Student ability to distinguish between superposition states and mixed states in quantum mechanics

We describe a method to create superposition states from a mixed state of a harmonic oscillator. If the initial state is described by a thermal state, then the resulting superposition state will be a "hot'' superposition state. Such a state can be distinguished from a statistical mixture by its coherence properties. Here we suggest how to demonstrate the coherence of the superposition state by observing interference fringes when the two parts of the wave packet are overlapped. In the case of the mixed superposition state, a partial overlap may not be sufficient to observe the presence of fringes. We introduce therefore the idea of a coherence length for the wave packet and demonstrate its relevance for interferometry of mixed states. We illustrate our ideas with the example of superposition states for the motion of a single trapped ion.

Solid-state physics is primarily concerned with the quantum mechanics of bulk materials and surfaces. Molecular physics and quantum chemistry are similarly the application of quantum mechanics to molecular problems. Bulk materials may be described as three-dimensional objects, and their spatial dimensions have a significant influence on the allowed solutions for quantum mechanical energy states or levels. These quantum mechanical levels in three dimensions give rise to electronic band structures which are commonly used to define a material as a metal, insulator, or semiconductor. Energy bands are formed from quantum mechanical states that are nearly continuous in energy. If the states that comprise a band are only partially filled with electrons, a metal is formed. For a fully occupied band separated by a relatively small energy gap, a semiconductor is the result. If the energy gap between a filled band and an empty band is large, the material is described as an insulator. Molecules are zero-dimensional objects with vanishing of the wave function in all three spatial directions and the bound electrons do not propagate. This gives rise to a discrete energy spectrum that is characteristic of molecules; the spacing between energy levels is large and there is no corresponding band picture of the electronic spectrum.

The original development of the formalism of quantum mechanics involved the study of isolated quantum systems in pure states. Such systems fail to capture important aspects of the warm, wet, and noisy physical world which can better be modelled by quantum statistical mechanics and local quantum field theory using mixed states of continuous systems. In this context, we need to be able to compute quantum probabilities given only partial information. Specifically, suppose that β is a set of operators. This set need not be a von Neumann algebra. Simple axioms are proposed which allow us to identify a function which can be interpreted as the probability, per unit trial of the information specified by β,of observing the (mixed) state of the world restricted to β to be σ when we are given ρ — the restriction to β of a prior state. This probability generalizes the idea of a mixed state (ρ) as being a sum of terms (σ) weighted by probabilities. The unique function satisfying the axioms can be defined in terms of the relative entropy. The analogous inference problem in classical probability would be a situation where we have some information about the prior distribution, but not enough to determine it uniquely. In such a situation in quantum theory, because only what we observe should be taken to be specified, it is not appropriate to assume the existence of a fixed, definite, unknown prior state, beyond the set β about which we have information. The theory was developed for the purposes of a fairly radical attack on the interpretation of quantum theory, involving many-worlds ideas and the abstract characterization of observers as finite information-processing structures, but deals with quantum inference problems of broad generality.

A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for phase difference between two different pure states to the case of mixed states in quantum mechanics is analyzed and the existence of phase singularities in the parameter space of an interference experiment with particles in mixed states pointed out. In the vicinity of such singular points the phase changes sharply and precisely at these points it becomes undefined. A closed circuit in the parameter space around such points results in a measurable phase shift equal to 2n\pi, where n is an integer. Such effects have earlier been observed in interference experiments with pure polarization states of light, a system isomorphic to the spin-1/2 system in quantum mechanics. Implications of phase singularities for the interpretation of experiments with partially polarized and unpolarized neutrons are discussed. New kinds of topological phases involving variables representing decoherence (depolarization) of pure states are predicted and experiments to verify them suggested.

The fabrication and control of macroscopic artificial quantum structures, such as qubits (Mooij et al., 1999; Nakamura et al., 1999; Friedman et al., 2000), qubit arrays (Johnson et al., 2011; Barends et al., 2014), quantum annealers (Boixo et al., 2013) and, recently, quantum metamaterials (Macha et al., 2014), have witnessed significant progress over the last 15 years. This was a surprisingly quick evolution from theoretical musings to what can now be called quantum engineering [the observation of such phenomena even in a single superconducting device was considered a truly challenging task in 1980 (Leggett, 1980)]. And today, we stand at the point where existing theoretical and computational tools become inadequate for predicting, analyzing, and simulating the behavior of such structures, in which quantum superposition and entanglement play the key role (Zagoskin et al., 2014). The long-known fundamental impossibility of simulating large enough quantum systems by classical means (Feynman, 1982), unfortunately, manifests itself already at the level of systems containing as few as several hundreds of qubits. Such a system is still too small to be used as an efficient quantum simulator of comparable systems, but already too large for us to tell with certainty, using the existing classical tools, whether it behaves as a quantum system should (Smolin and Smith, 2014). Furthermore, the complexity of already existing quantum processor prototypes confronts us with an engineering problem designing a reliable quantum device and testing its reliability. What is even worse, if there are fundamental corrections to the laws of quantum mechanics for large enough systems, we will be unable to discover them because of our inability to tell what exactly quantum mechanics would predict. Let us take the optimistic view that quantum computing is not fundamentally restricted by, for example, the size of a system capable of demonstrating quantum behavior (Penrose, 1999). In this scenario, it would be possible to create quantum computing devices that will allow us to design and fabricate ever bigger and better quantum computers, as well as other macroscopic quantum devices, of a character and use of which we cannot even imagine at the moment. Alternatively, we may find fundamental limits to the applicability of quantum mechanics. Nevertheless, this can happen only if the gap between our current ability to characterize large quantum systems and the capacities of the smallest workable quantum computers is bridged. Bridging this capacity gap is thus the immediate grand challenge for the field: a challenge that must be met if we hope to make further progress in quantum computing and quantum engineering or if we hope to discover fundamentally new physics, or both. While it is impossible to efficiently simulate a large quantum system by classical means by directly solving the appropriate equations of motion, it is feasible that essential quantum properties of an ensemble of such systems will be reflected in certain higher-level, global characteristics. These properties should be insensitive to details of a particular instance, computable by classical tools and accessible to experimental investigation. This view of a system of qubits as a quantum many-body system should be amenable to the approaches that have proven to work very well in numerous applications in condensed matter physics and quantum statistical mechanics. Therefore, with such earlier breakthroughs in mind, the task at hand will be difficult yet not impossible, and more than worth the effort.

We consider the Husimi Q(q, p)-functions which are quantum quasiprobability distributions on the phase space. It is known that, under a scaling transform (q; p) → (⋋q; ⋋p), the Husimi function of any physical state is converted into a function which is also the Husimi function of some physical state. More precisely, it has been proved that, if Q(q, p) is the Husimi function, the function ⋋2Q(⋋q; ⋋p) is also the Husimi function. We call a state with the Husimi function ⋋2Q(⋋q; ⋋p) the stretched state and investigate the properties of the stretched Fock states. These states can be obtained as a result of applying the scaling transform to the Fock states of the harmonic oscillator. The harmonic-oscillator Fock states are pure states, but the stretched Fock states are mixed states. We find the density matrices of stretched Fock states in an explicit form. Their structure can be described with the help of negative binomial distributions. We present the graphs of distributions of negative binomial coefficients for different stretched Fock states and show the von Neumann entropy of the simplest stretched Fock state.

A quantum ensemble $\{(p_x, \rho_x)\}$ is a set of quantum states each occurring randomly with a given probability. Quantum ensembles are necessary to describe situations with incomplete a priori information, such as the output of a stochastic quantum channel (generalized measurement), and play a central role in quantum communication. In this paper, we propose measures of distance and fidelity between two quantum ensembles. We consider two approaches: the first one is based on the ability to mimic one ensemble given the other one as a resource and is closely related to the Monge-Kantorovich optimal transportation problem, while the second one uses the idea of extended-Hilbert-space (EHS) representations which introduce auxiliary pointer (or flag) states. Both types of measures enjoy a number of desirable properties. The Kantorovich measures, albeit monotonic under deterministic quantum operations, are not monotonic under generalized measurements. In contrast, the EHS measures are. We present operational interpretations for both types of measures. We also show that the EHS fidelity between ensembles provides a novel interpretation of the fidelity between mixed states--the latter is equal to the maximum of the fidelity between all pure-state ensembles whose averages are equal to the mixed states being compared. We finally use the new measures to define distance and fidelity for stochastic quantum channels and positive operator-valued measures (POVMs). These quantities may be useful in the context of tomography of stochastic quantum channels and quantum detectors.

A comparison of methods to determine the flow angle components of atmospheric velocity vectors is presented. The atmospheric turbulence probes used to measure behavior of the atmosphere use different methods to obtain this information and there is some confusion as to which method is best. Although the methods are all based upon potential flow theory, the details about how the atmospheric and flight parameters are measured makes a difference to the design of an atmospheric turbulence probe and carrier research aircraft. This paper presents the mathematical theory behind five methods of obtaining atmospheric flow angle measurements from a moving aircraft. One of these methods, the Flush Air Data Sensing system developed by NASA has not previously been used in this particular application before but is found to be just as accurate as more commonly used methods that utilize a hemispherical sensing head. In fact, none of the methods presented show any statistical improvement at measuring flow angles over the others. It is therefore suggested that the best method is one that considers the probe and aircraft as a whole system rather than preferring one method over another.

I: Quantum Interference, Superposition States of Light, and Nonclassical Effects

The observation of quantum phenomena often necessitates sufficiently pure states, a requirement that can be challenging to achieve. In this study, our goal is to prepare a nonclassical state originating from a mixed state, using dynamics that preserve the initial low purity of the state. We generate a quantum superposition of displaced thermal states within a microwave cavity using only unitary interactions with a transmon qubit. We measure the Wigner functions of these “hot” Schrödinger cat states for an initial purity as low as 0.06. This corresponds to a cavity mode temperature of up to 1.8 kelvin, 60 times hotter than the cavity’s physical environment. Our realization of highly mixed quantum superposition states could be implemented with other continuous-variable systems, e.g., nanomechanical oscillators, for which ground-state cooling remains challenging.

Quantum mechanics is based on description of a state of a physical system in terms of wave function [1] (pure states) and of density matrix [2, 3] (mixed states). The attempts to find a classical-like interpretation of quantum mechanics [4, 5] and related constructions of quasidistribution functions in phase space of the system [6–9] give the idea that for quantum mechanics it is impossible to describe the state of the quantum system in terms of measurable positive probability analogously to the case of classical statistical mechanics, where the state of the system is described by the positive probability distribution due to presence of classical fluctuations.

This chapter discusses the dual nature of small particles, focusing on electrons. It begins by describing how the classical laws of physics, which were mostly established by the start of the twentieth century, did not work very well when applied on the atomic scale. In order to explain the atomic-scale phenomena, a new kind of physics was invented in the 1920s. This new physics was called quantum physics, or quantum mechanics. It applied to the world on the scale of the atom—the microscopic world, or microworld. The discussions then turn to an experiment involving ‘quantum’ coins; the difference between superposition state and mixed state; light, waves, and interference; and interference with electrons.

In his essay ‘Laws and States in Quantum Mechanics’, John Forge presents a case for considering laws of nature to be privileged sets of states, trajectories in the quantum mechanical analogue of phase space. Having presented an argument to show that states have to be taken with full ontological seriousness, Forge then uses those states to undergird his favourite account of laws and explanation — called the Instance View. On this view laws are a special sort of pattern, a certain kind of regularity, and explanations are instances of these regularities (so that you explain some phenomenon A by indicating that it is an instance of some regularity R). Forge then uses this account of laws and explanation to urge the acceptance of van Fraassen’s Modal Interpretation of Quantum Mechanics.

This paper presents the momentum map structures which emerge in the dynamics of mixed states. Both quantum and classical mechanics are shown to possess analogous momentum map pairs associated to left and right group actions. In the quantum setting, the right leg of the pair identifies the Berry curvature, while its left leg is shown to lead to different realizations of the density operator, which are of interest in quantum molecular dynamics. Finally, the paper shows how alternative representations of both the density matrix and the classical density are equivariant momentum maps generating new Clebsch representations for both quantum and classical dynamics. Uhlmann's density matrix [58] and Koopman wavefunctions [41] are shown to be special cases of this construction.

Quantum computing is a multidisciplinary field at the intersectionof quantum physics, computer science, and mathematics, aiming to har-ness quantum mechanical phenomena for computational advantage. Thisarticle provides a comprehensive review of both the foundational princi-ples and advanced theoretical models of quantum computing. We beginwith the basic framework of quantum computation, introducing the con-cept of qubits as two-level quantum systems, the linear algebra of Hilbertspaces, Dirac’s bra–ket notation, and the postulates of quantum mechan-ics (state superposition, unitary evolution, and projective measurement).We then discuss quantum logic gates and circuits, highlighting how clas-sical reversible computation principles extend to universal quantum gatesets. Next, we survey key quantum algorithms (from Shor’s factoring toGrover’s searching and beyond) and the complexity-theoretic implicationsof quantum computers, contrasting the class BQP with classical complex-ity classes. We cover the theoretical foundations of quantum error cor-rection and fault tolerance, explaining how quantum information can beprotected from decoherence using redundancy and how fault-tolerant pro-tocols can, in principle, allow scalable quantum computation despite noisyhardware. We then explore advanced models and paradigms of quantumcomputation, including measurement-based quantum computing (one-wayquantum computing), topological quantum computation with anyons, adi-abatic quantum computing and quantum annealing, continuous-variablequantum computing, and related approaches. We also provide an overviewof quantum information theory as it relates to computation, discussingentanglement, quantum entropy, and information-theoretic limits like theHolevo bound. Finally, we review the leading physical implementations(quantum hardware models) of quantum computers—trapped ions, super-conducting circuits, photonic systems, solid-state spin qubits, and oth-ers—outlining the challenges and achievements of each approach.