Diving Into The Quantum Realm

Going from the when-to-the-why-to-the how of Quantum Information

Published in

QVillage

8 min readJun 22, 2019

At the turn of the 20th century, a series of crises had left the Physics Community crippled with unanswered questions, from the existence of the “ultraviolet catastrophe” to electrons spiraling inexorably into the atomic nucleus, all hell had broken loose.

At first, such problems were resolved by the addition of ad hoc hypotheses to the corresponding classical theories. After almost a quarter century of turmoil, and a desperate need for a better understanding of the aforementioned phenomena, the crises came to an end with the creation of the modern theory of Quantum Mechanics.

The power of “weird”

If you asked me to describe quantum mechanics using a single word, that word would most definitely be weird.

“If you think you understand quantum mechanics, you don’t understand quantum mechanics.” ~ Richard P Feynman.

Quantum mechanics is as powerful as it is weird. So, how can we use this to our benefit? Much after the advent of quantum mechanics, someone thought, “Oh. What if we use quantum mechanical systems for processing of information?”

It was this notion that gave birth to the elegant field of Quantum Information.
Like many simple yet profound ideas, it was a long time before anybody thought of performing information processing using quantum mechanical systems.

Quantum Computation and Quantum Information is the study of the information processing tasks that can be accomplished using quantum mechanical systems.

Baby steps, one quantum bit at a time

The bit as we know, is the most fundamental concept of classical computation and classical information. It represents the smallest unit of information that can be processed classically.

The classical bit is binary, it can take only two possible values, 0 and 1. There is an ever-increasing need for more computational power, with independent classical systems in need of more power the number of bits associated with that system is increased. Till what extent can this be done?

There are problems such as simulating complex molecules which if done classically will need more bits than the number of atoms in The Milky Way Galaxy altogether.

Okay so, increasing the number of fundamental units of information isn’t feasible. But, what if we could increase the amount of information that is represented by a fundamental unit?

*Enter qubit*

A Quantum Bit or a Qubit is the fundamental unit of quantum information.

A qubit is essentially one of the simplest quantum mechanical systems, which we will describe as a mathematical object with certain specific properties.

You may be thinking that while it is true that in a practical scenario a qubit would be something physical, but for all further discussions, we will be treating qubits as abstract mathematical objects since it is easy for us to generalize all theories of quantum computation rather than being restricted to a particular physical system.

So then, what does a qubit give us? Just as a classical bit has a state, a qubit also has a state. Two possible states for a qubit are the states |0⟩ and |1⟩ which correspond to states 0 and 1 classically. The “| ⟩” notation is called the Dirac Notation and is a standard form of notation to represent quantum states.

The difference between a bit and a qubit is that a qubit can exist in a state other than |0⟩ and |1⟩. To better understand the concept of qubits, let’s try and represent classical bits as vectors.

It can be seen the two column vectors are orthogonal to each other. Since these two vectors are orthogonal, we can consider a vector lying in between these two as well, or in other words, it is possible to form linear combinations of states, which we refer to as superpositions.

Therefore, it would be fair to say that |ψ⟩ = α|0⟩ + β|1⟩ where α and β are complex numbers. To put it more formally:

The state of a qubit is a vector in a two-dimensional complex space.

The states of |0⟩ and |1⟩ are known as the computational standard basis and form an orthonormal basis for this vector space.

0 and 1?

Superposition is a fundamental property of quantum mechanics and it states that two quantum states can be “superposed” to give a third valid quantum state, much like the above-depicted state of |ψ⟩.

Here’s a thought experiment to better understand superposition:
Consider a coin with two distinct faces and let’s toss this coin. When the coin is in the air, we don’t know whether it is in a state of heads or tails. It's both heads and tails together.

Since the definition of superposition does not put any restriction on what the values of α and β can be, this would mean that any vector can be a qubit. But that is not the case. Remember that our qubit is a superposition of vectors which we constructed from classical bits. The length of the computational basis vectors is unity.

Therefore, the length of a vector for it to qualify as a qubit should be unity, or in other words:

Geometrically, we can interpret this condition as the length of the qubit vector being normalized to length 1. Thus, in general, a qubit’s state is a unit vector in a 2-dimensional vector space. To state it more formally, if we have n qubits then:

To be or not to be?

Coming to another super weird property of quantum mechanics, a quantum state is in a state of superposition until it is measured.

So, as soon as a quantum system is observed, it causes a change in the system and the superposition collapses giving us a definite value, that’s super weird, but, why does this happen? Well…no one knows! You can check out the infamous thought experiment of Schrödinger's Cat to better grasp this.

In the case of qubits, when we measure or observe a qubit the state of the qubit collapses, giving either of the computational basis vectors as the result. Therefore, even though qubits can have a large number of intermediate states, on measurement, it only gives either of the computational basis vectors. So then, where does quantum computing harness its glory from?

This dichotomy between the unobservable state of a qubit and the observations we can make lies at the heart of quantum computation and quantum information. In most of our abstract models of the world, there is a direct correspondence between elements of the abstraction and the real world.

The lack of this direct correspondence in quantum mechanics makes it difficult to intuit the behavior of quantum systems; however, there is an indirect correspondence, for qubit states can be manipulated and transformed in ways which lead to measurement outcomes which depend distinctly on the different properties of the state. Thus, these quantum states have real, experimentally verifiable consequences.

The bra-ket

The ket is the notation we’ve been using to represent quantum states so far
(|ψ⟩). The bra, on the other hand, is the conjugate transpose of the ket and is denoted by 〈ψ|.

Now that we know what the two notations mean, what happens when you combine one with the other?

Bra combined with a ket gives rise to what is known as the inner product and the inner product of vector A with vector B is denoted as 〈A||B⟩ which again is lazily denoted as 〈A|B⟩. Note that the inner product is not commutative.

Geometrically, the inner product represents the projection of the bra vector onto the ket vector.

Can we arrive at the condition for the validity of a qubit using the inner product?

If we take the inner product of |ψ⟩ = α|0⟩ + β|1⟩ with itself, we arrive at this condition.

Condition for validity of a qubit using inner-product.

Therefore, a more formal way of writing the condition for a vector to qualify as a qubit is 〈ψ|ψ⟩ = 1.

What happens when we measure a qubit?

Going back to our beloved coin, when the coin is tossed and is in the air, it is in a superposition of states, as soon as the coin lands on our hand and we look at it, we collapse this superposition and this gives us the result of either heads or tails.

Everything in the Quantum Realm is probabilistic in nature. Therefore, if we measure a qubit we get probabilities of it being in either of the computational bases.

The way we measure a qubit in a state|b⟩ is to take the inner product of the qubit with the state in which you want to measure it in and take the modulus of this result and square it. This gives us the probability that the post-measurement state of the qubit will be |b⟩.

Therefore, measuring a qubit in the standard basis would look something like this

But, the sum of probabilities should equal to 1. Huh. Isn’t that our condition for the validity of a vector to be a qubit? Damn.

Amazing job if you’ve made it this far! Wrapping up…

Hopefully, this gives you a good understanding of qubits. Pretty cool stuff, right? I hope you are now as excited about Quantum Information as I am.
*has a fanboy moment*.

Contributors to this article: Akshatha Laxmi and Mugdha Pattnaik.