Quantum Information Hidden in Quantum Fields

: We investigate a possible reduction mechanism from (bosonic) Quantum Field Theory (QFT) to Quantum Mechanics (QM), in a way that could explain the apparent loss of degrees of freedom of the original theory in terms of quantum information in the reduced one. This reduction mechanism consists mainly in performing an ansatz on the boson field operator, which takes into account quantum foam and non-commutative geometry. Through the reduction mechanism, QFT reveals its hidden internal structure, which is a quantum network of maximally entangled multipartite states. In the end, a new approach to the quantum simulation of QFT is proposed through the use of QFT's internal quantum network

physical meaning. From a mathematical point of view, this fact is due to the existence in QFT of unitarily inequivalent representations (uir) of the CCR (in the infinite volume or thermodynamic limit). The two particularly important cases of linear transformations, are the boson translation [2] [9] [23] [24] [25] and the Velatin-Bogolijubov transformation (for bosons) [26] [27].
In the context of conventional approach to Quantum Field Theory, when we try to explain the interacting theories, more than one class of representations is needed. The said phenomena was observed bay Haag and sometimes is called the Haag's no-go theorem, which states that free and interacting fields must necessarily be defined on different unitarily inequivalent Hilbert spaces.
The main problem is that when we try to construct a physical theory, by considering, e.g., the Poincare symmetry, we select just one of these classes and simply forget about the existence of others. This causes some problems such as Haag' s no-go theorem. On the other hand, the formulation of S-Matrix is such that one can find the final state by operating S-matrix on the initial state without taking into account the moment of interaction, regarding it as a black box. But it is the moment of interaction that all of these classes may become equally important.
The fact of ignoring the moment of interaction derives from the common attitude of the practitioners to adopt an ontology of events, instead of an ontology of processes.
In many domains of Physics, an ontology of events seems to be the only possible one, or at least the most convenient. The typical case is the scattering of particles. All that we can practically observe are the events before and after the scattering. All that happens in the meanwhile is unknowable, and the only theories we can make concern the correlations among input and output events. This way of reasoning is sufficient for many practical purposes. Let us now come to processes. A process is a temporal sequence of events. A process is ruled on by some dynamical law which characterizes the process itself .This is exactly the structural content of QFT, as stressed and explained in Refs. [1], [2] and [9], where it is clarified that the dynamics, expressed by the equations for the interacting fields (also called Heisenberg fields), defines and characterizes the theory under study and "manifests itself" in the observable physical fields at the level of the observations. Events are thus the "manifestations" of the underlying dynamics (the process).
For example, a calculation is ruled on by the implemented algorithm. An ontology of processes does not deny that observations are about events, but hold that events are explained only in terms of the underlying process, and that the descriptions of events and processes are somehow inseparable. The expression "ontology of processes" has been borrowed from information science, where it has been introduced within the context of space-temporal databases (see, for instance, Kuhn [28]). In particular, in the reduction mechanism of QFT to QM illustrated in this paper, it is extremely important to take due account of the moment of interaction, that is, to assume an ontology of processes, as we will see in Sect. 8 (and in the Conclusions). Avoiding doing so would lead to an internal classical computational structure of QFT, that is of no real help in the simulation of the latter.
Theoretical research on quantum simulation of QFT is very urgent nowadays, because it should support important experimental applications, mainly in high energy physics (HEP), and also for setting up the fundaments of theoretical computer science (TCS) [29].
The paper is organized as follows.
In Sect. 2, we make an ansatz on the boson field operator, we take the spatial slice at constant time equal to zero, and show that the centre of an open sphere of rational radius n r n / 1 = , where n is a positive integer, of the induced topology is an attractor, through which there is a flow of degrees of freedom of the boson field.
In Sect. 3, we describe a new metric within the attractive basin, which is quantized in Planck length units. This metric undergoes quantum fluctuations, which are maximum for an open sphere of unitary radius, and disappear when approaching the attractor. So, in the classical limit, the attractor would become a singularity in which all the degrees of freedom of the boson field would be lost.
In Sect. 4, we show that carrying out the ansatz on the boson field corresponds, in the momentum representation, to performing a boson transformation, which defines a new vacuum state. In the limit of infinite volume, the two representations are unitarily inequivalent, but in the finite volume of the attractive basin they are equivalent. This means that in the attractive basin, the original Quantum Field Theory has shrunk to Quantum Mechanics.
In Sect. 5, we hypothesize that, in the presence of quantum fluctuations of the metric, the surface of a sphere of radius n r , which incorporates the attractor basin, encodes quantum information (this will be formally demonstrated in Sections 6 and 7) and that in this case there is a relationship of uncertainty between the metric and quantum information. Due to the uncertainty relation, there are some missing qubits on the sphere, corresponding to the mixed states between two spheres of radius n r and 1 + n r respectively.
In Sect. 6, we illustrate the origin of the quantum information encoded by the surface of the sphere that incorporates the attractor basin, that had been hypothesized in the sect. 5. We show that when taking into account the ansatz (in the finite volume of the basin), the quantum fluctuations of the metric induce an uncertainty in the position state, giving rise to a superposed state that can be interpreted as a qubit. Furthermore, we show that two cat position states are maximally entangled (forming a Bell state) on the surface of the sphere, while the reduced state, which is completely mixed, is inside the sphere. Since the maximum entanglement entropy (mutual information) of the Bell state is equal to the maximum von Neumann entropy of the fully mixed state, there is no loss of quantum information. Hence, we extend this result to multipartite maximally entangled states such as the Greenberger -Horne -Zeilinger (GHZ) [30] states.
In Sect. 7, we show that in order to have n-partite maximally entangled states on the surface of the sphere enclosing the attractor basin, such a sphere should be a (modified) fuzzy sphere with rational radius n r n / 1 = , in the fundamental representation of SU(2) (with two elementary cells).
In fact, in the case of a usual fuzzy sphere, the latter would encode n qubits in the n N 2 = irreducible representation of SU(2) [22], each one of the N cells encoding a string of n bits, and such geometrical representation would be that of a separable n-qubits state. In the case of the modified fuzzy sphere, instead, the n-maximally entangled states are accommodated in two cells, each one encoding either a string n  0 or a string n  1 . Furthermore, in this way the one-to-one correspondence [19] between the Bloch sphere and the usual fuzzy sphere with unitary radius in the fundamental representation of SU(2) is maintained.
In Sect. 8, we show that a quantum computer can simulate both free and interacting fields once the quantum fields are reduced to a quantum network.
Sect. 9 is devoted to the conclusions.

The ansatz over the boson field operator
Let us consider the boson field operators ( ) is the boson number-density operator. Now, let us make an "ansatz" for where the subscript "A" stands for "ansatz", and ( ) x f is given by: In (2.2) we take the spatial slice at constant time 0 = t , and we choose: Then, the ansatz at constant time 0 = t takes the form: For future convenience, we make the following choice for the constant: A similar ansatz was used for the ) 2 ( SU gauge fields in [15], in order to reduce a pure non abelian gauge field theory to quantum information theory on the fuzzy sphere [18]. The complete metric space ( ) d R , 3 , where d is the Euclidean metric ( ) ' ' , x The open ball of radius n r centred at * x  is: Now, let us make the following natural choice for ( ) It is easy to check that  for large values of n (i.e., for smaller radius of the ball):

The ansatz and the quantum foam
It should be noted that the choice (2.
Then, the new metric inside the basin of the attractor is: The function ) (x   in (2.6) can be rewritten as: The absolute value of the variation of n g with respect to n is: The maximal fluctuation occurs for P L L = , that is at the Planck scale: The Wheeler relation was extended to the case of a quantum de Sitter space-time [32], which is discretized by spatial slices: where P t is the Planck time: . sec 10 3 . 5 In that context, the Wheeler relation takes the form: . This suggests that in absence of the quantum fluctuations of the metric, that is, in the classical limit, the attractor would become a singularity where all the degrees of freedom of the boson field would be lost.

From QFT to QM
In absence of the ansatz, the operator annihilates the vacuum 0 : The number operator is: and the v.e.v. of the number operator is zero: meaning that there are no particles in the vacuum state 0 .
In the k  -momentum space representation, the annihilation and creation operators ( ) The annihilation and creation operators ( ) By defining: where, in the case of homogeneous condensation, it is: We define the new vacuum: The number of quanta with momentum k  is: (4.12) The generator of the boson transformation (4.9) is a unitary operator U: is called the displacement operator [33] [2] [9].
Then the new vacuum is: The scalar product of the two vacua is given by: Now, let us consider, for example, the case of an homogeneous condensation, namely We have: In the infinite volume limit  → V the exponential tends to zero, and the two vacuum states are orthogonal: This means that the two representations are unitarily inequivalent. While this fact is admissible in QFT, it is forbidden in QM by the Stone-von Neumann theorem [7] [8].
In our case, we are considering a finite spatial volume, which is the basin of the attractor, that is the open ball of radius Then, the two vacuum states in this case are not orthogonal: (where the subscript A indicates the ansatz) and the two representations are now equivalent. More in detail, the volume of the ball is: (4.20) By replacing (4.20) in (4.17) we get: ( ) There is a countable set of vacuum states ( )

Metric-quantum information uncertainty relation
In [32] it was shown that the quantum information n I (the total number of qubits) stored by the n th cosmological horizon of a discrete quantum de Sitter space-time, with quantum fluctuations of the metric: Here we assume that the quantum information encoded by the surface of a sphere In [34] it was shown that n I  is the number of the "virtual qubits" (whose occurrence is due to the quantum fluctuations of the metric) which will be transformed into real qubits by Hadamard gates at each node.
Then, within the basin of the attractor, that is, the fluctuations of the metric get the maximal value, the maximal radius is the unit radius of the Bloch sphere, and quantum information In what follows, we will analyze the distribution of both the "real" qubits (pure states encoded on the surface of the sphere n S ) and "virtual" qubits (mixed states in the interior of a sphere, for example in between sphere n S of radius n r n / 1 = and sphere If we wanted to rebuild now the quantum fluctuations of the metric n g n / 1 ) ( =  at level n from virtual quantum information, we would see that the latter is too large because of the uncertainty relation (5.3) : This means that not all virtual qubits are transformed into pure states on the surface of the sphere n S , but some of them are transformed into mixed states below the surface.
Then we redefine the virtual quantum information as: where: This is the number of degrees of freedom released by the quantum field at level n . The remaining n states are n qubits at each level n .
Note that the redundancy of n I  is peculiar of the uncertainty relation (5.3), which instead was not valid in [34]. Now, the information n I  is not really lost, but simply transformed into the entanglement entropy of pure n-multipartite states maximally entangled at every level n, as we will see in Sect. 6.
More in detail, we will consider n-maximally entangled states: for  At this point one might argue that transforming quantum information into entropy means that there is anyway a loss of quantum information, which is forbidden by the conservation principle of quantum information. But this not the case, because entanglement entropy is strictly related to mutual quantum information. This relationship, which is fairly well known, will however be briefly discussed in Sect. 6.
However, it must be said that the transformation of "missing" information into mutual information is a process that can take place as long as the discrete quantum structure of space holds up, since in the classical limit this is no longer possible. In fact, we would get: n qubits are maximally entangled, with maximal entanglement entropy, which is 1, equating the quantum entropy of the fully mixed states. In more detail, at level 2, we have a Bell state with maximal entanglement entropy (equal to1), and there is a fully mixed state with quantum entropy equal to1.
At level 3, we have a maximally entangled 3 GHZ state, there are 2 missing qubits, there is a fully mixed state, which, together with the one at level 2, equates the quantum information balance. In fact, the sum of the maximal entanglement entropies of level 2 and level 3, is 2, which is equal to the sum of the quantum entropies of the mixed states at levels 2 and 3.
At level 4, we have a maximally entangled 4 GHZ state, there are 3 missing qubits, there is a fully mixed state, which, together with those of level 3 and level 2, equates the quantum information balance. In fact, the sum of the maximal entanglement entropies of level 2, level 3, and level 4, is 3, isequal to the sum of the quantum entropies of the mixed states at levels 2, 3 and 4, and so on.
The whole picture described so far can be formalized by a quantum network, which we call the "Hidden Quantum Network" (HQN) of (bosonic) QFT, with the following rules: 1.   The dotted connecting links are fully mixed states, converging in the attractor A.
The non-dotted connecting links are virtual states.
The nodes, labelled with integers n, represent the n spherical surfaces in the attractor basin.
The free outgoing links on each node n represent n qubit states, which are maximally entangled for any n> 1.
In terms of the n S spheres, the quantum network of Fig.1 can be visualized as in Fig.2. The dots on the circles represent the maximally entangled n-qubits encoded by the n th surface.
The n-1 arrows pointing from the n th surface to A, represent n-1 fully mixed states.
It might be worth pointing out the difference between the quantum network in Fig. 1 and that described in [34], and illustrated in Fig. 3, where all the virtual qubits have been transformed into real ones and have not been mixed. The quantum network of Fig. 3, which illustrates the quantum information content of the inflationary era was called the "Quantum Growing Network" (QGN).

Quantum fields and quantum entropy
The spatial degrees of freedom of the quantum field did "flow" inside the attractor's basin, and filled it with quantum entropy. The latter is the von Neumann entropy defined as: where  is the density operator of the quantum system: while for a generic mixed state it is: Let us consider now the pure quantum state  , which is the one-boson state in the position The density matrix is: (6.5) We remind that is the boson number-density operator: The quantum entropy is zero, as it should be for pure states: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 21 July 2020 Peer-reviewed version available at Quantum Reports 2020, 2, 33; doi:10.3390/quantum2030033 However, when the ansatz is taken into account (in the finite volume of the basin), the quantum fluctuations of the metric (6.7) The above expression of x   in terms of the metric fluctuations was first introduced in [32].
In this case, we can redefine the position state x  as the superposed state: Consider now a second position state ' x  , which can as well be redefined as: In the following, we will consider only the plus sign in both Eqs. (6.8) and (6.9) for simplicity.
Now, let us identify the position state x  and its uncertainty x   with the computational basis states of 2 C , that is, with the logical bits 0 and 1 , respectively: The two states in Eqs. (6.8) and (6.9) are then identified with the qubit cat-states, respectively: 14) The cases a) and b) cannot occur as they do not satisfy Eq. (6.12).
The bipartite entanglement entropy of a pure state In the case of maximal entanglement, then, the following relation holds between the entanglement entropy and the mutual information: Although a Bell state is a pure state: the reduced state is fully mixed: (6.20) Note that the reduced state (6.20) of case c) is identical to the reduced state in case b. Let us consider the metric for which is the same as the reduced state in case c) for  And so on.

Maximally entangled states on special fuzzy spheres
In this section we will formalize and demonstrate the assumption we made since the beginning that the n pure states are encoded by the spheres n S of rational radii n r n / 1 = , and are maximally entangled.
To this aim, we will consider a generalized version of the fuzzy sphere with radius n r r n FS Let us consider the ordinary, commutative sphere 2 S of radius r embedded in 3 R : (7.1) The fuzzy sphere is constructed replacing the algebra of polynomials on the sphere 2 S by the non commutative algebra of complex N × N matrices, which is obtained by quantizing the coordinates i x (i = 1, 2, 3), that is, by replacing the i x by the non-commutative coordinates i X : where ijk  is the three-dimensional anti-symmetric tensor, and k is a parameter called the noncommutativity parameter.
In terms of the new coordinates i X defined in (7.2), the relation in (7.1) becomes: The dimension N of the irreducible representation of SU(2), tht is In this case, the sphere is very poorly defined as only the North and the South poles can be distinguished. However, the higher is the dimensionality N of the representation, the lower is the fuzziness. From Eq. (7.10) The quantum version [36] [37] of the Holographic principle [38] [39] is strictly related to 2dimensional noncommutative geometry. In fact, for the case of one qubit, it was shown [19] that the geometrical representation of the qubit state space (the Bloch sphere) is in a one-to-one correspondence (a bijection) with the fuzzy sphere in the fundamental ( The area of each of the two cells is [15]: where ( ) i p  is the probability to find the i th string in the i th cell, and the fuzzy sphere encodes in total one qubit. See Fig. 4. As a whole, the N=2 fuzzy sphere encodes one qubit.
In the case of many qubits, the non-commutative C*-algebra is the algebra of logic quantum gates, which are  (2). By the GN [41] theorem, to this non-commutative C*-algebra, it is associated a quantum space which is the fuzzy sphere with n N 2 = cells. (For a review on C* algebras see for example [42] [43] [44]). The N cells are the N pure states, and then correspond to the N cyclic vectors, which are strings of n bits. The area of the i th cell is proportional to the probability of finding the i th string in that cell. The quantum background space (the fuzzy sphere) and the quantum computer are then in a oneto-one correspondence.
However, the one-to-one correspondence between the fuzzy sphere and the Bloch sphere is lost for In fact, let us consider, for example the case of two qubits,  To recover the one-to-one correspondence with the Bloch sphere, it will be necessary to slightly modify the definition of the area of the fuzzy elementary cells. To this purpose, we will take the radius of the fuzzy sphere, to be the radius of the sphere n S of rational radius n r n / 1 = .
The non-commutativity parameter in (7.5) becomes: In this case, the fuzzy sphere will be called "special fuzzy sphere", for any N. We see that, once the dimension N of the irreducible representation of SU(2) is fixed, k' depends on n. For

Quantum simulation of QFT: A new approach
In this Section, we propose a new theoretical approach to the topic of the quantum simulation of QFT, at the light of the arguments discussed in so far.
As it is well known, quantum simulation is needed since Feynman [45] showed that a classical computer would experience an exponential slowdown when simulating quantum phenomena, while a quantum computer would not.
Then Deutsch [46] described a universal quantum computer, and Lloyd [47] showed that a standard quantum computer can be programmed to simulate any local quantum system efficiently.
However, there is a conceptual, foundational problem in using quantum computers for simulating QFT, which arises from Haag's theorem [4], which states that free and interacting fields must necessarily be defined on unitarily inequivalent Hilbert spaces. So suppose that a scattering process must be simulated by a quantum computer. This is impossible in principle, because the quantum computer is a quantum mechanical system, with a finite number of degrees of freedom, which, according to von Neumann's theorem, has only unitarily equivalent representations. In other words, if the quantum computer simulates free input fields, let us say in representation F  (where the subscript F stands for "free") it cannot also simulate their interaction in representation I  (where the subscript I stands for "interacting") because it does not exist a unitary transformation Then the interaction process will be a black box for the quantum computer. See Fig. 8. Although the formulation of S-Matrix is such that one can find the final state by operating Smatrix on the initial state without taking into account the moment of interaction, regarding it as a black box, the interaction cannot be simulated by a QC. In fact, it is the moment of interaction that all of the classes of representations may become equally important, and instead the QC is endowed with only one class.
Hence, the "for all useful purposes" black box of the S matrix formulation becomes a true black box with regards to quantum simulation.
So, in principle, we will not be able to directly simulate interacting quantum fields. However, quantum simulation might become possible once quantum fields are reduced to a quantum network of qubits. A seemingly argument was proposed by Preskill [13] and by Jordan et al. [14], who recognized that in order to build up an efficient quantum algorithm to quantum simulate the 4  theory, the field should be "represented with finitely many qubits by discretization of space via a lattice, and discretization of the field value at each lattice site".
In this work, we have shown that a (bosonic) QFT has in itself a hidden quantum information IQFT. Extracting this quantum information, involves the reduction of QFT to a quantum-mechanical system, which is a quantum network, the Hidden Quantum Network (HQN) like the one shown in Fig. 1. Now, if we think of an external quantum simulator QCE, with which we would like to simulate the QFT, we would simulate in fact the HQN, not the original QFT, because the QFT ceased to exist once it revealed the quantum information IQFT it hid . However, in some sense, the quantum network HQN is a kind of "skeleton" of the original QFT, from which we could go back to QFT.
We state the following no-go theorem: "If it exists a representation QC  of the quantum simulator which is unitarily equivalent to the representation F  of the free fields, then it does not exist a representation ' QC  of the quantum simulator that is unitarily equivalent to the representation I  of the interacting fields". That means, in practice, that a quantum algorithm for simulating QFT will be incomplete, unless quantum QFT is reduced to a quantum network HQN, which has only one class of unitarily equivalent representations. Now, the question is: "How can an external quantum computer simulate two interacting quantum networks HQN 1 and HQN 2 ?". The answer is given by "gluing" the two quantum networks as shown in Fig. 9. The connected part of the network in Fig. 9 represents the interaction, and consists of fully mixed states (the dotted connecting links) and quantum fluctuations of the metric (the un dotted connecting links).
The connected part of the network is the most similar to a scale-free growing network, where free links are absent. However, the connected part is deterministic, in the sense that it follows some precise rules, as a lattice.
The disconnected part of the graph (outgoing free links), which consists exclusively of pure states that are maximally entangled, represents the free fields. The free links destroy the structure of a regular lattice, as the configuration of free links changes at each node.
Each node of the graph is associated with a quantum gate of the quantum simulator QC. The presence of maximally entangled states in the quantum network is crucial for quantum simulation, in fact entanglement was shown [48] to be necessary to achieve quantum computational speed-up.
By balancing the quantum entropy of the fully mixed states with the entanglement entropy of the maximally entangled states, it will be possible to simulate both the interacting fields and the free fields of the original QFT.
However, an important requirements is needed: The quantum simulator should be able to simulate mixed states. Actually, such a quantum computer has been described in [49], where the authors define a quantum circuit which is allowed to be in a mixed state and to use quantum operations as gates, not necessarily unitary.
It may be worth stressing the fact that the connected part of the graph in Fig. 9 is filled of fully mixed states, whose maximum von Neumann entropy indicates our total ignorance of the interaction process in the original QFT.
Ignoring the connected part of the quantum network in Fig. 9, is what one does in discarding the mathematical formalism of interactions in QFT, as often happens in constructivist formulations of QFT: only free fields in input and output are taken into account. If virtual states and mixed states were absent in the quantum network of Fig. 9, the computational speed would be much lower. In fact, in this case, the quantum network would be reduced to a n 2 Boolean lattice (n = 0,1,2, ....) represented by the regular tree graph (a binary tree) in Fig. 10. The dramatic consequences of ignoring the quantum-computational structure of the interaction process will be discussed, among other topics, in Sect. 9.

Discussion and conclusions
In this paper we have illustrated a particular process of reduction from QFT to QM. This reduction mechanism proved to be much more complex than just the reduction to finite volume. In fact, some unknown characteristics emerged and several topics were involved.
Among the unknown characteristics there is the quantum-computational structure which is intrinsically rooted in QFT, and the quantum-gravitational origin of this same structure. Among the topics involved, besides those of quantum information and quantum gravity, there are noncommutative geometry (the fuzzy sphere) and quantum simulation. The latter is of great importance as a possible theoretical support for practical applications in scattering processes of elementary particles at high energies. Until now, quantum algorithms to simulate QFT [13] [14] have mainly used lattices.
A lattice is a particular type of "regulator", which allows a computer simulation of QFT. However, while a lattice breaks Lorentz's invariance, our regulator does not, because a fuzzy sphere has the same rotational symmetry as the ordinary sphere. As Maas writes in his lectures [50]: "... it is an unfortunate consequence of our current understanding of quantum field theory that the need to have regulators always implies that some symmetries are broken, no matter what, until the regulator it is not removed". It should be noted, however, that Lorentz's invariance is restored on the fuzzy sphere.
A lattice is a mathematical artefact, as also Preskill says in [13]: " The lattice is an artifice introduced for convenience". In our case, instead, the regulator is physical, because the discretization is induced by the quantum fluctuations of the metric in the attractor basin.
Also, while in the case of a reticular regulator the limit of the continuum is reached (but not always) when the number of sites is huge and the spacing approaches zero, in our case there is the classic limit that is reached when the fluctuations of the quantum metrics vanish near the attractor.
Moreover, while in the case of a reticular regulator the reduction of QFT to QM is not mathematically explicit, in our case it is, since the ansatz corresponds to the execution of a boson translation (as it was illustrated in Sect. 4). This shows how the unitarily inequivalent representations of QFT are reduced to a single class of unitarily equivalent representations of QM.
In Preskill's algorithm for simulating scalar field theories, it was introduced a spatial lattice. To make the theory simulated, they replace the scalar field ( ) the quantum state of the field at a site can be encoded in n qubits. In our case instead, qubits are induced directly by the quantum fluctuations of the metric, due to the uncertainty relationship between metric and quantum information.
Both approaches can lead to satisfactory results, but ours is more physical, not only in a heuristic sense, since it is supported by a rigorous mathematical framework. As Preskill himself says in [13]: " There may be more clever ways of regulating that would improve the efficiency of the simulation... ..".
In any case, however, the profound philosophical meaning of the mathematical role of a regulator is that by reducing the infinite degrees of freedom of QFT to a finite number, allows the quantum simulator and the simulated quantum system to have unitarily equivalent representations even when interactions are present.
Then, it may not be quite true that a quantum computer can simulate any quantum system, and our doubt is shared by Preskill in [13]. We claim that a quantum computer can simulate the hidden quantum network (HQN) of the quantum system under study. More precisely, a quantum computer can be programmed to be in a one-to-one correspondence with the HQN.
We think that QFT is meta-logically described (work in progress) by a "quantum metalanguage" (QML) [51]. If that is true, then QFT is its own semantics (QFT interprets itself). In the reduction process illustrated in this paper, QFT would appear then as the semantics of the quantum logic underlying the quantum information hidden in it. The reduction process (in particular the regulator) would then play the role of a definitional equation [52], which allows the switch from a metalanguage to an object language (the logic). In particular, the quantum version [53] of the definitional equation allows to pass from a QML to the quantum logic of quantum information (QLI).
Hence, the metalinguistic links between assertions, which are interpretable as interactions of quantum fields, are sent to logical connectives between propositions, which correspond to quantum correlations such as quantum superposition and entanglement.
Then, the definitional equation corresponds to the regulator of QFT discussed in this paper. In this logical framework, Haag's theorem simply translates the fact that the QML contains the QLI, as every metalanguage contains the object language. Another important point to discuss is that of the serious consequences of the attitude of discarding the ontology of processes (in this case the ontology of interaction). As we have already pointed out in Sect. 8, the quantum network in Fig. 9 consists of a connected part (which describes the interaction) and a disconnected part (which describes the free fields). So, ignoring the interaction process is tantamount to eliminating the connected part of the quantum network in Fig. 9. The problem, unfortunately, is not only mathematical, but also physical. In fact, by eliminating the connected part of the graph, the balance between the quantum entropy of fully mixed states and the entanglement entropy of the maximum entangled states is lost. This involves the non-conservation of quantum information.
In fact, the result of neglecting the connected part of the quantum graph in Fig. 9 reduces the latter to a pair of binary trees like the one in Fig. 10, one for the input fields and the other for the output fields.
In this way, a Boolean lattice, a binary tree, which is an abstract data type used in computer science, would appear to be the classical computational "skeleton" of the original QFT, which makes no sense. In fact, as we said before, the computational "skeleton" of a quantum field theory is quantum. Using a binary tree to trace the original QFT would mean not recovering the quantum characteristics of the latter.
A fairly unexpected result of this article is that quantum gravity seems to be hidden in QFT just like quantum information does (remember that there is an uncertainty relationship between metric and quantum information). Indeed, the quantum fluctuations of the metric appear in this model within the attractive basin and induce uncertainty in the position states, leading to the definition of qubit states.
It would be worth further looking for a series of relationships between QFT, QM, quantum information, entangled space-time, quantum gravity, non-commutative geometry, quantum metalanguage and quantum logic, since these topics are closely intertwined.
QM and QFT are not at the same level, neither mathematical (due to the appearance of the uir of the CCR in QFT, while in QM this is prohibited by Stone-von Neumann's theorem) nor logical (QFT is described by a quantum metalanguage while QM is described by a quantum logic), neither physical (since Haag's theorem holds in QFT, and therefore the irreducible representations of free fields are unitarily inequivalent to those of interacting fields).
Moreover, QM has a classical space-time background, which is absolute. Instead, we believe that QFT should have an entangled quantum space-time [53] as a space-time background. If you remove the QFT from the background, what remains is the entangled space-time, which is itself a quantum network, quite similar to that depicted in Fig.1. Some relations between the entangled space-time background and meta-logic may be found in a recent paper [54].
To conclude, in this work we have shown that a (bosonic) quantum field theory T has in itself a hidden quantum information IT. Extracting this quantum information, however, involves the reduction of T to a quantum-mechanical system, which is a quantum network QT like the one shown in Fig. 1. Now if we think of an external quantum simulator QE, with which we would like to simulate T, we would actually simulate at least QT, not T, because the latter ceased to exist once it revealed the quantum information IT it hid. In a sense, however, QT is T's "skeleton", and from it we can go back to T, at least that's hope. It might be worth extending this reduction mechanism to fermionic QFT.