# Quantum Computing Series, Part 3: What is Quantum Mechanics?

Before we address how quantum computing might help solve some of the challenges plaguing IoT, it’s important to have a basic understanding of the fundamentals of quantum physics and quantum computing.

John von Nuemann developed the mathematically rigorous formulation of quantum mechanics. Unit vectors (called state vectors) represent the possible states (more precisely, the pure states) of a quantum mechanical system. These unit vectors reside in a complex separable Hilbert space, known as the state space, well defined up to a complex number of norm 1 (the phase factor). Therefore, the possible states can be regarded as points in the projectivization of a Hilbert space. This Hilbert space is usually known as the complex projective space.

The exact nature of this Hilbert space is system dependent. For example, the position and momentum states for a single non-relativistic spin zero particle is the space of all square-integrable functions, while the states for the spin of a single proton are unit elements of the two-dimensional complex Hilbert space of spinors. Each self-adjoint linear operator acting on the state space represents an observable. Each eigenstate of an observable corresponds to an eigenvector of the operator. The associated eigenvalue corresponds to the value of the observable in that eigenstate.

The inner product between two state vectors is the probability amplitude and is a complex number. While performing an ideal measurement of a quantum mechanical system, the probability of a system collapsing from a given initial state to a particular eigenstate is the square of the absolute value of the probability amplitudes between the initial and the final states. The possible results of such a measurement are the eigenvalues of the operator. This explains that the choice of self-adjoint operators for all the eigenvalues must be real. Computing the spectral decomposition of the corresponding operator for an observable gives us the probability distribution of that observable in a given state.

Typically, states are not pure for a general system. Instead, they are usually represented as statistical mixtures of pure states, or as mixed states, which are given by density matrices: self-adjoint operators of trace one on a Hilbert space. Moreover, for general quantum mechanical systems, the effects of a single measurement can influence other parts of a system in a manner that is described instead by a positive operator valued measure. Thus, the structure, both of the states and observables in the general theory, is far more complex than the idealization for pure states.

Quantum Mechanics vs Classical Physics

Quantum mechanics is a branch of physics that describes the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large (macroscopic) scales. Quantum mechanics is different from classical physics because in quantum mechanics energy, momentum, and other quantities have discrete, quantized values. In addition, quantum mechanics objects possess characteristics of both wave and particle. This is also known as the wave-particle duality. Further, per the Uncertainty principle, there are limits to the precision in measuring the quantities.

The word, quantum, derives from Latin and means “an amount”. In the field of quantum mechanics, quantum refers to a discrete unit assigned to a physical quantity, for example, the energy of an atom at rest. The discovery that particles are discrete packets of energy with wave-like properties led to quantum mechanics. Quantum mechanics is the branch of physics dealing with atomic and subatomic systems. It forms an important role in formulating the mathematical framework for many related fields in Physics and Chemistry. Some examples include condensed matter physics, solid-state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.

Why Do We Need Quantum Mechanics?

Quantum mechanics is essential to understanding the behavior of systems at atomic and even smaller scales. If one uses pure classical mechanics to describe the physical nature of an atom, electrons cannot orbit the nucleus. This is because orbiting electrons emit radiation due to their circular motion. Hence, the electrons would eventually collide with the nucleus due to the loss of energy by emitting radiation. This framework couldn’t explain the stability of atoms. Instead, electrons remain around the nucleus in a non-deterministic, uncertain, probabilistic wave-particle orbital. This defies the traditional assumptions of classical mechanics and electromagnetism.

The basis for development of quantum mechanics was to provide a better explanation and description of an atom. It was especially useful to describe the differences in the spectra of light emitted by different isotopes of the same chemical element as well as subatomic particles.

Classical Physics cannot account for the following four classes of Quantum mechanics phenomena:

• Quantization of certain physical properties
• Quantum entanglement
• Principle of uncertainty
• Wave–particle duality

Quantization of Certain Physical Properties

Quantization is the process of transition from a classical understanding of physical phenomena to a newer understanding of quantum mechanics.

Quantum Entanglement

A physical phenomenon that occurs after generation or during interaction between pairs or groups of particles so that determining only the quantum state of the whole system is possible. Measuring each particle independently of others is impossible. Noticeably, this holds true even when the particles are at a large distance from each other. Therefore, a quantum state describes that state of the system as a whole and not of individual particles.

An entangled system is defined as one whose quantum state cannot simply be regarded as a product of states of its local constituents. It means that its constituents are one wholesome entity and are not individual particles. In entanglement, it’s impossible to fully describe a constituent without considering the other(s). A composite system’s state can always be expressed as a sum or superposition of products of individual states of the local constituents. If the sum has more than one term, the system is entangled.

Various types of interaction may result in quantum systems becoming entangled. To determine some ways to achieve entanglement for experimental purposes, see the section below on methods. Entanglement breaks when the entangled particles decohere through interaction with the environment; for example, when making a measurement.

An Entanglement Example

Let us consider an example of entanglement where a subatomic particle undergoes decay and results into an entangled pair of other particles. The decay events follow the various conservation laws. As a result, the measurement outcomes of one daughter particle must correlate highly with those of the other daughter particle. This is to ensure the total momenta, angular momenta, energy, and so forth remain roughly the same before and after. For instance, a spin-zero particle decays into a pair of spin-½ particles. To ensure conservation of angular momentum, the total spin before and after this decay must be zero. Therefore, whenever the first particle is measured as spin up on an axis, the other particle is always found spin down when measured on the same axis. This is called the spin anti-correlated case. If the prior probabilities for measuring each spin are equal, the pair is in the singlet state.

Separating the two particles results in better observation of the phenomenon of entanglement. For experiment sake, imagine a particle is put in the White House in Washington and the other one is put in UC Berkeley. Now, we measure a particular characteristic of one of these particles (say, for example, spin) and obtain a result. We then measure the other particle using the same criterion (spin along the same axis). What do we find out? The result of the spin measurement of the second particle will match the spin measurement result for the first particle. It matches in a complementary sense in that the spin measurements will be opposite in their values.

Interpreting the Results of Our Example

The above result may or may not come across as surprising. Based on conservation of angular momentum in classical and quantum mechanics alike, a classical system would display the same property, and a hidden variable theory is certainly required to do so. While the classical system has definite values for all the observables all along, the quantum system does not. The quantum system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measuring the first particle. This might be surprising for the entangled particles that are spatially separated and distant.

Measuring Entangled Particles

Experiments conducted in the recent years have measured entangled particles within a distance that’s less than 10,000th of time it would take for light to travel between the entangled particles. According to the quantum theory, the effect of measurement of any particle happens instantly to affect the entire system. It is impossible, however, to use this effect to transmit classical information at speeds faster than light.

Quantum entanglement might give a mind-numbing-though-false impression of allowing communication at speeds faster than light. Per the no-communication theorem, these phenomena do not allow true communication. They only let two observers in different locations, howsoever distant, see the same system simultaneously. But, there is no way to control what does each of these observers see.

Wavefunction collapse is an epiphenomenon (secondary effect or byproduct) resulting from quantum decoherence. This, in turn, is just an effect of the underlying local time evolution of the wavefunction of a system and all of its environment. Since the underlying behavior does not violate local causality or allow faster-than-light communication, it follows that neither does the additional effect of wavefunction collapse, irrespective of it being real or apparent.