CC BY
4
DOI: 10.33693/2313-223X-2021-8-3-29-35
Tri-State+ Communication Symmetry Using the Algebraic Approach
E. Gerck ©
Planalto Research, Mountain View, CA, USA
E-mail: ed@gerck.com
Abstract. This work uses the algebraic approach to show how we communicate when applying the quantum mechanics (QM) concept of coherence, proposing tri-state+ in quantum computing (QC). In analogy to Einstein's stimulated emission, when explaining the thermal radiation of quantum bodies in communication, this work shows that one can use the classical Information Theory by Shannon (with two, random logical states only, "0" and "1", emulating a relay), and add a coherent third truth value Z, as a new process that breaks the Law of the Excluded Middle (LEM). Using a well-known result in topology and projection as a "new hypothesis" here, a higher dimensional state can embed in a lower-dimensional state. This means that any three-valued logic system, breaking the LEM, can be represented in a binary logical system, obeying the LEM. This satisfies QC in behavior, offering multiple states at the same time in GF (3m), but frees the implementation to use binary logic and LEM. This promises to allow indeterminacy, such as contingency, reference failure, vagueness, majority voting, conditionals, computability, the semantic paradoxes, and many more, to play a role in logic synthesis, with a much better resolution of indeterminate contributions to obtain coherence and help cybersecurity. We establish a link between Einstein's and Shannon's theories in QM, hitherto not reported, and use it to provide a model for QC without relying on external devices (i.e., quantum annealing), or incurring in decoherence. By focusing on adequate software, this could replace the emphasis in QC, from hardware to software.
Key words: QuIC, interconnects, communication, bit, qubit, qutrit, qudit, qtrust, tri-state+, information, algebraic, validation, quantum, classical, meaning, coherence
Acknowledgments. The author is indebted to Software Engineer Andre Gerck, Tiffany Gerck Project Manager of Planalto Research, Edgardo V. Gerck doctorate student, and three anonymous reviewers. Research Gate discussions were also used, for "live" feedback, important due to the physical isolation caused by COVID.
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FOR CITATION: Gerck E. Tri-State+ Communication Symmetry Using the Algebraic Approach. Computational Nanotechnology. 2021. Vol. 8. No. 3. Pp. 29-35. DOI: 10.33693/2313-223X-2021-8-3-29-35
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1. INTRODUCTION
In the past few decades, the qubit - a two-level quantum-mechanical system - has attracted considerable attention for its mysterious quantum properties [1; 2]. In trying to open the "black box" in the quantum state of a qubit, with further, better, analysis of the interaction process, one can hope to find a "new hypothesis" where the data make a wider causal sense, bringing much higher speed, cybersecurity, and scalability. The development of nanoelectronics devices also needs to involve quantum computing in allowing prediction of the desired structure of matter. Whether we want to use binary logic, ternary logic, or other is still open. Fast Galois field calculations are available in current binary chips, as well as spintronic methods, all with finite integer fields. They are especially useful for processing-in-memory and neural networks. Digital computers can emulate floating point numbers, continuous results, and CDs can play apparently continuous music from discrete files, in any rhythm, better than analogue. Analogue and continuity have become the approximation, while digital has become the true result. Code is the exact result we see everywhere, while analogue has been more and more deprecated. Work on nanoelectronics may require the development of quantum computers with a fundamentally new architecture. One
is starting to see the possibility of increasing the logic level of representation. This has led to the proposals of three-level systems (qutrit) [3], four-level systems (qudit) [4; 5], and three or more logic level systems called qtrust. The algebraic approach of this work is illustrated by qtrust, that uses a variable number of logic states, over extended finite integer fields with at least a ternary base in GF(3n) [6]. The results suggest that qutrits, qudits, and qtrust offer a promising path toward extending the frontier of quantum computersand possibly nanotechnology. Theoretical work [7] suggests that quantum processors based on three-level logic systems, or qutrits, might require fewer resources to build than one based on qubits. A similar result is offered for qudits [4] and qtrust [6]. Logic does not have to be binary, or incomplete. Ambivalence, e.g., is a valid result in a ternary system.
Here, at this very Moscow State University, Setun (Russian: Сетунь) was a three-level logic computer developed in 1958, as well-known. It was arguably the most modern ternary computer, using the symmetric ternary number system and three-valued ternary logic instead of the two-valued binary logic prevalent in other computers. In 1965, a regular binary computer was used to replace it. But in 1970, a new ternary computer architecture, the Setun-70, was developed; it was implemented as a simulation program running on a binary computer.
QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
This demonstrated that while ternary logic may have computational advantages, binary computers seem to be sufficient. We can extend to more physical arguments, in further topics. For example, the first protocol in quantum cryptography was the BB84, which however may not have taken advantage of the full potential of multiple superposition states [8]. Using three-or-more logic is suitable for describing a quantum cryptography protocol which may have a number of advantages compared to the "binary" BB84 protocol. This is to be published elsewhere. A two-day NSF workshop, held Oct. 31 - Nov. 1, 2019, changed the focus to "Quantum Interconnect" (QuIC), leading to a roadmap focusing on components and introducing QuICs, which report was published by a group with Awschalom and including 34 others [2]. In two-state systems given by qubits, Awschalom [2] seem to present special challenges for QuICs.
Comparatively, the current quantum theory of qubits is linked, however, to the classical "bit", following Boolean or classical logic laws, such as the Law of the Excluded Middle (LEM), which carry only two possible values, "0" and "1", to emulate the workings of a relay circuit, and uses a formless "fluid" analogy of classical information, that can only be blocked (relay open), routed or replicated (relay closed).
It is therefore highly desirable to investigate the model of communication, especially in the quantum regime, of an expected quantum communication system, as key to quantum computation (QC), quantum speed, and cybersecurity.1 In particular, whether an algebraic approach with a three-or-more valued logic in software can satisfy the postulates of quantum mechanics and can be encoded in the standard binary hardware. This also, in reversal of the usual Bohr correspondence principle, which states that the behavior of systems described by the theory of quantum mechanics must reproduce classical physics in the limit of large quantum numbers. Here, the quantum regime dictates the rules of the classical regime, imposing tri-state, and can improve the binary theory of Shannon [9], as considered in Section 5. A shorter version was used in the actual presentation, and is available online at [6].
2. BREAKING THE LEM
The LEM is broken in the double-slit experiment [1] in QM. This is shown experimentally, as confirmed by all experiments to date [1], with such low light intensities so that only one photon would enter the apparatus.
Theoretically, the general state is given by as the one-dimensional Schrodinger equation for bound states in QM [10]:
^d^(x) =[e _ ^ ( x )] x ),
2m dx2
(l)
where E is the energy and V(x) is the potential, with the boundary conditions ^(0) = = 0. Here, ^ is also the coherent superposition of the solutions ^ and where only slit a or slit b are open at the same time:
(a Ь )•
(2)
Thus, the behavior of systems described by the Niels Bohr interpretation of QM 2 does not reproduce classical physics in the limit of small quantum numbers, although it reproduces
for high quantum numbers, being counter-intuitive [1] to our usual observed experience, in those small numbers.
The reason the double-slit experiment is counter-intuitive is because it breaks the LEM.3 Freedom from LEM would save cost, as discussed in Section 5, by using three or more logical states. And in order to allow a better resolution of indeterminate contributions (such as in 1-out-of-3 voting), more than two states can be used to obtain coherence.
While many are considering a far-future and expensive hardware solution for QC, such as quantum annealing, this work sees 'breaking the LEM' as an opening to a "new hypothesis" here. Freedom from LEM would save cost, as discussed in Section 5, by using three or more logical states. And in order to allow a better resolution of indeterminate contributions (such as in 1-out-of-3 voting), more than two states can be used to obtain coherence.
If the world is quantum or not, anyway Newton's calculus and real numbers lead to infinitesimals and infinities, which were shown by Brillouin [13] to be non physical. No one can physically push spacetime arbitrarily close to zero, no without limit, or exclude particle creation or annihilation at low frequecies. Galois fields and finite differences can be used, however, to build an alternative to conventional calculus, without infinitesimals or the limit concept, so that the conventional approach of analysis need not to be the only approach, for physicists [14]. The derivative and integral formulas, however, remain the same. The new finite difference theory can be perfectly accurate, and yet there is always a space between integers, which represent different points in spacetime.
However, physics is showing that, although non physical, one can keep using infinitesimals and infinities in mathematics, and not change analysis (i.e., calculus) or limits. This is because a higher dimensional state can embed in a lower dimensional state, as well-known in projection. In physics, the universe can have singularities, be quantum at the core, and yet reality should be the consequence of a continuous-looking universality [12; 13] -where we observe this through what can only ever be a far-away reference frame. The details of the microscopic, even breaking the LEM, should not be so relevant to the macroscopic behavior and asserting the LEM, in universality. Can the same be affecting communications, that we are not seeing the microscopic?
In 1916-1917, Einstein [15; 16] famously argued that, in addition to the random processes of spontaneous absorption and spontaneous emission in Eq. (1), a third, new, and coherent process of stimulated emission must exist microscopically for physical bodies, as a result providing experimental evidence for the quantum, reproducing exactly the experimental studies of the thermal radiation of bodies in quantum communication, and providing the basis for the later invention of the laser (light amplification by simulated emission of radiation).4 This is the so-called black-body radiation law, macroscopic, and even normal light from a candle, a lamp, or, a radio wave, have a stimulated emission component. This has been extended recently, as well-known, with collective effects, such as superradiance and superabsorption, into 5 states, but with no essentially new process. Following the ternary pattern, one can readily predict that superstimulated emission should also exist, as a collective effect, and this pattern should go further.
How can one prevent a next SolarWinds, Microsoft Exchange, and the Colonial Pipeline cyberattacks? 2 We do not support the Copenhagen interpretation [9; 10].
3 One cannot split the photon at the double-slit experiment, notwithstanding Huygens and all classical considerations, such as the Maxwell equations. It would not be one particle anymore [1].
4 More than 55 000 laser-related patents have been granted in the United States.
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This work's "new hypothesis" here, in trying to open the "black box" in the quantum state, with further, better, analysis of the interaction process, that we hope to find, where the data make a wider causal sense. It is GF (3n) ^ GF (2m), for suitable m > n e Z/Zp, where p is a prime, meaning that any three-valued logic system, breaking the LEM, can be represented (i.e, embed) in a binary logical system, obeying the LEM. Therefore, Einstein's "stimulated emission" provides coherence in universality, and applies not only to bodies that we must use to transmit and receive information, but also to how we communicate.
For example, an answer stimulates someone to emit a reply in coherence (stimulated emission), or anti-coherence, and it must be in coherence to be effective, and so on. To communicate, we realize in day-to-day experience that one needs not only information, as surprise, but also coherence, as that which both sides know.
We establish a unity, with this hypothesis, between Einstein's [14; 15] and Shannon's [16; 17] theories, hitherto not reported, with plenty of physical examples, both referring to quantum communication processes of bodies, and extend it to be applicable also when bodies are not used, but communication exists.
3. NETWORK CODING
Network coding, originally proposed in 2000 [18], can now be considered for coherent and secure traffic. Between the source and any of the receivers of an end-to-end communication session, one is not only capable of stopping, relaying and replicating data messages, as Shannon considered, but also of coding incoming messages to produce coded outgoing ones, which has been used classically for attacks and, beneficially, for network coding [19], in preventing attacks, and for peer-to-peer content distribution, since it eliminates the need for content reconciliation, and is highly resilient to peer failures.
The fundamental insight of network coding is that information to be transmitted from the source in a session can be inferred, by the intended receivers, and does not have to be transmitted verbatim. A similar concept is found in the well-known spread spectrum techniques, and in cybersecurity [20], where anchors can be used to correct the information received from the source using, e.g., majority voting (1-out-of-3).
The significant aspect in QC, as the result of coherent superposition in Eq. (2), is still that the actual message is one selected from a set of possible messages. This is achieved by coherence, whereby the message is qualified. As a consequence, it has the proper semantics [21], e.g., the proper meaning.
From this perspective, one is not rejecting Shannon's IT because it is binary, two-state, obeys the LEM, uses binary computers, or uses the "fluid" model, in implementation, but proposing Shannon's IT as part of implementing a larger quantum theory of IT with three or more states in logic behavior in QC.
4. SHANNON - A MATHEMATICAL THEORY OF COMMUNICATION
In 1948, Claude Shannon published "A Mathematical Theory of Communication" [16]. Communication is defined [22] as the process whereby information is transferred from one point in spacetime, called the source, to another point in spacetime, called the destination. Information is what is transferred from source to destination, if nothing is transferred the information
is zero and there is no communication; information can also be seen as surprise, as to what is received. This relates to uncertainty, and information is a measure of uncertainty, which is then related to entropy. The average information is called the source entropy [22].
The once fuzzy concept of "information" was proposed in a precise way, as stated above, to quantify the fundamental unit of classical information, the "bit", and using binary logic, with the LEM being valid.
Previous work has been included, selectively, in the references given so far. However, we do not criticize any of such previous references, that are necessarily wrong when applied to quantum information systems, but say that the symmetries of a binary system, that must use the LEM and binary logic, are insufficient for a suitable quantum communication process.
5. TRI-STATE VERSUS TWO-STATE
This work advances experimentally in binary logic, the observation that, for the same function, computation can be accomplished better even classically, by using three logical states, rather than one can do with binary logic, which necessarily includes the LEM.
This is perhaps surprising but well-known experimentally in complex digital systems [23], that allow designers to separate behavior from implementation at various levels of abstraction, in order to achieve, routinely, million gate chip designs while working with tri-stateTM using [24], a ternary logic system as in Fig. (2).
Our argument, in "modus ponens", is that, a coherent logic state, building a "coherent channel", should exist also in classical Information Theory, although embedded in a binary logic system, in order to be able to model the communication that must exist, analogous to the experimental fact that a physical state of simulated emission must microscopically exist in quantum communication using physical systems of atoms, molecules, and plasma, as well-known by Einstein [15; 16].
The "coherent channel" provides coherence, as behavior in both cases, surprisingly, even in the classical computing implementation using binary logic, LEM computers.
A natural question is then satisfied, whether three-or-more valued logic systems can be embedded in a binary logical system. The answer is yes, as well-known in topology and projection. Achieving freedom from the LEM in the behavior while the implementation can obey the LEM, and saving cost. Cobreros et all. [25] and Fedorov et al. [4] have also analysed it, positively. As shown [23], this would work experimentally, but at the expense of performance - in cost, speed, and noise rejection - and scalability.
But this further establishes, in "modus tollens", a physical unity between Einstein's and Shannon's theories in the quantum regime. We can use it to provide a model for QC without relying on external devices (i.e., quantum annealing), or incurring in decoherence [26].
The question is, can a software breaking the LEM, run on LEM hardware? We discuss it positively, and tri-state can then offer many more discriminating channels than binary logic, allowing a much better resolution of indeterminate contributions to obtain coherence, allowing them to be much better discriminated for and filtered by correlation, not just by clipping.
To those who question that tri-state or more would be somehow "illogical" to consider, one notes that, in unpublished notes, before 1910, Charles Sanders Pierce is well-known
QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
to have soundly rejected the idea that all propositions must be either True or False, as in Boolean logic, the same as Frege in semantics [21]. Pierce developed well-understood rules where the LEM is not valid, including some truth tables. A modern treatment can be seen in the results by Jones [27], and our works in publication.
The two-state logic levels are given in Fig. (1) in the next page, offering: (1) a low-level state "0" when the lower transistor is on and the upper transistor is off; and (2) a highlevel state "1" when the upper transistor is on and the lower transistor is off.
vcc т/
vcc
7
GND
GND
Fig. 1. Example of two-state levels in a circuit, 0 and 1
VCC т/
VCC
vcc т/
7
7
GND
GND
GND
Fig. 2. Example of three states logic: 0, 1, Z
To implement three state logic, a physical possibility is a conventional tri-state buffer or gate.5 This can be seen in Fig. (2), showing the three cases in positive logic:
The solution found for the third logic level, and implemented in [23] devices, was to use a high-impedance state "Z", that allows a direct wire connection of many outputs (e.g., routinely with up to a hundred outputs), to a common line, a bus. This exemplifies a programmable (and coherent) interconnect, by the semantics and a challenge-response system, with the ability to move information between different systems that serve distinct tasks at the same time, with freedom from the LEM in the behavior while the implementation can obey the LEM, saving cost.
Using state Z to behave as a coherent interconnect, current information technologies using challenge-response systems, as in a SystemVerilog design [23; 24], can have a semantics to connect to different systems, can avoid race-conditions, handle faults, and maintain a coherent design across different systems. These aspects can also be programmed dynamically at operation time, using tri-stateTM designs [23].
The complications of using tri-state logic in implementation of the logic synthesis leads to different benefits/drawbacks in each design, such as using more complex three-valued logic gates or simple two-valued logic gates [23]. As explained above, and as further reviewed in the next Section, "any three-valued logic system, breaking the LEM therefore, can right-represent (i.e, embed) in a suitable two-valued logical system, obeying the LEM."
However, they are not equal, not even entirely equivalent. Any hardware description language (HDL) such as SystemVerilog will eventually be synthesized and different vendors offer different synthesis tools to create devices of their making from the same HDL behavior description. Of interest here, an FPGA vendor, e.g., could code the HDL in a module with a bus they designed. However, when the FPGA is actually synthesized from the code, it would have to use a tri-state buffer because an FPGA cannot output tri-state. Solutions by FPGA vendors such as Actel [28] describe other procedures, since there is no physical tri-state logic inside an FPGA. In this case, an Actel FPGA implements internal multipledrivers on a net with multiplexers instead of three-state logic.
But the states are in different dimensions, and a continuous path in the higher dimension (tri-state) would necessarily map into a discontinuous path in the lower dimension (two-state). This happens due to a well-known theorem in topology and projection, important in communication [22], that we call TR, standing for Topological Reduction. Chiral information (3D), e.g., is well-known not be represented in a projection to 2D, but can in a projection to GF(23).
In QC, one can be more precise than physical QM if one makes the model, as the behavior, be more inclusive for coherence, even though implementation should be limited, for practical reasons, to use GF (2m) and use the LEM. Hence, QC promises to be easier to realize than QM.
Three-valued logic, even in GF (2m) implementations, besides contingency, reference failure, and vagueness, have been associated with at least four other phenomena of interest -namely, conditionals, majority voting, computability, and the semantic paradoxes [25; 29]. These mathematical processes relate to coherence and are inversely related to indeterminacy.
The addition of a third truth value in ternary logic using GF(3n), or tri-state+, is calculated with n in Table 1. Higher n orders promise to open the floodgates to a large and near unlimited number of outputs that can be simultaneously considered, for speed and cybersecurity, with many more distinct operators, whatever base one uses. As shown in Table 1, using tri-state+ offers many more discriminating channels, as near 6e + 347 possible outputs exist with n = 3, for a possible choice of n, than GF (2), or binary logic, with two-states, allowing a much better resolution of contributions, with a much improved correlation.
One feels the need to introduce more symmetries than GF (2), or binary logic, in Shannon IT [16]. No longer should we be forced to regard information as a formless "fluid", which can only be blocked, routed, or replicated, obeying the LEM as a "Procrustean bed".6 For example, one can use majority voting, or 2-out-3 function, even in GF(22).
6. THE ALGEBRAIC APPROACH IN QC
This work is based on the algebraic properties of modular arithmetic in number theory, following QC in three or more states, as in GF (3n) or GF (2m), and breaking the LEM.
Such as the 74LS241 octal buffer.
6 Where binary logic, an arbitrary standard, is used to measure success, while completely disregarding obvious harm that results from the effort.
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Modular arithmetic, which is widely known, is a system of arithmetic for finite integers, where numbers "wrap around" when reaching a certain value, called the modulus. One considers two integers x and y to be the same if x and y differ by a multiple of n, and write this as x = y mod n, and say that x and y are congruent modulo n. Intuitively, division should 'undo multiplication', that is 'to divide' x by y means to find a unique number z such that z times y is x. A unique z exists modulo n only if the greatest common divisor of y and n is 1, and we say that y and n are co-prime.
But while the foregoing is clear, as a subject, Galois fields has also to do with the structure of groups and the relationship with the structure of fields, and how the roots of a polynomial relate to one another. For example, it is easy to implement how finite integer division works, see above, using direct Galois numbers such as GF (2), but it is more involved with extended Galois numbers, such as GF (2m). Due to universality in physics, seen above, these perhaps confounding factors are of no direct concern here, in communication at a higher level, and are perhaps of interest only to some mathematicians.
In mathematics, a 'field' is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. The group of finite integers modulo p, where p is a prime number, is denoted in mathematics by Z/Zp. It is well-known that Z/Zp:
1) is an abelian group under addition;
2) is associative and has an identity element under multiplication;
3) is distributive with respect to addition, under multiplication;
4) is a field.
A mathematical field with a finite number of members is known as a finite field or Galois field. This name is the only property of Galois fields that interest us here.
Finite fields, as GF(pn), has been useful in the fields of cybersecurity [20], error-correction [30], and encryption, with the well-known AES (Advanced Encryption Standard), where GF (28) is used to translate computer data as they are represented in binary, syntactic forms, using Galois extended finite integer fields GF (2m), with m = 8, as well-known.
This work provides for implementation in a binary gate multi-agent environment, while keeping the ternary behavior, and extending it, offering 3, 9, 27... states. It is then possible that other finite integer numbers could be used, and they all would be mathematical fields, but three states seems supported by the formation of the atomic line with "stimulated emission" [14; 15], universality [12; 13] justifies using any number higher than GF (2), breaking the LEM with QM and Eq. (1) requires at least three states, and using Galois fields already extends exponentially any chosen finite integer, as in Table 1, while offering fast hardware support in today's processors with GF (2m) [30].
With binary logic and diadic operators (2 inputs, binary), there are 16 functions. In electrical engineering, gates implement these functions, notably the AND, OR, NAND, NOR, and XOR (exclusive or) gates. The NAND or XOR gates are functionally complete (meaning that any digital logic circuit can be constructed from either one of these gates) [23]. The XOR function (half adder circuits are implemented with XOR gates) needs brute-force for reversal, and this is basic in cryptographic applications, such as the well-known Advanced Encryption Standard (AES) with GF(28). Table 1, summarizes the results, below.
Table 1
Two-state versus GF(3")
Order Logic states Monadic input operators Dyadic input operators
Binary 2 4 16
1 3 33 33 x 3
2 32 332 332 " 32
3 33 333 333 " 33
n 3n 33n 33" x 3n
We see in Table 1, that for diadic functions, the number increases considerably for ternary operators, reaching 33 x 3 or 19,683, compared with 16 for binary operators, though not all functionally complete. The diadic ternary functions can help reduce indeterminacy, with a better correlation. There are, clearly, too many functions to enumerate. We will refrain from exploring them, because we can already achieve freedom from the LEM in the behavior while the implementation can obey the LEM.
As the number of states in GF(3n) advance, Table 1 shows that the number of possible operators increase exponentially and, while many are trivial and not functionally independent, a total of near 6e + 347 diadic operators exists with n = 3, to be implemented in QC.
We now remember the statement of the last Section, that "any three-valued logic system, breaking the LEM, can represent (i.e, embed) in a two-valued logical system, obeying the LEM." This calls us to separate behavior from implementation, so that computation, or physical realization, of tri-state+ logic, breaking the LEM, is able to use known binary logic, with LEM components or gate circuits. Proof: the number of binary states in GF (2m) can increase more than the number of tri-states+ in GF(3n), with m > n.
In other words, three states break the LEM, but GF (3) can be realized in GF (2m), which obeys the LEM, can use binary functions, and has already more functions than GF(3), for m = 3. With ternary logic, the number of monadic functions is 33 or 27, while this is exceeded by GF (23), with 256 monadic operators. We achieved freedom from the LEM in the behavior while the implementation can obey the LEM.
Stimulated emission is seen as a necessary, ternary manifestation of coherence, and we propose it (e.g, see our "new hypothesis' in Sections 1 and 6). We call it tri- state+, and it extends itself in a ternary pattern to ever higher orders, captured here by the GF(3n) symmetries, using an algebraic approach where the number of states is not fixed a priori, and coherence effects can be used, further and further, while in communication using GF(2m) implementations.
Here, therefore, the role of an added mathematical apparatus as discussed here for Galois fields, is not to create unnecessary complications in a description of reality, but implies that there exist more adequate and representative pictures of reality where these other number fields can be used as basic elements of the mathematical description [31].
Accordingly, one moves from the classical Shannon Boolean analogy of circuits with relays, valid for the LEM and a formless and classical "fluid" model of information, with a syntactic expression called 'bit', to a quantum tri- state+, where information is given by an algebraic approach with ternary object symmetry, modeled by GF (3n) and implementable as GF (2m).
КВАНТОВЫЕ СТРУКТУРЫ И КВАНТОВОЕ МОДЕЛИРОВАНИЕ QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
7. DISCUSSION
By providing a well-known "Procrustean bed", binary logic shuts off indeterminacy, without processing it, and arrives at a classical, apparently clean and determinate, result that does not take any indeterminacy into account, though reducing design effort.
The properties of such theory have two quantum fatal flaws affecting the "bit" model for QC, in addition to fatal consequences such as the LEM, already disproved in the double-slit experiment in QM. Quantum information is not a formless "fluid", modeled by a simple object, the "bit", and that can simply be blocked, routed or replicated; Shannon's IT is thus not able to take network coding into account. Quantum information further does not always obey binary logic. The Shannon thesis of similitude of communication circuits with relay theory, and with binary logic, thus LEM, is valid only in the classical.
This work's "new hypothesis", called tri-state+, in trying to open the "black box" of QM, is then that any three-valued logic system, breaking the LEM, can be represented (i.e, embed) in a suitable binary logical system, obeying the LEM. This agrees with all the theoretical and experimental evidences, dating from 1916, with Einstein and the existence of the quantum. In this "new hypothesis", where the data can make a wider causal sense with Sannon's IT, the LEM has limited validity in QC, due to useful indeterminacy contributions, that must remain indeterminate in Eq. (2), the QM solution sought by QC. Thus, coherence effects should be used in communication. This is another example of universality in physics.
A new type of industry, of cybercrime, has been developing profitably also since 2000 [32; 33], which can now be checked
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by the "new hypothesis", where coherence effects are used in communication, for cybersecurity7 not just speed.
This work has discussed a possible new, quantum future, with the 'new hypothesis". How communication, as a system device or process, should need the quantum symmetries that we call tri-state+, proposing more states than the current binary logic, and without a "Procrustean bed" (i.e. LEM) in behavior.
This suggestion is directly applicable to QC without relying on external devices (i.e., quantum annealing), or incurring in decoherence. Then, QC behavior must break the LEM, as the double-slit experiment in QM already does, and one can implement it today with QM in a binary computer, even a cell phone, and adequate software, in logic synthesis.
To communicate, hence, one needs not only information, as surprise, as that which the receiving side ignores, but also coherence, as that which both sides know. Building coherence is a task that QC can provide with this model, as described, without any special hardware, providing not just speed but cybersecurity.
As another task opened by this work, multilevel logic and mathematics formulas, and software, need to be described and implemented to take full advantage of tri-state+, yet using binary, LEM computers as we know today to implement; this is being published. This could replace the emphasis in QC, from hardware to software, saving cost.
This research received no external funding. The author also declares no conflict of interest.
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Статья поступила в редакцию 11.06.2021, принята к публикации 16.07.2021 The article was received on 11.06.2021, accepted for publication 16.07.2021
ABOUT THE AUTHOR
Ed Gerck, PhD (Physics), Planalto Research. Mountain View, CA, USA. ORCID: 0000-0002-0128-5875 D. E-mail: ed@gerck.com
СВЕДЕНИЯ ОБ АВТОРЕ
Эд Герк, PhD (Physics) Planalto Research, Маунтин-Вью, Калифорния, США. ORCID: 0000-0002-0128-5875D. E-mail: ed@gerck.com
