scholarly journals Multiplicative functions arising from the study of mutually unbiased bases

10.53733/99 ◽  
2021 ◽  
Vol 51 ◽  
pp. 65-78
Author(s):  
Berthold-Georg Englert ◽  
Heng Huat Chan
Keyword(s):  
Hilbert Space ◽  
Multiplicative Function ◽  
Quantum Physics ◽  
Exponential Sum ◽  
Mutually Unbiased Bases ◽  
Multiplicative Functions

We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Quantum Physics ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Classical Physics ◽  
Core Set ◽  
The World ◽  
Subordinate Role

It now appears that quantum computers are poised to enter the world of computing and establish its dominance, especially, in the cloud. Turing machines (classical computers) tied to the laws of classical physics will not vanish from our lives but begin to play a subordinate role to quantum computers tied to the enigmatic laws of quantum physics that deal with such non-intuitive phenomena as superposition, entanglement, collapse of the wave function, and teleportation, all occurring in Hilbert space. The aim of this 3-part paper is to introduce the readers to a core set of quantum algorithms based on the postulates of quantum mechanics, and reveal the amazing power of quantum computing.

scholarly journals The Topological Origin of Quantum Randomness

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 581
Author(s):  
Stefan Heusler ◽  
Paul Schlummer ◽  
Malte S. Ubben
Keyword(s):  
High School ◽  
Hilbert Space ◽  
Group Theory ◽  
Lorentz Group ◽  
Quantum Physics ◽  
Time Development ◽  
Mathematical Structures ◽  
Stochastic Nature ◽  
Quantum Randomness ◽  
And Topology

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘4π-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘2π-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a distinction(A,B) using group theory and topology. Crucially, the 2:1-mapping from SL(2,C) to the Lorentz group SO(3,1) turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction (A,B). While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes.

ON A CERTAIN GENERAL EXPONENTIAL SUM

2005 ◽  
Vol 01 (02) ◽  
pp. 183-192 ◽  
Author(s):  
H. MAIER ◽  
A. SANKARANARAYANAN
Keyword(s):  
Rational Number ◽  
Upper Bound ◽  
Exponential Sum ◽  
Multiplicative Functions ◽  
Sum Formula

In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum [Formula: see text] and obtain a non-trivial upper bound when α is a certain type of rational number.

scholarly journals Mutually unbiased unitary bases of operators on d-dimensional hilbert space

2020 ◽  
Vol 18 (01) ◽  
pp. 1941026 ◽  
Author(s):  
Rinie N. M. Nasir ◽  
Jesni Shamsul Shaari ◽  
Stefano Mancini
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Unitary Operator ◽  
Dimensional Subspace ◽  
Mutually Unbiased Bases ◽  
Unitary Operators ◽  
Dimension Formula ◽  
Dimensional Hilbert Space ◽  
Prime Dimension

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

scholarly journals Some characterizations of totients

1996 ◽  
Vol 19 (2) ◽  
pp. 209-217 ◽  
Author(s):  
Pentti Haukkanen
Keyword(s):  
Multiplicative Function ◽  
Arithmetical Function ◽  
Multiplicative Functions ◽  
Dirichlet Convolution

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

scholarly journals Correlations of multiplicative functions and applications

2017 ◽  
Vol 153 (8) ◽  
pp. 1622-1657 ◽  
Author(s):  
Oleksiy Klurman
Keyword(s):  
Asymptotic Formula ◽  
Multiplicative Function ◽  
Divided Difference ◽  
Circle Method ◽  
Partial Sums ◽  
Multiplicative Functions ◽  
Image Position ◽  
Bounded Partial Sums

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

scholarly journals An Application of Quantum Logic to Experimental Behavioral Science

2021 ◽  
Vol 3 (4) ◽  
pp. 643-655
Author(s):  
Louis Narens
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Additive Function ◽  
Quantum Physics ◽  
Behavioral Science ◽  
Slight Modification ◽  
Quantum Probability ◽  
Rational Approach ◽  
Dutch Book Argument ◽  
Von Neumann

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Qubits and the Framework of Quantum Mechanics

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Information ◽  
State Space ◽  
Quantum Physics ◽  
Complex Hilbert Space ◽  
The State ◽  
Mathematical Framework ◽  
Infinite Dimensional

In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.

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