Quantum theory was radically different from the theories of nature which came before it. One key difference was its use of complex numbers. This opened a longstanding debate over whether quantum theory fundamentally requires complex numbers -- or if their use is merely a convenient choice. Until recently, this question was considered open. However, in a 2021 Nature article, a decisive argument was presented asserting that quantum theory needs complex numbers since real-number quantum theory is inconsistent with the postulates of quantum theory. In this work, we show that this conclusion was premature, and in actual fact, a real-number quantum theory is consistent with the postulates of quantum theory. Our theory retains key features such as representation locality (i.e. local physical operations are represented by local changes to the states). A direct consequence of our results is that quantum theory based on real or complex numbers are experimentally indistinguishable.Quantum theory was radically different from the theories of nature which came before it. One key difference was its use of complex numbers. This opened a longstanding debate over whether quantum theory fundamentally requires complex numbers -- or if their use is merely a convenient choice. Until recently, this question was considered open. However, in a 2021 Nature article, a decisive argument was presented asserting that quantum theory needs complex numbers since real-number quantum theory is inconsistent with the postulates of quantum theory. In this work, we show that this conclusion was premature, and in actual fact, a real-number quantum theory is consistent with the postulates of quantum theory. Our theory retains key features such as representation locality (i.e. local physical operations are represented by local changes to the states). A direct consequence of our results is that quantum theory based on real or complex numbers are experimentally indistinguishable.

arXiv:2504.02808Apr 3, 20250 citations

Quantum mechanics and number theory

WORLD SCIENTIFIC eBooksMay 30, 20160 citations

A quantum number theory

Lucas Daiha, Roberto Rivelino

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ( $q$ -numbers) of a Hilbert space that generate classical numbers ( $c$ -numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$ -number $\textbf{N}$ , such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$ , satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$ -numbers $n \in \mathbb{N}$ . A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$ -number $\textbf{Z}$ , such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$ , $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$ , albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$ -numbers as ` $q$ -number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu $n$ its. To interconnect QNSV of different dimensions, associated to the same $c$ -number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$ , the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators ($q$-numbers) of a Hilbert space that generate classical numbers ($c$-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural $q$-number $\textbf{N}$, such that $\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}$, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural $c$-numbers $n \in \mathbb{N}$. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer $q$-number $\textbf{Z}$, such that $\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}$ and obeys a Lie algebra structure. The eigenvalues of each $\textbf{Z}$ component generate a set of classical integers $m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}$, $\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}$, albeit all components do not generate $\mathbb{Z}^3$ simultaneously. We interpret the eigenvectors of the $q$-numbers as `$q$-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu$n$its. To interconnect QNSV of different dimensions, associated to the same $c$-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset $W \subseteq \mathbb{Q}^{*}$, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.

arXiv:2108.10145Aug 18, 20210 citations

Quantum Computational Number Theory

Song Yuan Yan

Springer eBooksJan 1, 201524 citations

A Realistic Deterministic Quantum Theory Using Borelian-Normal Numbers

T. N. Palmer

The elements of a deterministic quantum theory are developed, which reformulates and extends standard quantum theory. The proposed theory is `realistic' in the sense that in it, a general M-level quantum state is represented by a single real number r. Surprising as it may seem, this real number is shown to contain the same probabilistic information as the standard Hilbert-space state, plus additional information from which measurement outcome is determined. A crucial concept in achieving this is that of Borelian (number-theoretic) normality. The essential role of complex numbers in standard quantum theory is subsumed by the action of a set of self-similar permutation operators on the digits and places of the base-M expansion of a base-M Borelian-normal r; these permutation operators are shown to have complex structure and leave invariant the normality of the underlying real number. The set of real numbers generated by these permutations defines not only the Hilbert space of standard quantum theory, but also, in addition, the sample space from which quantum measurement outcomes can be objectively determined. Dynamical real-number state reduction is precisely described by deterministic number-theoretic operators that reduce the degree of normality of r; from the degree of normality one can infer the standard quantum-theoretic trace rule for measurement probability. All the foundational difficulties of standard quantum theory are described in terms of the proposed theory. It is shown that the real-number states of the proposed theory are precisely its beables.The elements of a deterministic quantum theory are developed, which reformulates and extends standard quantum theory. The proposed theory is `realistic' in the sense that in it, a general M-level quantum state is represented by a single real number r. Surprising as it may seem, this real number is shown to contain the same probabilistic information as the standard Hilbert-space state, plus additional information from which measurement outcome is determined. A crucial concept in achieving this is that of Borelian (number-theoretic) normality. The essential role of complex numbers in standard quantum theory is subsumed by the action of a set of self-similar permutation operators on the digits and places of the base-M expansion of a base-M Borelian-normal r; these permutation operators are shown to have complex structure and leave invariant the normality of the underlying real number. The set of real numbers generated by these permutations defines not only the Hilbert space of standard quantum theory, but also, in addition, the sample space from which quantum measurement outcomes can be objectively determined. Dynamical real-number state reduction is precisely described by deterministic number-theoretic operators that reduce the degree of normality of r; from the degree of normality one can infer the standard quantum-theoretic trace rule for measurement probability. All the foundational difficulties of standard quantum theory are described in terms of the proposed theory. It is shown that the real-number states of the proposed theory are precisely its beables.

arXiv:quant-ph/0205053May 10, 20021 citation

Quantum Mechanics and Number Theory: an

Giuseppe Mussardo

Jan 1, 20100 citations

q

-numbers) of a Hilbert space that generate classical numbers (

c

-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural

q

-number

\textbf{N}

, such that

\mathbf{N}^{2} \equiv N_{1}^{2} + N_{2}^{2}

, satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural

c

-numbers

n \in \mathbb{N}

. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer

q

-number

\textbf{Z}

, such that

\mathbf{Z}^{2} \equiv Z_{1}^{2} + Z_{2}^{2} + Z_{3}^{2}

and obeys a Lie algebra structure. The eigenvalues of each

\textbf{Z}

component generate a set of classical integers

m \in \mathbb{Z}\cup \frac{1}{2}\mathbb{Z}^{*}

\mathbb{Z}^{*} = \mathbb{Z} \setminus \{0\}

, albeit all components do not generate

\mathbb{Z}^3

simultaneously. We interpret the eigenvectors of the

q

-numbers as `

q

-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qu

n

its. To interconnect QNSV of different dimensions, associated to the same

c

-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset

W \subseteq \mathbb{Q}^{*}

, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.