Introduction to Quantum Physics

Wave Particle Duality: A Closer Look at Matter

Wave-particle duality is the fundamental concept in quantum mechanics that states every particle or quantum entity, such as a photon or electron, exhibits both wave and particle characteristics. This idea was a radical departure from classical physics, which strictly separated the two concepts. 

First, let us take photons (light) as an example. During propagation, photons act like waves, which is proven by their diffraction at the slits, and also because of the formation of interference patterns on the final screen (Young’s Double Slit Experiment). These two phenomena are key characteristics of waves. 

However, when photons interact with subatomic particles such as an electron, photons act like particles (Einstein's Photoelectric Effect). What was observed is that photons with high enough frequency (energy) like UV light are able to eject electrons instantly from a metallic surface when they are shone. Meanwhile, low frequency photons like infrared light are unable to eject electrons, no matter how long they are shone. This shows that photons are like particles: discrete, countable entities, that are able to transfer energy/momentum in a "lump" instantly (analogous to linear collision between billboards on a pool table). 

⇒ These two results prove that photons (light) display characteristics of both waves and particles.

Second, let us take a look at electron particles. Something very interesting happens when we shoot them through a double slit. In the first figure above, we “observe” the electrons as they pass through the slits, specifically, we identify whether they pass through the top or bottom slit (by putting a detector at the entrance of the slits). Essentially, we are performing an act of measurement on the electron's location in space. What was observed are, two bright spots on the final screen. This suggests that the electrons were behaving as discrete particles.

However, in the second figure, we did not “observe” (measure) the electrons as they pass through the slits, so we do not know whether they went through the top or bottom slit, and by extension, their position in space. Amazingly, an interference pattern was formed! This suggests that the electrons were behaving as waves rather than particles! Actually, what causes this is called the Heisenberg’s Uncertainty Principle, which will not be discussed here as it is a little out of topic. 

⇒ The key takeaway of this is that, subatomic particles like electrons also display characteristics of both waves and particles.

In summary, wave–particle duality is a central concept in quantum mechanics, evident in both photons and electrons. Typically, their dynamics are most effectively described through their wave nature, meaning under the right conditions, we could model a subatomic particle like an electron as a wave. Therefore, we will use some mathematics to describe these waves, and within this framework, particles are represented by a mathematical construction called the wavefunction, to be discussed below.

Wavefunction

In quantum mechanics, the wavefunction (Psi, Ψ) is a mathematical description of the quantum state of a system (e.g. a single particle, group of particles, or photons). To determine the wavefunction Ψ of a quantum system, we would need to solve something called the Schrödinger equation.

The significance of modelling electron as a mathematical wave is that Ψ contains all the information that can be known about that system such as position, momentum, energy and more. The wavefunction Ψ is generally a complex-valued function of space and time:

In Dirac notation, we write the wavefunction in a slightly different way:

However, both notations refer to the same thing, the quantum particle's wavefunction Ψ,  just written in different formalisms. In quantum computing, we almost explicitly use Dirac notation as it is cleaner, more succinct and simplifies calculations. (See Learning: Quantum Computing) 

Superposition

In high-school physics, we learnt that the principle of superposition of waves is the vector sum of the amplitudes between the two waves at a point (the resulting wave's amplitude). However, in quantum computing, the term is generally used to describe the fact that a quantum system is simultaneously (at the same time) existing as two different states. To understand what this means, let's carefully re-examine the electron double slit experiment.

In the classical regime, electrons are tiny, discrete particles, whose trajectory follow a fixed path (like throwing a ball or marble). Following the experiment, if we shoot these electrons one-by-one through the double slit, they either go through the top slit or bottom slit. Thus, they form only two bright spots since the electron either went up or down.

However, in the quantum regime things are much different. The interference pattern formed on the screen is due to the interference between the two waves emerging from top slit and bottom slit. But wait, aren't electrons tiny, discrete particles? Also, didn't we send them in one-by-one? Moreover, we know that interference can only occurs between two waves. So, the questions we have now are: How do electrons being tiny discrete particles interfere? More importantly, since they are sent in one-by-one, what exactly do they interfere with? The answer is surprising... Each electron interferes with itself!

Before we measure a quantum system, they exist as a superposition between states. In the experiment above, we say that the electrons exist in a superposition of going through the top slit, and the bottom slit. Due to superposition, the electrons interfere with itself! Mathematically, we write a superposition state like this:

In quantum computing, any quantum system with two energy levels can be chosen as a qubit because they are analogous to bit 0 and bit 1 in a normal computer (but with the additional quantum properties). In the electron double slit experiment above, we could have arbitrarily chosen "travel bottom slit" as |0⟩ while "travel upper slit" as |1⟩, or vise-versa. In general, we write a qubit's superposition state as: 

where α and β are the probability amplitudes, which are complex.

To recap, when we have superposition of states say |0⟩ and |1⟩, the quantum system exists as both the states simultaneously. Through the act of measurement, we force the quantum system to take on a definite state, either |0⟩ or |1⟩ (as proven when we "observe"/measure whether the electrons pass through the top or bottom slit). One might ask if the probabilities of |0⟩ or |1⟩ the same? The only way to accurately determine these probabilities is to measure a large assemble of the exact identical quantum system (maybe 100, 1000 or more samples). Finally, by analysing the total number of  |0⟩ and |1⟩'s measured, we can obtain the probabilities.

Let's take a look on how these probabilities relate to α and β in the section below.

Born Rule

Born rule is a fundamental postulate in quantum mechanics that relates the mathematical wavefunction Ψ to the probabilities of outcomes of a measurement on that system. For our purposes, a suffice definition is: the absolute value squared of a wavefunction |Ψ|² at a point in space is equal to the probability density of finding the particle at that point. By extension, this means that the area under the curve of |Ψ|² represents the probability of finding the particle within that region in space. Therefore, given two points in space say a=3.0nm and b=9.0nm, the probability of finding the particle within this region could be 5%, 18%, 30%, depending on the area beneath the curve:

An extremely important condition that comes with this postulate is that the total integral of the absolute value squared of a wavefunction |Ψ|² over space must equal to 1 (since the particle must be found somewhere).  

So, Born rule is a very important postulate which allows us the extract the probability of finding a particle within a certain region in space, given its wavefunction Ψ. For our case, we only need to understand that:

• The amplitude squared of a wavefunction |Ψ|² is equal to its probability density.

• The area under the curve of |Ψ|² is equal to the probability of finding the particle within that region.

• The total integral over |Ψ|² must equal to one.

In Dirac notation, Born rule applies the same. This time, the definition is: For a quantum system that is described by a state vector ∣Ψ⟩, to measure an observable corresponding to a specific state ∣ϕ⟩, the probability of finding the system in that state is the inner product squared of the two state vectors:

Remember that the inner product in an arbitrary vector space is a generalized dot product between two vectors (i.e. how much two vectors are aligned with each other). So, the equation above is quantifying how much ∣Ψ⟩ aligns with ∣ϕ⟩, which turns out to be the probability of obtaining ∣ϕ⟩ when measuring ∣Ψ⟩. Suppose in previous electron double slit experiment, we wish to determine the probability of observing the electron in the |0⟩ state (traveling through the bottom slit). It is mathematically written like this:

By expanding the equation above and applying Born rule, we will obtain the relationship between complex coefficients α and β with probability:

Again, when we observe the quantum system, we will only either find to be in state |0⟩ and |1⟩. Therefore, the total probability must still equal to one: 

In summary:

• The absolute value squared of a vector state's coefficient (e.g. |α|² or |β|²) represents the probability of finding that vector state when measuring the quantum system ∣Ψ⟩.

• The sum of absolute value squared (e.g. |α|² + |β|²) of all vector state coefficients must equal to one.

Entanglement

Before talking about entanglement, we need to lay down some definitions:

• Correlation — Two systems or variables are correlated if there is a discernible relationship or connection between them. When we measure one, we gain some information or predictive power about the state of the other. For example, if there is one red and one blue ball inside a basket, Alice randomly selects a ball and finds out it is red. Since the balls are correlated, Alice instantly knows that the other ball in the basket is blue.

• Locality — All influences and interactions are limited by the speed of light, even information. A theory is local if the description of a system at a specific point in spacetime depends only on its immediate surroundings and the physical fields at that point. Any change to a system can only affect another system after a time delay. For example, throwing a rock in a pond. The ripples spread outward at a fixed speed. A duck on the far side of the pond is unaffected until the ripples reaches it.

Quantum entanglement can be imagined as a stronger correlation that cannot be simulated by classical physics. This occurs when two or more particles form a single quantum system, and their individual properties become inextricably linked. For example, suppose two persons Alice and Bob share an entangled pair of particles. Let's assume the current entanglement is such that both particles will have the same spin upon measurement (i.e. if one particle was measured to be spin up, the other would also be spin up). If Alice travels to a different galaxy, then measures her particle (collapsing its wavefunction) to find it to be spin up, she will immediately know that Bob's particle is also spin up. 

The key difference between entanglement and classical correlation is that classical correlation is local while entanglement is not. In the ball experiment, we already knew that one ball is red and the other blue. These ball's colors are already defined locally, and they can't suddenly change their color. This is in contrast to entanglement, where both particles are in a superposition between spin up and down. It is the act of measurement on one particle, which causes it wavefunction to collapse, which forces the particle to take on the state spin up or down. After this measurement, since they are entangled, the other entangled particle will instantly follow suite and will have the same spin. This is non-locality, the particles had no fixed value/property of spin up or down throughout the experiment, and the measurement of one particle immediately "informs" the other entangled particle to take on a state.

One problem with non-locality is that, if Alice and Bob were truly galaxies apart, then information must have been transferred faster than the speed of light because Alice instantly knew Bob's particle state. This violates special relativity where nothing, not even information can travel faster than light. However, luckily (or unluckily) for us the No-Communication Theorem proves that it is impossible to use quantum entanglement to transmit information or classical data faster than the speed of light.

In order to mathematically represent entanglement, we first need to define a two-qubit system. Below shows is how we write two-qubit system who's particles are both spin down |00⟩ , both spin up  |11⟩, one spin down the other spin up |01⟩ and vise-versa |10⟩:

If two particles are entangled to be same state, then this quantum system's superposition can be written as follows:

The coefficient in front is a normalization constant and can be ignored, while the two states |00⟩ and |11⟩ as a whole can be thought of as single quantum state, meaning if we measure ∣Ψ⟩, it will collapse to either |00⟩ state or |11⟩ state (makes sense since we established that both particles are entangled to have the same spin). 

Two particles can also be entangled in the following way:

Conclusion

Quantum physics reveals a world that defies our everyday intuition. Through concepts introduced in this learning section, we see that light and matter cannot be described simply as waves or particles, but rather as quantum systems that embody both characteristics depending on how we observe them, especially the double-slit experiments with photons and electrons demonstrate which show us that observation itself shapes reality. These foundations where they lead to superposition, entanglement, and the probabilistic nature of measurement, forming the essential framework for quantum mechanics and, ultimately, the principles that make quantum computing possible.