No Preferred Reference Frame at the Foundation of Quantum Mechanics

Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance and Continuity, at the foundation of those axiomatic reconstructions, maps to “no preferred reference frame” (NPRF, aka “the relativity principle”) as it pertains to the invariant measurement of Planck’s constant h for Stern-Gerlach (SG) spin measurements. This is in exact analogy to the relativity principle as it pertains to the invariant measurement of the speed of light c at the foundation of special relativity (SR). Essentially, quantum information theorists have extended Einstein’s use of NPRF from the boost invariance of measurements of c to include the SO(3) invariance of measurements of h between different reference frames of mutually complementary spin measurements via the principle of Information Invariance and Continuity. Consequently, the “mystery” of the Bell states is understood to result from conservation per Information Invariance and Continuity between different reference frames of mutually complementary qubit measurements, and this maps to conservation per NPRF in spacetime. If one falsely conflates the relativity principle with the classical theory of SR, then it may seem impossible that the relativity principle resides at the foundation of non-relativisitic QM. In fact, there is nothing inherently classical or quantum about NPRF. Thus, the axiomatic reconstructions of QM have succeeded in producing a principle account of QM that reveals as much about Nature as the postulates of SR.

On the Stochastic Mechanics Foundation of Quantum Mechanics

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.

Philosophy of Quantum Mechanics: Dynamical Collapse Theories

Quantum Mechanics is one of the most successful theories of nature. It accounts for all known properties of matter and light, and it does so with an unprecedented level of accuracy. On top of this, it generated many new technologies that now are part of daily life. In many ways, it can be said that we live in a quantum world. Yet, quantum theory is subject to an intense debate about its meaning as a theory of nature, which started from the very beginning and has never ended. The essence was captured by Schrödinger with the cat paradox: why do cats behave classically instead of being quantum like the one imagined by Schrödinger? Answering this question digs deep into the foundation of quantum mechanics. A possible answer is Dynamical Collapse Theories. The fundamental assumption is that the Schrödinger equation, which is supposed to govern all quantum phenomena (at the non-relativistic level) is only approximately correct. It is an approximation of a nonlinear and stochastic dynamics, according to which the wave functions of microscopic objects can be in a superposition of different states because the nonlinear effects are negligible, while those of macroscopic objects are always very well localized in space because the nonlinear effects dominate for increasingly massive systems. Then, microscopic systems behave quantum mechanically, while macroscopic ones such as Schrödinger’s cat behave classically simply because the (newly postulated) laws of nature say so. By changing the dynamics, collapse theories make predictions that are different from quantum-mechanical predictions. Then it becomes interesting to test the various collapse models that have been proposed. Experimental effort is increasing worldwide, so far limiting values of the theory’s parameters quantifying the collapse, since no collapse signal was detected, but possibly in the future finding such a signal and opening up a window beyond quantum theory.

An Intricate Quantum Statistical Effect and the Foundation of Quantum Mechanics

An intricate quantum statistical effect guides us to a deterministic, non-causal quantum universe with a given fixed initial and final state density matrix. A concept is developed on how and where something like macroscopic physics can emerge. However, the concept does not allow philosophically crucial free will decisions. The quantum world and its conjugate evolve independently, and one can replace fixed final states on each side just with a common matching one. This change allows for external manipulations done in the quantum world and its conjugate, which do not otherwise alter the basic quantum dynamics. In a big bang/big crunch universe, the expanding part can be attributed to the quantum world and the contracting one to the conjugate one. The obtained bi-linear picture has several noteworthy consequences.

Symbol Correspondence for Euclidean Systems

The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.

Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors

Bohr proposed the complementarity principle in 1927 as the foundation of quantum mechanics, since then relevant debates have been critically discussed for many years. Applying a pair of spin particles, Einstein proposed the EPR paradox in 1935. Using nonlocal potential properties, Aharonov and Bohm proposed the AB effect to test complementary measurement results. Under locality conditions, Bell established a Bell inequality under classical logic. Using a pair of particles and double sets of ZMI devices for complementarity measurements, Hardy proposed the Hardy paradox in 1992. During the past 50 years, locality and nonlocality tests on complementarity were hot-topics among the advanced quantum information, computing and measurement directions with various theoretical extensions and solid experimental results.These complementarity approaches separated local/nonlocal parameters to form different equations without an integrated logic framework to describe these equations including both local and nonlocal features consistently. The main results provide a series of paradoxes that conflict with each other. This paper uses conjugate transformation. Based on the m+1 kernel form of 0-1 states, n pairs of conjugate partitions were established. Under a given configuration in N bits, a set of 2n 0-1 feature vectors are applied to construct conjugate transformation operators in logic levels with intrinsic measurements to be a set of measurement operators.The key results of the paper are listed in Theorem 5. Two special functions of vector logic (CNF or DNF expression) and four equivalent expressions of the elementary equation are examples to show local and nonlocal variables in equations consistently. Applying two pairs of conjugate sets <A, B> and their complementary sets <A', B''>, 4 meta measures are established corresponding to <±aA;±bB> quantitative features under measurement operators. The main results of the paper are represented in Lemma (1-4), Theorem (1-5) and Corollary (1-7). From a vector logic viewpoint, conjugate complementary scheme can organize local and nonlocal variables to satisfy the comprehensive properties of modern logic constructions on completeness, non-conflict and consistence in a united logic framework.

No-cloning Theorem and Quantum Copying

Heisenberg’s uncertainty relation and Bohr’s principle of complementarity form the foundations of quantum mechanics. If these are violated then the edifice of quantum mechanics can come crashing down. In this chapter, it is shown how cloning or perfect copying of a quantum state can potentially lead to a violation of these sacred principles. A no-cloning theorem is proven showing that the cloning of an arbitrary quantum state is not allowed. The foundation of quantum mechanics is therefore protected. It is also shown how quantum cloning can lead to superluminal communication. It is also discussed that, if making a perfect copy of a quantum state is forbidden, how best a copy of a state can be made.

Birth of Quantum Mechanics—Planck, Einstein, Bohr

In this chapter, three problems whose resolution laid the foundation of quantum mechanics are discussed. First the pioneering work of Max Planck is described who explained the spectrum of light emitted from a so-called blackbody by making a bold ansatz that the energy associated with the oscillations of electrons comes in packets or quanta of energy. Second it is shown how Einstein invoked Planck’s hypothesis to explain the photoelectric effect by arguing that light should come in packets or quanta of energy and this energy should be proportional to the frequency. Electrons are emitted if the energy of the quantum of light, which has come to be known as a photon, is higher than a critical value. The third problem relates to the atomic structure of hydrogen. A full description is given of how Bohr applied a quantization condition to explain the emission of light at certain specific frequencies.

Quantum Mechanics for Beginners

Quantum mechanics is a highly successful yet a mysterious theory. Quantum Mechanics for Beginners provides an introduction of this fascinating subject to someone with only a high school background in physics and mathematics. This book, except the last chapter on the Schrödinger equation, is entirely algebra-based. A major strength of this book is that, in addition to the foundation of quantum mechanics, it provides an introduction to the fields of quantum communication and quantum computing. The topics covered include wave–particle duality, the Heisenberg uncertainty relation, Bohr’s principle of complementarity, quantum superposition and entanglement, Schrödinger’s cat, Einstein–Podolsky–Rosen paradox, Bell theorem, quantum no-cloning theorem and quantum copying, quantum eraser and delayed choice, quantum teleportation, quantum key distribution protocols such as BB-84 and B-92, counterfactual communication, quantum money, quantum Fourier transform, quantum computing protocols including Shor and Grover algorithms, quantum dense coding, and quantum tunneling. All these topics and more are explained fully but using only elementary mathematics. Each chapter is followed by a short list of references and some exercises. This book is meant for an advanced high school student and a beginning college student and can be used as a text for a one semester course at the undergraduate level. However it can also be a useful and accessible book for those who are not familiar but want to learn some of the fascinating recent and ongoing developments in areas related to the foundations of quantum mechanics and its applications to quantum communication and quantum computing.