A Deep Dive Into the Perplexities of Classical Systems

Introduction

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!