Last week, we went into great detail about the fundamentals of general, multi-state classical systems, grounding their mathematical representation in the spirit of linear algebra and basis vectors. To ease a smoother transition into quantum concepts, we introduced the concept of Dirac notation, which is a convenient and shorthand way of representing the basis vectors corresponding to distinct states.

Before we move on to quantum computing, let us reanalyze the Lighdie analogy from a new angle.

Lighdie’s Classical Library Analogy

In our original analogy, Lighdie was faced with the daunting task of brushing aside her teenage angst by selecting a book from the existing catalog of quantum books on Shor’s Algorithm (SA), Grover's Algorithm (GA), and the Phase Kick Back technique (PKB). Recall that Lighdie’s probability distribution \((0.3, 0.6, 0.1)\)of selecting the \(SA\), \(GA\), and \(PKB \\\), respectively, led to the following classical state:

$$\psi = 0.3 \ket{PKB} + 0.6 \ket{SA} + 0.1 \ket{GA}$$

Where \(\ket{SA}, \text{and } \ket{GA}\)are the Dirac-notation equivalent of the following unit vectors

$$\quad |PKB \rangle = \pmatrix{ 1 \\ 0 \\ 0},\quad |SA\rangle = \pmatrix{ 0 \\ 1 \\ 0},\quad |GA \rangle = \pmatrix{ 0 \\ 0 \\ 1}$$

, and \(\psi\) is the classical system.

Understanding Post-Measurement Determinism

Dirac ket notation is concise and elegant, but it falls short in explaining the underlying meaning behind the obscure 1s and 0s in each basis vector. To bridge this conceptual gap, we now introduce the idea of post-measurement determinism by revisiting the second rule of Lighdie’s angsty library spree:

Every time Lighdie decides to select a book from the existing catalog, her rays of light destroy the other existing books in the catalog.

If taken at face value, the rule seems absurd — clearly the product of a sleep-deprived individual (don’t worry, guys, I get plenty of sleep!). However, we quickly realize that this statement reflects the fact that classical systems become deterministic once measured.

In other words, before Lighdie makes a selection, \(\psi\) is uncertain and unmeasured. After a selection is made, the system \(\psi\) collapses to that choice with a probability of \(1\), which is reflected. For example, if Lighdie selects the book (SA), then \(\psi\) collapses to the state:

$$\ket{SA} = \pmatrix{ 0 \\ 1 \\ 0}$$

Where the 1 represents that Lighdie has made her selection to SA with absolute certainty.

Similarly, if Lighdie selects the book PKB, then \(\\psi\) collapses to the state

$$\ket{PKB} = \pmatrix{1 \\ 0 \\ 0}$$

with absolute certainty.

The Mathematics of Classical Probability

Keep in mind that there is an infinite number of classical systems one can construct. These classical states can represent anything tangible, from fruits to cards. For the sake of generality, if we take \(\mathbf{n}\) general classical states \(c_1, c_2, \dots, c_n\)​ with corresponding probabilities \(p_1, p_2, \dots, p_n\), then we can describe an arbitrarily large classical system \(\phi\) such that:

$$\phi = p_1\ket{c_1} + p_2\ket{c_2} + \dots + p_n\ket{c_n} = \sum_{i=1}^n p_i \ket{c_i}$$

Notice that the represent all valid outcomes in the system \(\phi\), which means that:

$$\sum_{i=1}^{n} p_i = 1, \text{all p_i are non-negative}$$

Now that we have constructed a complete mathematical description of any classical system, we are ready to investigate the peculiarities of quantum randomness.

Introducing Quantum Randomness

We will begin by repurposing our previous analogy to explain quantum randomness or superposition.

Consider an alternate dimension to Lighdie’s universe, where an individual named Mighty resides. Mighty is also an extremely angsty teen who is motivated to visit the library, where he discovers a peculiar book adorned by three unique titles — Shor’s Algorithm, Grover’s Algorithm, and Phase Kick Back technique.

Mighty and the mScanner: A New Dimension of Uncertainty

Curious and suspicious of the front cover, Mighty flips to the back cover, where he finds an even stranger description:

When you scan this book, you can expect:
A γ² chance of receiving Shor’s Algorithm book,
A β² chance of receiving Grover’s Algorithm,
A α² chance of eceiving the Phase Kick Back book.

Utterly flabbergasted by this blurb, Mighty decides to seek assistance from the librarian. She simply tells him,

“That’s how quantum books work — you don’t know which book you will receive till you measure it.”

Reluctantly, Mighty scans the book using a mysterious mScanner and is shocked when it morphs into a red 500-page text on the Phase Kick Back Algorithm. He quickly checks his receipt, which reads:

The Quantum Receipt

Quantum Book:
Original State: \(\ket{book} = \alpha \ket{PKB} + \beta \ket{SA} + \gamma \ket{GA}\)
Amplitudes: \(\alpha, \beta, \gamma\)
Possible Books: \(\ket{PKB}, \ket{SA}, \ket{GA}\)

Measuring Device: mScanner
Result: \(\ket{PKB}\)

This receipt captures the true meaning of quantum states. The original state of Mighty’s mysterious book was in a superposition of SA, GA, and PKB. In other words, when Mighty first encountered the book, it was truly all of these three books at the same time.

In particular, the state is formally:

$$\alpha \ket{PKB} + \beta \ket{SA} + \gamma \ket{GA}$$

where the coefficients \(\alpha, \beta , \gamma \) are called amplitudes, whose square magnitudes give the probability of observing each outcome:

$$P(|PKB\rangle) = |\alpha|^2, \quad P(|SA\rangle) = |\beta|^2, \quad P(|GA\rangle) = |\gamma|^2$$

Similar to Lighdie’s classical randomness, applying the mScanner measurement device to the quantum book collapsed it into one definite state. Unlike the classical version, amplitudes encode the system's randomness instead of direct probabilities.

Looking Ahead

That concludes this week's blog and the final part of my all-immersive quantum computing introduction!

In the following weeks, we will be exploring the strange mathematical rules governing amplitudes and explaining why quantum states must always be normalized.

In last week's blog, we covered the basics of discrete sets and explored the conceptual implications of deterministic,fixed-state classical systems. To make the necessary leap to quantum information, however, we must explicitly define the mathematical nature of non-deterministic, multi-state classical systems using probabilistic vectors. If this seems overwhelming at first, then don’t worry. We will be using analogies and simple allusions to ground these abstract concepts in a more concrete perspective.

Meet Lighdie: A Library Analogy

We begin our discussion of multi-state classical systems and vectors with a library-focused analogy. Picture yourself as an individual named Lighdie, a teenager known for producing blinding blasts of light. It’s Saturday, and Lighdie has had a dreadfully uneventful day of lying on her bed and scrolling through TikToks. Fed up with her perpetual laziness, Lighdie decides she needs to challenge herself with some difficult quantum textbooks. She heads to the library and quickly finds the computing-focused catalog. To her dismay, there is a set of three different quantum books on the shelf. Lighdie’s hours upon hours of screen time has made her angsty, so she struggles to select a book - should she borrow the quantum book on Shor’s Algorithm(SA), Grover’s Algorithm(GA), or the Phase-Kick Back Technique (PKB)? Although teenager angst can vary based on one’s emotional state, Lighdie’s angst is constant and strangely specific, adhering to a particular set of rules.

1. Lighdie has a set probability of choosing each book.

2. Every time Lighdie decides to select a book from the existing catalog, her rays of light destroy the other existing books in the catalog.

3. The existing Catalog is renewed at the end of each day, meaning that Lighdie will encounter the same 3 books each day.

After visiting the library for 10 days, Lighdie’s choices are:

1. \(\text{SA - 3 times}\)

2. \(\text{PKB - 6 Times}\)

3. \(\text{GA - 1 times}\)

Modeling the 3-State Classical System

Now for the moment of truth: How does this analogy demonstrate the primary characteristics of multi-state classical systems?

To answer this question, we first explicitly define our 3-state classical system \(\psi\) as the book Lighdie selects on any given day, which could be any option or state from the set \(\{PKB, SA, GA\}\). Next, we describe the probability of observing each state within \(\psi\) by calculating basic ratios from Lightdie’s choices over 10 days.

1. \(P(SA) = 3/10 = 0.3\)

2. \(P(PKB) = 6/10 = 0.6\)

3. \(P(GA) = 1/10 = 0.1\)

Lastly, we know that \(\psi\) is a deterministic system defined by constant probabilities because Lighdie is limited to the same 3 books every day, where she can check out exactly 1 book.

Therefore, we can confidently state that \(\psi\) is a 3-state classical system with possible states \(\\{PKB, SA, GA\\}\) and state probabilities

$$\left(P(SA) = \frac{3}{10}, P(PKB) = \frac{6}{10}, P(GA) = \frac{1}{10} \right)$$

While our current representation of \(\psi\) is a harmonious mixture of qualitative and quantitative characteristics, quantum researchers favor a more compressed representation that can be manipulated with Algebra - column vectors.

From Probabilities to Vectors

Define a \(3 \times 1\) column vector \(\mathbf{z}\):

$$\mathbf{z} = \begin{pmatrix} 0.6 \\ 0.3 \\ 0.1\end{pmatrix}$$

Each row of vector \(\mathbf{z}\) represents each possible book Lighdie could select, while the entries themselves are the probability that she selects’ that row’s book. Most importantly, though, is that the entries of \(\mathbf{z}\) add up to \(1\) because Lighdie must select exactly one book every day.

Unit Vectors and Dirac Notation

Our vector representation of \(\psi\) is nifty, but introducing unit vectors is a necessary step in understanding the uncertainty surrounding quantum systems (no pun intended!). In linear algebra, a unit vector is a deterministic vector that is synonymous with one particular state. In our case, the three unit vectors represent Lighdie choosing one book with absolute certainty.

$$e_{pkb} = \begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}, e_{sa} = \begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix}, e_{ga} = \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}$$

With these unit vectors, we can now define the system as:

$$Z = 0.6 e_{pkb} + 0.3 e_{sa} + 0.1 e_{ga}.$$

This representation is significant as it allows us to frame classical systems not just as a list of probabilities, but as linear combinations of foundational basis states.

To make a seamless transition into quantum mechanics, physicists and computer scientists often use a special notation for these basis states, called Dirac Notation (alternatively “bra-ket” notation). Instead of writing our basis states as unit vectors, we write them as kets:

$$\ket{PKB} = \begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}, \ket{SA} = \begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix}, \ket{GA} = \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix}$$

Now, \(\psi’s\) dirac notation equivalent is:

$$\psi = 0.6 \ket{PKB} + 0.3 \ket{SA} + 0.1 \ket{GA}.$$

This is the same system we’ve been discussing, but it is written in a way that smoothly transitions to the quantum world. In the classical world, the coefficient set \(\{0.6,0.3,0.1\}\) represents the direct probabilities of observing each basis state in \(\psi\). In next week’s quantum mechanics blog, the coefficients become complex amplitudes whose squared magnitudes yield desired probabilities.

Whew, that concludes my blog for this week. Stay tuned for next week’s deep dive into the inner workings of quantum systems, and thank you!

Currently, quantum computing is being touted as the next frontier in information processing, with the potential to enhance the problem-solving capabilities of modern computers. However, understanding quantum computing can be considerably challenging because it combines high-level computing concepts with considerable amounts of linear algebra.

Rest assured, though, as we will use clear examples throughout this blog series to make complex, math-heavy ideas easier to picture. We will begin with the most fundamental building block of all: the concept of a set.

Introduction to Sets

Sets are convenient and shorthand ways of writing out a list of objects. You can think of them as the mathematical analogue to a basket of items. For example, if we have a basket with the following items — an apple, an orange, and a mango — then we can define a set as:

$$A = \{0,1\}$$

Each member of set \(A\), or any set in general, is also called an element. While our example used fruits, elements can also represent numbers, symbols, or any other object.

Before we can meaningfully explore the quantum world, we must understand how sets and information are used to conceptualize the classical world.

Sets in Classical Computing

In classical computing, information is commonly expressed in binary, which is a number system consisting entirely of \(0\)s and \(1\) s. Therefore, in a similar vein to our fruit basket analogy, we can define the standard classical set as:

$$\sigma = \{0,1\}$$

where σ is a commonly used Greek letter to represent sets. Each element of any set describing classical phenomena is called a state.

Now that we have clearly defined sets in relation to classical computing, we are ready to tackle the concepts of classical systems.

Classical System: Definition and Properties

A classical system is any physical or digital system consistent with the principles of classical physics — think of a traditional pendulum. One of the governing properties of classical systems is the exclusivity of classical states — meaning that the classical system can only be in one state at any given moment. For example, the set \(\sigma\) has two possible states, \(0\) and \(1\) , but the corresponding system \(\psi\) will be in exactly one of them. This state determinism is what separates classical systems from quantum systems.

Certain vs. Uncertain Classical Systems

Classical systems can be characterized as certain or uncertain, an important distinction that will solidify later once we compare them to quantum systems.

• A certain classical system is one whose state is fixed and unchanging.

• Uncertain classical systems never have constant states and are instead described in terms of probabilities until they are measured.

Wrapping Up

Yay! You’ve made it to the end of my very first quantum blog. Today, we have made major inroads towards understanding quantum computing by covering the basic concepts of sets and connecting them to classical systems.

As we move forward, we will be using vectors, matrices, and complex numbers to ground quantum states and sets in mathematical rigor. If you are unfamiliar with linear algebra, I highly recommend checking out 3Blue1Brown’s Essence of Linear Algebra series. He enhances the abstract concepts of linear algebra using dynamic animations and engaging visuals. For now, focus on the first two videos in the series.

You can start watching here: Essence of Linear Algebra.

To learn more about complex numbers, check out 3Blue1Brown’s excellent introduction, Complex Number Fundamentals. The first free videos should be enough to follow along with my future posts.

Get ready for a quantum leap into my next post!