American Journal of Physics, July 2024July 2024 Issue,

Volume 92, No. 7

Geometric visualizations of single and entangled qubits

The Bloch Sphere visualization of the possible states of a single qubit serves as a useful pedagogical and conceptual tool, offering a one-to-one map between qubit states and points in a 3D space. However, understanding many important concepts of quantum mechanics, such as entanglement, requires developing intuitions about states with a minimum of two qubits, which map one-to-one to unvisualizable spaces of six dimensions and higher. In this paper, we circumvent this visualization issue by creating maps of subspaces of 1- and 2-qubit systems that quantitatively and qualitatively encode properties of these states in their geometries. For the 1-qubit case, the subspace approach allows one to visualize how mixed states relate to different choices of measurement in a basis-independent way and how to read off the entries in a density matrix representation of these states from lengths in a simple diagram. For the 2-qubit case, a toroidal map of 2-qubit states illuminates the non-trivial topology of the state space while allowing one to simultaneously read off, in distances and angles, the level of entanglement in the 2-qubit state and the mixed-state properties of its constituent qubits. By encoding states and their evolutions through quantum logic gates with little to no need of mathematical formalism, these maps may prove particularly useful for understanding fundamental concepts of quantum mechanics and quantum information at the introductory level.


In this issue: July 2024 by Jesse Kinder; Claire A. Marrache-Kikuchi; Raina Olsen; Beth Parks. DOI: 10.1119/5.0221011

Footnotes (and other editorial innovations), by Beth Parks. DOI: 10.1119/5.0221171


Computing the shape of planet Earth, by Stephen J. Norton. DOI: 10.1119/5.0145569
Editor's note: This short paper presents a new derivation of the Earth's eccentricity using the minimization of the Earth's total energy that includes both the gravitational and the centrifugal energies. It employs the Green's function of the Laplace operator, a mathematical tool that is commonly used, for instance in electrostatics. Appropriate for advanced undergraduate mechanics courses or for illustrating the utility of Green's functions.

Torricelli's experiment and conservation of momentum, by D. Alvaro-Berlanga; R. Planet; A. Fernandez-Nieves. DOI: 10.1119/5.0145991
Editor's note: This paper shows how a familiar example from introductory fluid mechanics—the speed of a fluid as it exits a hole—contains subtleties that can be used to illustrate more advanced concepts in fluid mechanics. You'll want to adopt this as an example when you teach about control volumes.

Maxwell and the development of electromagnetic theory, by Alfred M. Bork. DOI: 10.1119/5.0183355
Editor's note: Maxwell's papers on electromagnetism are foundational to physics, and modern physicists have much to learn from reading them. This paper shares a nearly 50-year-old introduction to the papers written by Alfred Bork but never published. The manuscript was transcribed and edited by Kirk McDonald, who also added extensive footnotes and citations. The paper will help both students and faculty who wish to read and learn from Maxwell's original papers.

Exploring counterclockwise thermodynamic cycles Featured, by Randall D. Knight. DOI: 10.1119/5.0152547
Editor's note: In this paper, author Randall Knight explores the rich behavior of counterclockwise thermodynamic cycles. A counterclockwise cycle can function as a heat pump or refrigerator, but this is not the only possibility. The choice of thermal reservoirs and the choice of which reservoir heat is absorbed from and exhausted to during the cycle affect the coefficient of performance and even the basic function of the device. The author illustrates these possibilities through an analysis of counterclockwise Carnot, Kelvin, Brayton, and Stirling cycles. The analysis includes interesting possibilities, such as a cycle that operates with a single reservoir, as well as practical applications like the role of heat exchangers. The surprising results of this analysis will likely be of interest to anyone who has studied thermodynamics, and the material can easily be incorporated into any course that includes thermodynamics, from introductory physics to graduate study.

Insight into the gas–liquid transition from the Berthelot model, by Li-Qin Mi; Dandan Li; Shanshan Li; Zhong-Heng Li. DOI: 10.1119/5.0094686
Editor's note: While the van der Waals equation of state provides a simple model for phase transitions, it fails to achieve a good quantitative fit for properties near phase transitions in most substances. A closely related model, the Berthelot model, still has only two free parameters, but it allows the attraction between molecules to depend not only on volume but also on temperature. This paper builds on the parametric expressions for the van der Waals gas derived in a 1982 paper in this journal by John Lekner. It shows that similar expressions derived from the Berthelot model provide a much better fit to the data. This derivation could be shared with students in intermediate or advanced thermodynamics courses.

Geometric visualizations of single and entangled qubits Featured, by Li-Heng Henry Chang; Shea Roccaforte; Ziyu Xu (徐子瑜); Paul Cadden-Zimansky. DOI: 10.1119/5.0137901
Editor's Note: While single qubit states can be visualized with a Block Sphere, 2-qubit states are much more challenging to visualize. This paper presents a new way to visualize these states using toroids. While the representation is limited to states with real coefficients, it can illustrate both separable and entangled states, allowing students to develop intuition for their use in quantum information. This representation can be used in introductory classes in a quantum information curriculum but will also be valuable in more standard quantum mechanics courses at a variety of levels.

The experiment: Student exploration into systematic uncertainty, by Nicholas P. Gray; Tanisha K. Rutledge; Leigh Parrott; Christopher A. Barns; Kevin B. Aptowicz<. DOI: 10.1119/5.0190546
Editor's Note: This work will be very interesting to two audiences. Its most obvious audience is those who teach the e/m experiment. These readers may have observed that the statistical measurement uncertainties are seldom large enough to make the result consistent with the accepted value. This paper explains these deviations. The paper may be even more useful to the less obvious audience: those who teach upper-level laboratories or projects. This paper may inspire these instructors to involve their students in extended investigations into this and other classic experiments.


Spin coating experiments and theory for undergraduate physics and engineering students—A connection to microfabricatio, by M. Foster; J. Mendoza Cortes; H. C. Mayer. DOI:10.1119/5.0169090
Editor's note: The semiconductor industry, which is predicted to reach US$1 trillion by 2030, is a major employer of physicists. Semiconductor processing relies on photolithography, and photolithography relies on spin coating of photoresists. These facts on their own might inspire instructors to teach students about spin coating, even if the phenomenon was not such a beautiful application of fluid mechanics. This paper shows you how to teach students about spin coating using inexpensive equipment and safe fluids (corn syrup, castor oil, and olive oil). Students can predict how and whether the film thickness depends on variables such as density, speed, viscosity, and time, and can test these predictions in the laboratory.


Computing scattering cross sections for spherically symmetric potentials, by Anil Khachi. DOI: 10.1119/5.0176369
Editor's Note: This Note shows how Scilab, a free software package, can be used to find phase shifts within the Variable Phase Approach (VPA) to scattering, allowing the scattering cross sections to be calculated. Students may find the VPA approach to be conceptually appealing, as it shows how the phase builds up due to the scattering potential.


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