Theory of Quantum Gravity Information Processing

The theory of quantum gravity is aimed to fuse general relativity with quantum theory into a more fundamental framework. The space of quantum gravity provides both the non-fixed causality of general relativity and the quantum uncertainty of quantum mechanics. In a quantum gravity scenario, the causal structure is indefinite and the processes are causally non-separable. Here, we provide a model for the information processing structure of quantum gravity. We show that the quantum gravity environment is an information resource-pool from which valuable information can be extracted. We analyze the structure of the quantum gravity space and the entanglement of the space-time geometry. We study the information transfer capabilities of quantum gravity space and define the quantum gravity channel. We reveal that the quantum gravity space acts as a background noise on the local environment states. We characterize the properties of the noise of the quantum gravity space and show that it allows the separate local parties to simulate remote outputs from the local environment state, through the process of remote simulation.


Introduction
In general relativity, processes and events are causally non-separable because the causal structure of space-time geometry is non-fixed. In a non-fixed causality structure, the sequence of time steps has no interpretable meaning. In our macroscopic world, events and processes are distinguishable in time and, thus, causally separable because the space-time geometry has a deterministic causality structure. The meaning of time evolution is also non-vanishing and has an interpretable notion in the microscopic world of quantum mechanics. It is precisely the reason why classical and quantum computations are evolved by a sequence of time steps and why the term time has an interpretable and plausible meaning in the macro-and microscopic levels. A fundamental difference between the nature of events of general relativity and quantum mechanics is that although the theory of general relativity provides a non-fixed causal space-time structure with deterministic events, in quantum mechanics, the space-time geometry has a fixed, deterministic causality structure whereas the events are nondeterministic. Quantum gravity is provided to fill the gap between these two fundamentally different theories. The theory of quantum gravity combines the results of general relativity with quantum mechanics to construct a more general framework. In quantum gravity, the causal structure is non-fixed, and the events are probabilistic [1][2][3][4][5][6][7]. In the quantum gravity space, the computations and the information processing steps are interpreted without the notion of time evolution. This space-time structure allows us to perform quantum gravity computations and to build quantum gravity computers, which fuse the extreme power of quantum computations and the non-fixed causality structure of general relativity [4]. The space of quantum gravity can be further exploited in quantum communication protocols, in quantum AI, in quantum error correction, and particularly in the development of quantum computers , [41][42][43][44][45][46][47][48][49][50][51].
Besides the attractive properties of quantum gravity theory, the appropriate characterization of the information processing structure of the quantum gravity space is still missing. In this work, our aim was to provide a model for the information processing structure of quantum gravity. We show that the quantum gravity space acts as an information resource-pool and reveal that the quantum gravity space stimulates a noisy map on the local environment states of independent, physically separated local maps. This background noise of the quantum gravity space allows the local parties to simulate remote, physically separated processes in the quantum gravity space, in a probabilistic way. We call this process remote simulation, an event that can be accomplished only as a coin tossing in a fixed causality structure. We also study the entangled space-time structure of quantum gravity and define the partitions over which the information flow between the separated processes is possible. We characterize the properties of the quantum gravity channel and the information transmission capability of the quantum gravity space by the tools of quantum Shannon theory. We introduce the terms quantum gravity memory and stimulated storage, which allow for the generation and storage of qubit entanglement exploiting the information resource-pool property of the quantum gravity space. This paper is organized as follows. Section 2 provides the entanglement structure of the quantum gravity space, the information resource-pool property of quantum gravity, and the structure of the quantum gravity channel. Section 3 studies the information flow through the quantum gravity environment and characterizes the correlation measures. Section 4 provides a quantum gravity memory and introduces the term stimulated storage. Finally, Section 5 concludes the paper.

Information Processing of Quantum Gravity
and in matrix form as whereas 1 E E r  can be expressed in as This partial transpose is non-negative; hence,  will be used in the right-hand side in the equations throughout.) However, in the quantum gravity scenario, the information transmission through the partitions cannot be described by an ideal (i.e., noiseless) map; thus, the local degrading map can be applied only with success probability p . Thus, the remote simulation is a noisy process, that is, it is probabilistic. If the I identity map is realized on i E , then the remote simulation is not possible from i E . This outcome has probability 1 p -.
In particular, the probabilistic remote simulation process can be characterized by a CPTP map   , defined as ( ) and the output of this map is as follows: It is trivial that if the parties have no information about each other, then the remote output j B can be simulated from the local environment i E only with probability and ( ) This is precisely the case in a standard scenario, where the quantum gravity effects are not present. The situation changes if we step into the quantum gravity space, which leads to success probability where AA r ¢ refers to an entangled input system, and where A ¢ and A ¢¢ refer to the input and output systems, and the Kraus-operators where ( ) X p x represents an arbitrary probability distribution, and r A x and r B x are the separable density matrices of the output system. The logical channel performs a complete von Neumann measurement on its input system r and outputs ( )  , which outputs the density matrix s x , together called conditional state preparation: Introducing the notation X P for the X-basis and Z P for the Z-basis, let the local  channels be defined as follows: Let the local  be defined as follows: performs a projective measurement in the Z-basis on the local environment state i E .
Using Equations (31) and (32) along with the local channels For this setting, the state of r  is evaluated as follows: Thus, if 1 X P = P , then Bob simulates Alice's output from his local environment 2 E through the The action of Equations (27) where X and Z are the Pauli operators. By applying the proof of Appendix E from [10], immediately yields that this process matrix identifies a causally non-separable process; and, the . From these arguments, the main conclusion regarding the information resource-pool property of the quantum gravity environment can be derived. In the quantum gravity setting, the local map Thus, from the local environment i E , the remote output j B can be simulated via the local map In particular, the quantum gravity environment acts as a noisy map on the local environment state and behaves as an information resourcepool for the local parties.
The model of remote simulation in the quantum gravity environment is summarized in Fig. 2.
The quantum gravity acts as a noise on the local environments; thus, it behaves as an information resource-pool for the local parties about the remote CPTP maps.
These results confirm that, in the quantum gravity setting, there exists local independent CPTP maps, for which the local environments can be used to simulate the remote outputs with success probability 1 2 p > . The quantum gravity environment, indeed, acts as an information resourcepool for the local parties. ■ In Theorem 3, we reveal the structure of the quantum gravity channel that allows to model the quantum gravity space as an information transmission device between the i E local environment and the remote output j B .
Theorem 3 (The structure of the quantum gravity channel) The local CPTP maps, In Theorem 2, we have seen that by exploiting the extra resources of quantum gravity, Alice can simulate Bob's output with probability 1 2 p > , above the standard limit Here, we show that it leads to a well-defined channel structure-called the quantum gravity channel-between Alice and Bob. The causality structure of quantum gravity space-time geometry leads to an interesting configuration, namely, it brings alive a so-called remote simulation map, which acts locally at the parties, on their local environment states. The quantum gravity channel is referred by the CPTP map  is summarized in Fig. 3.
From Equation (39) where the anti-degradability of the qubit gravity channel where X, Y, and Z are the Pauli operators. One can get the condition sin cos v u ³ , which is analogous to Equation (44)  . Taking the superset  of these gravity channels, the result is a convex set because  formulates a supergravity channel as with complementary channel where F and G are elements of the Stinespring representation.
is as follows: Using Lemma 17 from [40], one can readily see that the super gravity channel  is antidegradable because applying map F Tr on Equation (50)   . This degrading map arises from the extra informational resource-pool property of quantum gravity, and the realization of this map is trivially not possible with probability 1 2 p > in the standard scenario, where the causality is fixed and non-vanishing.

Information Transfer of Quantum Gravity Theorem 4. (Information transfer of quantum gravity) The quantum gravity environment allows the transfer of classical and quantum information between the local maps
where r and s are the Bloch vectors, s s s s é ù = ë û  , , The eigenvalues + -+ -, , , From these eigenvalues, the -£ £ 1 1 and ( ) As one can readily check, for these parameters, the relations + + £ where ( ) S ⋅ is the von Neumann entropy and The amount of purely classical correlation ( ) 2 d = is the dimension of system E r  , and k q makes up a normalized probability distribution in the rank-one POVM elements = k k E q k k [50].
The purely classical correlation can also be expressed by the following formula: where the functions 1 2 , f f , and 3 f are defined as follows [36][37]50]: can be given as the maximization of the coherent informa- The results of the correlation measure analysis are summarized in Fig. 4.

Stimulated Storage in Quantum Gravity Memories
The quantum gravity scenario allows us to build quantum memories with a non-fixed causality.
In this section, we propose an example for this statement. Our quantum gravity memory is a quantum SR latch (S-set, R-reset), built from a pair of cross-coupled Toffoli-NOR quantum gates.
In classical computer architectures, the SR latch (flip-flop or bistable multivibrator) is one of the most basic and fundamental storage elements and building blocks of digital electronics devices. An SR latch consists of two cross-coupled NOR gates for the storing of one-bit information, and it operates with two stable states. The SR latch has two control inputs and two signal inputs, which are the back-looped outputs of the neighboring NOR gate (called cross coupling). The output of the classical SR latch is controlled by the S and R inputs, which allows only one stable output realization, Q , or its complement, Q . The state transitions of the cross-coupling structure have a fixed causal structure in a classical SR latch.
In particular, in a quantum gravity SR latch, both output realizations are simultaneously allowed as stable state, which makes possible the stimulated storage of a qubit entanglement The proposed quantum gravity SR latch exploits the information resource-pool property (see Theorem 2) of the quantum gravity space to preserve the entanglement. The The NOR Toff C Toffoli-NOR quantum circuit is shown in Fig. 5. where where 1 3 k £ , following the structure of (10).
The main contribution of the SR  quantum gravity SR latch is that the non-fixed causality of the E  quantum gravity structure leads to the simultaneous realizations of the Q and Q outputs, which can be used as the stimulation and storage of qubit entanglement, utilizing the resource-pool property of quantum gravity (see Theorem 2). The active S and R control commands are : and and in terms of the control state formalism, the realizations of the local maps is 0 : The truth table of the SR  quantum-gravity SR-latch is as follows: thus, the resulting output of SR  is evaluated as The SR  quantum gravity SR latch with quantum gravity control is depicted in Fig. 6 and the entanglement of Q and Q . The resource for the stimulation and storage processes is provided by the quantum gravity environment.
In this example, we showed that the information resource-pool property of the quantum gravity environment can be exploited in quantum memories. We proposed a quantum gravity memory device and introduced the term stimulated storage, which allows the stimulation and storage of qubit entanglement, exploiting the information resource-pool property of the quantum gravity environment.
The results indicate that the structure of the quantum gravity space can be further exploited in the development of quantum devices and quantum computers.

Conclusions
The theory of quantum gravity integrates the fundamental results of quantum mechanics with general reality. This fusion injects and adds several benefits to quantum mechanics, most importantly the non-fixed causality structure of space-time geometry and the existence of causally nonseparable processes. In this work, we provided a model for the information processing structure of the quantum gravity space. We analyzed the connection of the gravity environment with the local processes and revealed that the quantum gravity environment is an information transfer device. This property makes the use of quantum gravity space as an information resource-pool available for the parties. We introduced the term remote simulation and showed that the quantum gravity space induces noise on the local environment states, which allows the parties to simulate locally separated remote systems. We investigated the terms of quantum gravity memory and stimulated storage, which allows for the generation and preservation of the entanglement of qubits exploiting the information resource-pool property of quantum gravity. The information processing structure of quantum gravity can be further exploited in quantum computations, in quantum error correction, in quantum AI, in quantum devices, and particularly in the development of quantum computers.

S.1 Notations
The notations of the manuscript are summarized in Table S.1.