ∗ ∗ Pw = TwTw = Tw1 ...Twm Twm ...Tw1 . (7) σx, σy, σz measurements. The no-signalling condition im- plies that the probability of 0 on Alice’s side is indepen- At the low temperature β > R there exists a unique dent of Bob’s settings, and vice versa. Hence the strings type I∞ KMS-state φR on the statistical system of codes, resemble Bernoulli distributions with a Kolmogorov com- which is a Toeplitz-Cuntz algebra with time evolution: plexity equal to the binary entropy of the probability of 0, plus a correction due to the existence of non-zero mutual itn σt(Tw)= q Tw. (8) information between Alice’s and Bob’s outputs. Since both sides enter symmetrically in the CHSH inequality, The partition function is: this correction to Kolmogorov complexity is a priori the same on Alice’s and Bob’s sides. We use this argument to (R−β)n −1 ZC(β)=(1 q ) . (9) replace Eq. (4) with a class of Hamiltonians assumed to − describe a ‘bipartite system’ in the framework of codes. However the isometries in the algebra do not add up to The Kolmogorov order is an arrangement of words unity. Only at the critical temperature β = R, where a a C in the increasing order of complexity [36]. It i ∈ Sr r phase transition occurs for all codes Cr, is there a unique is not computable and it differs radically from any num- KMS state on the Cuntz algebra, i.e., an algebra such bering of ai based on the Hamming distance in the codes that, importantly for our argument, isometries add up to Cr. Words that are adjacent in the Kolmogorov order ∗ unity: Pa TaTa =1. have the same complexity. We now select an Ising-type Critical behavior of the original discrete linguistic Hamiltonian: model is described at β = R by a field theory on the ˆ H2 = X ai aj , (10) metric space SR, which obeys unitarity. By construc- − × tion, this fractal also has scaling symmetry. This yields a ij field theory respecting scaling and unitarity. While there as a dynamical model on the language that describes has been some discussion of models that are scale in- bipartite quantum systems. The sum is taken over N variant but not conformal, we assume that, in agreement neighbors in the Kolmogorov order, i.e. all strings of with Polyakov’s general conjecture [34], this field theory identical complexity. The result of multiplication on bi- is conformal. The field it describes is clearly an emer- nary words is a new word with letters isomorphic to mul- gent phenomenon, for its underlying dynamics is given tiplication results in a two-element group 1 . Hence, in terms of codes. However, within the conformal field for a two-letter alphabet a,b , {± } theory this field, now a basic object, is to be considered { } fundamental. Due to the properties of continuity, uni- a a = b b = b, a b = b a = a. (11) × × × × tarity and to the geometric character of its state space, the conformal model becomes a tentative candidate for a A binary language with N = 6 using H2 gives rises to in- reconstruction of quantum theory. formation dynamics which is, on the one hand, equivalent to information dynamics of a bipartite quantum system and, on the other hand, equivalent to the dynamics of a 3-dim Ising model. This is because a class of Hamilto- V. AMOUNT OF CORRELATIONS nians with N = 6 has the same number of terms as in three spatial dimensions, although the codes that belong Since quaternionic quantum mechanics or, in some lim- to this class are uncomputable due to the properties of ited cases, real-number quantum mechanics can be repre- Kolmogorov complexity. Plainly, one cannot tell which sented in the Hilbert space [35], one should expect that binary codes give rise to the N = 6 situation nor should continuity and unitarity alone do not single out quan- one expect that Hamiltonians Hstat and H2 belong to tum theory. In other words, the conformal model of Sec- the same universality class. However, the equivalence of tion IV likely contains more than a description of ‘quan- (10) with a 3-dim Ising model suggests that, just like tum’ languages. In this section, we do not seek to provide the Ising model itself, the Hamiltonian H2 also exhibits a necessary and sufficient condition that selects only code critical behaviour described by a conformal field theory. words generated by quantum theory. Rather, we pick As it is usually the case in statistical mechanics, criti- out a particular example, namely a class of models corre- cal regime can be studied without knowing the details of sponding to the critical regime of binary codes describing the dynamics. Correlations of order 2 in this regime are measurements on bipartite quantum systems in the usual described by the lowest-dimensional even primary scalar 3-dimensional Euclidean space. ǫ = σ σ in the conformal field theory. This field is sym- First we define an informational analog of ‘bipartite.’ metric;× hence it presents a good candidate to describe the In quantum theory, subsystems that are entangled can be symmetry of bipartite correlations in the CHSH inequal- materially different, but they are described by the same ity under the switch between Alice and Bob. Following 5 the above intuition, we assume that it provides a descrip- VI. CONCLUSION tion of ‘bipartiteness’ within the conformal model. The operator dimension of ǫ is Historically, quantum logical reconstructions of quan- 1 tum theory drive home the importance of the assump- ∆ =3 , (12) ǫ − ν tions of continuity and composition rule. These are two pillars of quantum theory. Freely interpreting Bohr’s dic- where ν is a well-known critical exponent describing the tum that the unambiguous communication of measure- correlation length [37]. ment results requires a mathematical formulation, we The 3-dim Ising equivalence has its limitations since proposed a mathematical framework that sits on these the true metric space of code evolution is not flat space pillars and the idea that the observer is defined by the but the fractal SˆC . Still, it provides significant evidence limited string complexity. The result is a conjecture that H2 has a critical regime. Further, exponential char- on the amount of bipartite correlations slightly differ- acter of the mapping that links the fractal embedded in ent from the Tsirelson bound, but consistent with avail- the unit cube with flat Euclidean space hints at the ex- able experimental results S = ∆ǫ + 2 3.41267 istence of a connection between the critical behaviours 3.426 0.016 [39] and 2∆ 2.82534 2.827≃ 0.017 [40].≤ ± ǫ ≃ ≤ ± of the Ising model and the code. The correlation length Our argument crucially relies on the assumption that in the fractal representation of a language describes a the number of strings possessing the same complexity logarithmic distance from which words are brought in after uncomputable Kolmogorov reordering is N = 6. clusters of equal complexity by the Kolmogorov reorder- This does not need to be so for all codes. Codes with ing. If H2 exhibits a critical behaviour similar to that N = 4 correspond to 2-dim Ising interaction (ν = 1) and of Hstat, then correlations in the critical regime at string give rise to the classical bound on bipartite correlations complexity β = R come from the entire fractal. The Ising 2∆ǫ =2 1 = 2. Codes with N = 8 correspond to 4-dim analogy with the scaling of the correlator of the lowest Ising interaction· (ν =1/2) and give rise to the Popescu- primary even field suggests a power law for the amount Rorlich maximum correlation 2∆ =2 (4 1 )=4. It ǫ · − 1/2 of correlations on the words of equal complexity: is not clear whether binary codes endowed with critical − dynamics exist for other values of N and, if they do, what ǫ(a)ǫ(0) a 2∆ǫ . (13) h i∼ meaning they may have. We conjecture that, due to the exponential mapping be- Although our model is highly speculative, we believe tween spaces, the corresponding correlations in the frac- that it demonstrates the interest to explore quantum the- tal are limited by the logarithm of the RHS of (13). Their ory via novel mathematical formalisms. As Wittgenstein maximum strength 2∆ǫ can be computed based on the said, “A particular method of symbolizing may be unim- value ν =0.62999(5) in [38]: portant, but it is always important that this is a possible method of symbolizing. [This] possibility. . . reveals some- 2∆ǫ =2.82537(2). (14) thing about the nature of the world” [41].

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