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On consistency of the quantum-like representation algorithm

Peter Nyman

Published Jan 1, 2009• 8 citations
OpenAlex

Abstract

In this paper we continue to study so called “inverse Born’s rule problem”: to construct representation of probabilistic data of any origin by a complex probability amplitude which matches Born’s rule. The corresponding algorithm – quantum-like representation algorithm (QLRA) was recently proposed by A. Khrennikov [1]–[5]. Formally QLRA depends on the order of conditioning. For two observables a and b, b|a- and a|b conditional probabilities produce two representations, say in Hilbert spaces H b|a and H a|b. In this paper we prove that under natural assumptions these two representations are unitary equivalent. This result proves consistency QLRA. Keywords quantum-like representation algorithm, inverse Born’s rule problem, order of conditioning, unitary equivalence of representations. 1

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On hyperbolic interferences in the quantum--like representation algorithm for the case of triple-valued observables

Peter Nyman

The quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov \cite{K1,K2,K3,K4,K5} to solve the "inverse Born's rule problem", i.e. to construct a representation of probabilistic data - measured in any context of science - and represent this data by a complex or more general (A Clifford algebra is introduced for this more general representation) probability amplitude which matches a generalization of Born's rule. The outcome from QLRA will introduce the formula of total probability with an additional term of trigonometric, hyperbolic or hyper-trigonometric interference and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical datas are collected from measurements of two trichotomous observables and the complexity of the problem increased eventually compared to the case of dichotomous observables. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum-like representation. In this paper we will present a class of statistical data which satisfy these nonlinear constraints and have a quantum-like representation. This quantum-like representation induces trigonometric-, hyperbolic- and hyper-trigonometric interferences representation.

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On relations between classical and quantum theories of information and probability

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In this thesis we study quantum-like representation and simulation of quantum algorithms by using classical computers.The quantum--like representation algorithm (QLRA) was introduced by A. Khrennikov (1997) to solve the ``inverse Born's rule problem'', i.e. to construct a representation of probabilistic data-- measured in any context of science-- and represent this data by a complex or more general probability amplitude which matches a generalization of Born's rule.The outcome from QLRA matches the formula of total probability with an additional trigonometric, hyperbolic or hyper-trigonometric interference term and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical data is collected from measurements of two dichotomous and trichotomous observables respectively. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum--like representation. We also study simulations of quantum computers on classical computers.Although it can not be denied that great progress have been made in quantum technologies, it is clear that there is still a huge gap between the creation of experimental quantum computers and realization of a quantum computer that can be used in applications. Therefore the simulation of quantum computations on classical computers became an important part in the attempt to cover this gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation. The second part of this thesis is devoted to adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulations on classical computers. Concretely we represent Simon's algorithm, Deutsch-Josza algorithm, Shor's algorithm, Grover's algorithm and quantum error-correcting codes in the Mathematica symbolic language. We see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include future algorithms in this framework.

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The inverse Born problem in contextual probability theories: quantum spin and continuous random variables

Stefano Bacchi, Luca Curcuraci, Giulio Gasbarri

We revise contextual probability theory and the associated inverse Born problem, solved by the Quantum-Like Representation Algorithm (QLRA) in case of two discrete random variables. After pointing out the special feature and limitations of QLRA for two binary random variables, we generalize the QLRA procedure to solve the inverse Born problem to the case of three binary random variables. We analyze the quantum spin model in the light of our results and showed that can be easily obtained from a contextual probability model. We furthermore study the inverse Born problem in the case of two continuous random variables, exploiting the general idea underlying QLRA.

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Andrei Yu. Khrennikov

We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.

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The Inverse Born Rule Equivalence. On the Informational Limits of Real-Valued Amplitude Encodings and the Measurement of Quantum Advantage in Data Embeddings

Sebastian Zając, Jacob L. Cybulski, Bartosz Dziewit +1 more

When does quantum data encoding provide genuine quantum advantage, and when does it merely rephrase a classically solvable problem? We prove an \emph{Equivalence Theorem} demonstrating that any encoding mapping classical data to real-valued amplitudes, $\vertψ_c\rangle = \sum_i c_i \vert i\rangle$ with $c_i \in \mathbb{R}$ and $\sum_i c_i^2 = 1$, composed with a data-independent parameterised unitary and computational-basis measurement, yields exactly the class of classical quadratic forms. We identify the geometric mechanism driving this collapse: the restriction to $\mathbb{R}$ forces a vanishing Berry connection, removing the complex phases required for data-dependent quantum interference. To operationalize this boundary, we introduce encoding diagnostics -- phase complexity $C[Φ]$ and mode-wise von Neumann mutual information $I[Φ]$ -- and link them to the information-geometric excess $Δg$. We show that for all real-valued encodings, $Δg = 0$ identically. We term the misidentification of such models as evidence of quantum computational power the \emph{Inverse Born Rule Fallacy}. Supported by numerical experiments, our results establish that complex-phase structure is a strictly necessary condition for data-driven (Type~B) quantum advantage.

quant-pharXiv:2602.21350•Feb 26, 2026•0 citations

Quantum-like Representation of Macroscopic Configurations

Andrei Yu. Khrennikov

The aim of this paper is to apply a contextual probabilistic model (in the spirit of Mackey, Gudder, Ballentine) to represent and to generalize some results of quantum logic about possible macroscopic quantum-like (QL) behaviour. The crucial point is that our model provides QL-representation of macroscopic configurations in terms of complex probability amplitudes -- wave functions of such configurations. Thus, instead of the language of propositions which is common in quatum logic, we use the language of wave functions which is common in the conventional presentation of QM. We propose a quantum-like representation algorithm, QLRA, which maps probabilistic data of any origin in complex (or even hyperbolic) Hilbert space. On the one hand, this paper clarifyes some questions in foundations of QM, since some rather mystical quantum features are illustrated on the basis of behavior of macroscopic systems. On the other hand, the approach developed in this paper may be used e.g. in biology, sociology, or psychology. Our example of QL-representation of hidden macroscopic configurations can find natural applications in those domains of science.

arXiv:0705.3898•May 26, 2007•1 citation
In this paper we continue to study so called “inverse Born’s rule problem”: to construct representation of probabilistic data of any origin by a complex probability amplitude which matches Born’s rule. The corresponding algorithm – quantum-like representation algorithm (QLRA) was recently proposed by A. Khrennikov [1]–[5]. Formally QLRA depends on the order of conditioning. For two observables a and b, b|a- and a|b conditional probabilities produce two representations, say in Hilbert spaces H b|a and H a|b. In this paper we prove that under natural assumptions these two representations are unitary equivalent. This result proves consistency QLRA. Keywords quantum-like representation algorithm, inverse Born’s rule problem, order of conditioning, unitary equivalence of representations. 1
The quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov \cite{K1,K2,K3,K4,K5} to solve the "inverse Born's rule problem", i.e. to construct a representation of probabilistic data - measured in any context of science - and represent this data by a complex or more general (A Clifford algebra is introduced for this more general representation) probability amplitude which matches a generalization of Born's rule. The outcome from QLRA will introduce the formula of total probability with an additional term of trigonometric, hyperbolic or hyper-trigonometric interference and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical datas are collected from measurements of two trichotomous observables and the complexity of the problem increased eventually compared to the case of dichotomous observables. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum-like representation. In this paper we will present a class of statistical data which satisfy these nonlinear constraints and have a quantum-like representation. This quantum-like representation induces trigonometric-, hyperbolic- and hyper-trigonometric interferences representation.
In this thesis we study quantum-like representation and simulation of quantum algorithms by using classical computers.The quantum--like representation algorithm (QLRA) was introduced by A. Khrennikov (1997) to solve the ``inverse Born's rule problem'', i.e. to construct a representation of probabilistic data-- measured in any context of science-- and represent this data by a complex or more general probability amplitude which matches a generalization of Born's rule.The outcome from QLRA matches the formula of total probability with an additional trigonometric, hyperbolic or hyper-trigonometric interference term and this is in fact a generalization of the familiar formula of interference of probabilities. We study representation of statistical data (of any origin) by a probability amplitude in a complex algebra and a Clifford algebra (algebra of hyperbolic numbers). The statistical data is collected from measurements of two dichotomous and trichotomous observables respectively. We see that only special statistical data (satisfying a number of nonlinear constraints) have a quantum--like representation. We also study simulations of quantum computers on classical computers.Although it can not be denied that great progress have been made in quantum technologies, it is clear that there is still a huge gap between the creation of experimental quantum computers and realization of a quantum computer that can be used in applications. Therefore the simulation of quantum computations on classical computers became an important part in the attempt to cover this gap between the theoretical mathematical formulation of quantum mechanics and the realization of quantum computers. Of course, it can not be expected that quantum algorithms would help to solve NP problems for polynomial time on classical computers. However, this is not at all the aim of classical simulation. The second part of this thesis is devoted to adaptation of the Mathematica symbolic language to known quantum algorithms and corresponding simulations on classical computers. Concretely we represent Simon's algorithm, Deutsch-Josza algorithm, Shor's algorithm, Grover's algorithm and quantum error-correcting codes in the Mathematica symbolic language. We see that the same framework can be used for all these algorithms. This framework will contain the characteristic property of the symbolic language representation of quantum computing and it will be a straightforward matter to include future algorithms in this framework.
We revise contextual probability theory and the associated inverse Born problem, solved by the Quantum-Like Representation Algorithm (QLRA) in case of two discrete random variables. After pointing out the special feature and limitations of QLRA for two binary random variables, we generalize the QLRA procedure to solve the inverse Born problem to the case of three binary random variables. We analyze the quantum spin model in the light of our results and showed that can be easily obtained from a contextual probability model. We furthermore study the inverse Born problem in the case of two continuous random variables, exploiting the general idea underlying QLRA.
We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability amplitudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible'' mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.
When does quantum data encoding provide genuine quantum advantage, and when does it merely rephrase a classically solvable problem? We prove an \emph{Equivalence Theorem} demonstrating that any encoding mapping classical data to real-valued amplitudes, ∣ψc⟩=∑ici∣i⟩\vertψ_c\rangle = \sum_i c_i \vert i\rangle∣ψc​⟩=∑i​ci​∣i⟩ with ci∈Rc_i \in \mathbb{R}ci​∈R and ∑ici2=1\sum_i c_i^2 = 1∑i​ci2​=1, composed with a data-independent parameterised unitary and computational-basis measurement, yields exactly the class of classical quadratic forms. We identify the geometric mechanism driving this collapse: the restriction to R\mathbb{R}R forces a vanishing Berry connection, removing the complex phases required for data-dependent quantum interference. To operationalize this boundary, we introduce encoding diagnostics -- phase complexity C[Φ]C[Φ]C[Φ] and mode-wise von Neumann mutual information I[Φ]I[Φ]I[Φ] -- and link them to the information-geometric excess ΔgΔgΔg. We show that for all real-valued encodings, Δg=0Δg = 0Δg=0 identically. We term the misidentification of such models as evidence of quantum computational power the \emph{Inverse Born Rule Fallacy}. Supported by numerical experiments, our results establish that complex-phase structure is a strictly necessary condition for data-driven (Type~B) quantum advantage.
The aim of this paper is to apply a contextual probabilistic model (in the spirit of Mackey, Gudder, Ballentine) to represent and to generalize some results of quantum logic about possible macroscopic quantum-like (QL) behaviour. The crucial point is that our model provides QL-representation of macroscopic configurations in terms of complex probability amplitudes -- wave functions of such configurations. Thus, instead of the language of propositions which is common in quatum logic, we use the language of wave functions which is common in the conventional presentation of QM. We propose a quantum-like representation algorithm, QLRA, which maps probabilistic data of any origin in complex (or even hyperbolic) Hilbert space. On the one hand, this paper clarifyes some questions in foundations of QM, since some rather mystical quantum features are illustrated on the basis of behavior of macroscopic systems. On the other hand, the approach developed in this paper may be used e.g. in biology, sociology, or psychology. Our example of QL-representation of hidden macroscopic configurations can find natural applications in those domains of science.