We introduce the notion of a p-adic quantum bit (p-qubit) in the context of the p-adic quantum mechanics initiated and developed after the seminal paper of Volovich [Theor. Math. Phys. 71, 574 (1987)]. In this approach, physics takes place in a three-dimensional p-adic space rather than Euclidean space. Based on our prior work describing the p-adic special orthogonal group [Di Martino et al., arXiv:2104.06228 [math.NT] (2021)], we outline a program to classify its continuous unitary projective representations, which can be interpreted as a theory of p-adic angular momentum. The p-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes p.
We introduce the notion of a p-adic quantum bit (p-qubit) in the context of the p-adic quantum mechanics initiated and developed after the seminal paper of Volovich [Theor. Math. Phys. 71, 574 (1987)]. In this approach, physics takes place in a three-dimensional p-adic space rather than Euclidean space. Based on our prior work describing the p-adic special orthogonal group [Di Martino et al., arXiv:2104.06228 [math.NT] (2021)], we outline a program to classify its continuous unitary projective representations, which can be interpreted as a theory of p-adic angular momentum. The p-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes p.
Composing p-adic qubits: from representations of SO(3)p to entanglement and universal quantum logic gates
In the context of p-adic quantum mechanics, we investigate composite systems of p-adic qubits and p-adically controlled quantum logic gates. We build on the notion of a single p-adic qubit as a two-dimensional irreducible representation of the compact p-adic special orthogonal group SO(3)p. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)p mod pk. Then, we tackle the problem of p-adic qubit composition and entanglement, fundamental for a p-adic formulation of quantum information processing. We classify the representations of SO(3)p mod p, and analyse tensor products of two p-adic qubit representations lifted from SO(3)p mod p. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For p=3, we construct a set of gates from 4-dimensional irreducible representations of SO(3)p mod p that we prove to be universal for quantum computation.
This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.
We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.
We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.
It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this paper, we suggest an information-theoretic analysis of this relationship, by proving a particular mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that carry 'minimal amounts of direction information', interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d = 3 and quantum theory on two qubits (including entanglement and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.
The theory of angular momentum and spin in quantum mechanics seems to defy common-sense intuition.We render the theory intelligible again by pointing out that this apparent impenetrability merely stems from an {\em undue} parallel interpretation of the algebraic expressions for the angular-momentum and spin operators in the group representation theory of SO(3) and SU(2). E.g. the correct meaning of the quantumoperator L^z=ℏ(x∂y∂−y∂x∂) is not that it is the operatorfor the z-component Lz of the angular momentum L, but rather the expression of the operator for the angular momentum L when it is aligned with the z-axis. Hence what we are used to note (erroneously) as L^z is not a scalar but a vector operator.The same applies{\em mutatis mutandis} for the spin operators. In the correct interpretation, the wholealgebraic formalism is just the group representation theory for the rotationsof three-dimensional Euclidean geometry. It is thus mere,elementary high-school mathematics (in a less usual, more technical guise) and as such totally exempt of any physics, let alone quantum mysteries.The change of interpretation has no impact on the algebraic results, such that they remain in agreement with experimental data and the algebra quantum mechanics remains correct. It is all only a matter of the correct geometrical meaning of the algebra.All these statements are proved within the framework of the group representation theory for SO(3) and SU(2) which is the basic tool used to describe rotational motion in quantum mechanics.