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The Quantum Hamilton–Jacobi Equation and the Link Between Classical and Quantum Mechanics
Full text linkWe study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure.Quanta 2022; 11: 42–52
The Physical Meaning of the Holographic Principle
Full text linkWe show in this pedagogical review that far from being an apparent law of physics that stands by itself, the holographic principle is a straightforward consequence of the quantum information theory of separable systems. It provides a basis for the theories of measurement, time, and scattering. Utilizing the notion of holographic screens, which are information encoding boundaries between physical subsystems, we demonstrate that the physical interaction is an information exchange during which information is strictly conserved. Then we use generalized holographic principle in order to flesh out a fully-general quantum theory of measurement in which the measurement produces finite-resolution, classical outcomes. Further, we show that the measurements are given meaning by quantum reference frames and sequential measurements induce topological quantum field theories. Finally, we discuss principles equivalent to the holographic principle, including Markov blankets and the free-energy principle in biology, multiple realizability and virtual machines in computer science, and active inference and interface theories in cognitive science. This appearance in multiple disciplines suggests that the holographic principle is not just a fundamental principle of physics, but of all of science.Quanta 2022; 11: 72–96
Clifford Algebras, Spin Groups and Qubit Trees
Full text linkRepresentations of Spin groups and Clifford algebras derived from the structure of qubit trees are introduced in this work. For ternary trees the construction is more general and reduction to binary trees is formally defined by deletion of superfluous branches. The usual Jordan–Wigner construction also may be formally obtained in this approach by bringing the process up to trivial qubit chain (trunk). The methods can also be used for effective simulation of some quantum circuits corresponding to the binary tree structure. The modeling of more general qubit trees, as well as the relationship with the mapping used in the Bravyi–Kitaev transformation, are also briefly discussed.Quanta 2022; 11: 97–114
Dual Instruments and Sequential Products of Observables
Full text linkWe first show that every operation possesses an unique dual operation and measures an unique effect. If a and b are effects and J is an operation that measures a, we define the sequential product of a then b relative to J. Properties of the sequential product are derived and are illustrated in terms of Lüders and Holevo operations. We next extend this work to the theory of instruments and observables. We also define the concept of an instrument (observable) conditioned by another instrument (observable). Identity, state-constant and repeatable instruments are considered. Sequential products of finite observables relative to Lüders and Holevo instruments are studied.Quanta 2022; 11: 15–27
"Mysteries" of Modern Physics and the Fundamental Constants c, h, and G
Full text linkWe review how the kinematic structures of special relativity and quantum mechanics both stem from the relativity principle, i.e., "no preferred reference frame" (NPRF). Essentially, NPRF applied to the measurement of the speed of light c gives the light postulate and leads to the geometry of Minkowski space, while NPRF applied to the measurement of Planck's constant h gives "average-only" projection and leads to the denumerable-dimensional Hilbert space of quantum mechanics. These kinematic structures contain the counterintuitive aspects ("mysteries") of time dilation, length contraction, and quantum entanglement. In this essay, we extend the application of NPRF to the gravitational constant G and show that it leads to the "mystery" of the contextuality of mass in general relativity. Thus, we see an underlying coherence and integrity in modern physics via its "mysteries" and the fundamental constants c, h, and G. It is well known that Minkowski and Einstein were greatly influenced by David Hilbert in their development of special relativity and general relativity, respectively, but relating those theories to quantum mechanics via its non-Boolean Hilbert space kinematics is perhaps surprising.Quanta 2022; 11: 5–14
Steady State in Ultrastrong Coupling Regime: Expansion and First Orders
Full text linkUnderstanding better the dynamics and steady states of systems strongly coupled to thermal baths is a great theoretical challenge with promising applications in several fields of quantum technologies. Among several strategies to gain access to the steady state, one consists in obtaining approximate expressions of the mean force Gibbs state, the reduced state of the global system-bath thermal state, largely credited to be the steady state. Here, we present analytical expressions of corrective terms to the ultrastrong coupling limit of the mean force Gibbs state, which has been recently derived. We find that the first order term precisely coincides with the first order correction obtained from a dynamical approach—master equation in the strong-decoherence regime. This strengthens the identification of the reduced steady state with the mean force Gibbs state. Additionally, we also compare our expressions with another recent result obtained from a high temperature expansion of the mean force Gibbs state. We observe numerically a good agreement for ultra strong coupling as well as for high temperatures. This confirms the validity of all these results. In particular, we show that, in term of coherences, all three results allow one to sketch the transition from ultrastrong coupling to weak coupling.Quanta 2022; 11: 53–71
On the Connection Between Quantum Probability and Geometry
Full text linkWe discuss the mathematical structures that underlie quantum probabilities. More specifically, we explore possible connections between logic, geometry and probability theory. We propose an interpretation that generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories. We stress the relevance of developing a geometrical interpretation of quantum mechanics.Quanta 2021; 10: 1–14
Revisiting the Quantum Open System Dynamics of Central Spin Model
Full text linkIn this work, we revisit the theory of open quantum systems from the perspective of fermionic baths. Specifically, we concentrate on the dynamics of a central spin half particle interacting with a spin bath. We have calculated the exact reduced dynamics of the central spin and constructed the Kraus operators in relation to that. Further, the exact Lindblad type canonical master equation corresponding to the reduced dynamics is constructed. We have also briefly touched upon the aspect of non-Markovianity from the backdrop of the reduced dynamics of the central spin.Quanta 2021; 10: 55–64
George Sudarshan: Perspectives and Legacy
Full text linkGeorge Sudarshan has been hailed as a titan in physics and as one who has made some of the most significant contributions in several areas of physics. This article is an attempt to highlight the seminal contributions he has made in physics and the significant developments that arose from his work.Quanta 2021; 10: 75–104
How Do the Probabilities Arise in Quantum Measurement?
Full text linkA satisfactory resolution of the persistent quantum measurement problem remains stubbornly unresolved in spite of an overabundance of efforts of many prominent scientists over the decades. Among others, one key element is considered yet to be resolved. It comprises of where the probabilities of the measurement outcome stem from. This article attempts to provide a plausible answer to this enigma, thus eventually making progress toward a cogent solution of the longstanding measurement problem.Quanta 2021; 10: 65–74