# Non-Kolmogorovian Probabilities and Quantum Technologies

## Abstract

**:**

## 1. Introduction

## 2. The Ideas behind Quantum Probability

#### 2.1. In the Beginning, There Was Ontological Probabilities

“I should like only to say this: the determinism of classical physics turns out to be an illusion, created by overrating logical-mathematical concepts. It is an idol, not an ideal in scientific research and cannot, therefore, be used as an objection to the essentially indeterministic statistical interpretation of quantum mechanics.”

“It is no longer the objective events but rather the probabilities for the occurrence of certain events that can be stated in mathematical formulae. It is no longer the actual happening itself but rather the possibility of its happening—the potential, to employ a concept from Aristotle’s philosophy—that is subject to strict natural laws.”

He continues with a stronger assertion:“It was quantum mechanics that first assumed the existence of primary probabilities in the laws of nature, which could not be reduced, by means of auxiliary hypotheses, to deterministic laws, as is possible, for instance, with the thermodynamical probabilities of classical physics. This revolutionary development is considered as final by the large majority of modern physicists, first of all by Born, Heisenberg, and Bohr, with whom I myself agree.”([26], page 46)

“The state of a system (object) being given, only statistical predictions can in general be made about the results of future observations (primary probability), whereas the result of the single observation is not determined by laws, being thus an ultimate fact without cause. This is necessary in order that quantum mechanics may be regarded as the rational generalization of classical physics, and complementarity as the generalization of causality in the narrower sense.”([26], page 46, our emphasis)

“In throwing dice we do not know the fine details of the motion of our hands which determine the fall of the dice and therefore we say that the probability for throwing a special number is just one in six. The probability wave of Bohr, Kramers, Slater, however, meant more than that; it meant a tendency for something. It was a quantitative version of the old concept of “potentia” in Aristotelian philosophy. It introduced something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality.”

#### 2.2. Experimental Evidence Supports the Ontological Probabilities Assumption

#### 2.3. How to Avoid Empirical Evidence and Return to an Ignorance Interpretation of Probabilities?

## 3. The Main Features of Quantum Probability

#### 3.1. Kolmogorov’s Axioms

- (1)
- $\mu \left(\mathrm{\Omega}\right)=1$
- (2)
- For any denumerable family of pairwise disjoint sets ${\left\{{A}_{i}\right\}}_{i\in I}$$$\mu (\bigcup _{i\in I}{A}_{i})=\sum _{i}\mu \left({A}_{i}\right)$$

#### 3.2. Quantum Contexts and Quantum States

- One can perform different experiments on a quantum system (trivial).
- Some of these experiments can be jointly performed. Others cannot (this is a non-trivial quantum feature).
- There is a joint probability distribution (that obeys Kolmogorov’s axioms) for experiments that can be jointly performed. However, the physics changes radically from context to context (consider the wave-particle behavior in a Mach–Zehnder interferometer): the strong dependence on the context is an expression of what can be called Bohr contextuality [57].
- There is no joint (Kolmogorovian) probability distribution for all possible contexts. This is a mathematical expression of the notion of contextuality.
- Experiments with the same content appear as repeated in different (incompatible) contexts (for quantum models of dimensions greater or equal to 3). This happens with a high degree of regularity, and can be given a very precise mathematical description. This is a highly non-trivial feature of quantum physics, which has no classical analog; a sort of “identification principle” seems to be at work here. As a result, empirical contexts are “intertwined” in a complex way. The characterization of this intertwining is one of the main challenges for understanding quantum contextuality.
- The marginal probabilities associated with concrete experiments do not depend on the other experiments that can be jointly performed. This is a highly non-trivial feature of quantum theory and is known as the no-signal condition.

- For each ${f}_{i,j}$ in Table 1, we have a probability space $({\mathrm{\Omega}}_{ij},{\mathrm{\Sigma}}_{ij},{\mu}_{ij})$.
- For each row i of the matrix, we have a joint probability space $({\mathrm{\Omega}}_{i},{\mathrm{\Sigma}}_{i},{\mu}_{i})$.
- ${f}_{1,1}$ and ${f}_{4,1}$ have the same content, but they should not be a priori identified (and a similar conclusion holds for the rest of the repeated random variables).

- (1)
- ${\mu}_{\rho}\left(\mathbf{1}\right)=1$
- (2)
- For any denumerable family of pairwise orthogonal projections ${\left\{{P}_{i}\right\}}_{i\in I}$$${\mu}_{\rho}(\bigcup _{i\in I}{P}_{i})=\sum _{i}{\mu}_{\rho}\left({P}_{i}\right)$$

#### 3.3. Negative Probabilities

- Alternative 1: paste the Boolean algebra and end up with a non-Boolean structure as we did in the previous section (and define the states as usual in the quantum logical approaches).
- Alternative 2: keep using Boolean algebras, but with negative probabilities.

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

## 4. Quantum Information Theory and Quantum Technologies: A Non-Kolmogorovian Perspective

**Definition 5.**

**Definition 6.**

#### 4.1. What Is a Computer?

#### 4.2. Deterministic Classical Computing

**Definition 7.**

#### 4.3. Probabilistic Classical Computing

#### 4.4. Quantum Computing in a Schematic Way

**Step 1**- We initialize the computer in the state $|{\psi}_{0}\rangle =|0\cdots 0\rangle $ (or any other conveniently chosen state).
**Step 2**- Next, we apply a collection of gates represented by unitary operators ${\left\{{U}_{i}\right\}}_{i=1,...,m}$, and obtain a final state $|\psi \rangle ={U}_{m}\cdots {U}_{2}{U}_{1}|{\psi}_{0}\rangle $.
**Step 3**- We perform measurements on a selected set of qubits. Depending on the results obtained, we decide which other steps to follow.

**Step 1’**- We initialize the computer in the state ${\rho}_{0}=|0\cdots 0\rangle \langle 0\cdots 0|$ (or any other conveniently chosen state).
**Step 1’**- Next, we apply a collection of gates. Real quantum gates are only approximately unitary. Thus, a final non-pure state $\rho $ is obtained.
**Step 1’**- We perform measurements on a selected set of qubits. Depending on the result obtained, we decide which other steps to follow. Measurements (readouts) are also noisy (and this has to be considered too).

**Step 1**. Start with an initial reference state $\nu \in \mathcal{C}$.**Step 2**. Apply a collection of automorphisms ${\left\{{U}_{i}\right\}}_{i=1,...,m}$ to reach a desired final state $\mu (-)=({U}_{m}\cdots {U}_{2}{U}_{1})\left(\nu \right)(-)$.**Step 3**. Measure the system when state $\mu $ is reached, check the result obtained; depending on the result, stop the process or continue with the the other steps of the algorithm.

## 5. The Extended Church–Turing Thesis and Quantum Supremacy

The ECTT can be interpreted as follows: every physical evolution can be efficiently modeled using a classical computer. R. P. Feynman was one of the first people who conjectured that there could be quantum systems for which time evolution could not be modeled efficiently using classical computers. Nowadays, we have good reasons to believe that this conjecture is true and that the ECTT is false. Quantum computers seem to outperform their classical cousins. Recent experiments support this idea [5,6,7,8,9]. Related to the failure of the ECTT, we must mention quantum supremacy. It can be defined as follows:All computational problems that are efficiently solvable by realistic physical devices are efficiently solvable by a probabilistic Turing machine.

**Definition 8.**

#### 5.1. Quantum Random Circuits and Cross-Entropy Benchmarking

One may wonder to what extent algorithmic innovation can enhance classical simulations. Our assumption, based on insights from complexity theory, is that the cost of this algorithmic task is exponential in circuit size. Indeed, simulation methods have improved steadily over the past few years. We expect that lower simulation costs than reported here will eventually be achieved, but we also expect that they will be consistently outpaced by hardware improvements on larger quantum processors.

#### 5.2. Why Is There a Quantum Speed-Up?

**Theorem 1.**

- Preparation of qubits in all possible computational basis states;
- All possible Clifford gates (Hadamard, controlled-NOT, and phase gate S); and
- Performing measurements in the computational basis.

- The power of quantum computers seems to rely on their capability of generating a large part of the richness of the quantum state space. Due to the Gottesman–Knill theorem, it is not enough to generate superposed and entangled states. Universal quantum computers need to generate a rich enough subset of the quantum state space.
- Generating the quantum state space is equivalent to realizing all possible rotations implemented by unitary operators (quantum gates) in a coherent way. For this task, a universal set of gates is needed.
- (Quantum) contextuality is directly related to the non-Kolmogorovian nature of the quantum probability calculus.

Quantum systems are the only sources of a true non-Kolmogorovian probability known to us. It is exponentially hard for a classical entity to efficiently simulate a sufficiently rich non-Kolmogorovian probability space. Therefore, quantum advantage can be defined as the ability of a quantum device in generating genuine quantum contextuality, which is directly related to its ability in exploring the quantum probability space by the application of a rich enough set of quantum gates.

## 6. Conclusions

- The probabilistic characteristics of quantum phenomena seem to have an ontological nature (this is the main working hypothesis underlying quantum theory). All empirical evidence to date supports this assumption, and it lies at the basis of the development of technologies that rely on genuine randomness.
- As such, the ontological probabilities involved in quantum phenomena are described by a highly non-classical probabilistic calculus. Quantum systems are, to our knowledge, the only examples of entities obeying those very precise mathematical laws. Therefore, quantum systems are the only genuine sources of a non-Kolmogorovian probability.
- Quantum information theory possibly emerges from the assumption that the devices used to store, process, and transmit information are entities obeying a genuine (not simulated) non-Kolmogorovian probability calculus.
- Quantum computing can be considered a non-Kolmogorovian version of classical probabilistic computing. From this perspective, the orthomodular lattice formed by the projection operators of the Hilbert space is an essential algebraic structure for understanding quantum advantage and quantum contextuality (see also [87]).
- The degree to which a quantum device is able to generate a non-classical probability space can be quantified by appealing to measures, such as the cross-entropy benchmarking (used in recent experiments attempting to demonstrate quantum advantage).

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Four different experiments on a quantum system. Each one is assumed to have a different content from the others.

**Figure 2.**Some experiments can be jointly performed, meaning that the actions needed to implement them can be made at the same time in the same system. For other experiments, this cannot be done. This is an example of incompatibility between experiments.

**Figure 3.**There is a joint probability distribution for the observables contained in every row (i.e., they are compatible). However, experiments taken from different rows cannot be jointly performed (due to the existence of incompatible observables). Furthermore, there is no global probability distribution for all possible experiments (this is an expression of contextual behavior).

${\mathit{f}}_{1,1}$ | ${\mathit{f}}_{1,2}$ | ∅ | ∅ |

∅ | ${f}_{2,2}$ | ${f}_{2,3}$ | ∅ |

∅ | ∅ | ${f}_{3,3}$ | ${f}_{3,4}$ |

${f}_{4,1}$ | ∅ | ∅ | ${f}_{4,4}$ |

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Holik, F.H.
Non-Kolmogorovian Probabilities and Quantum Technologies. *Entropy* **2022**, *24*, 1666.
https://doi.org/10.3390/e24111666

**AMA Style**

Holik FH.
Non-Kolmogorovian Probabilities and Quantum Technologies. *Entropy*. 2022; 24(11):1666.
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**Chicago/Turabian Style**

Holik, Federico Hernán.
2022. "Non-Kolmogorovian Probabilities and Quantum Technologies" *Entropy* 24, no. 11: 1666.
https://doi.org/10.3390/e24111666