Quantum computing is one of the first great technologies of the 21st century, but the details are still shrouded in mystery. I can explain conventional digital computing down to the electron in a MOSFET, and with this newsletter, I have made it my mission to do the same for quantum computing.

Welcome to the Quantum Edge newsletter. Join me in my quest to translate the mysteries of the quantum world to the language of the dinner table and the coffee shop.

In today’s newsletter: Continuing to look into superposition, this time with math

Throughout this work, you have read numerous times that this quantum stuff doesn’t work the way we describe it, but that metaphor and analogy are decent ways of gaining an understanding. I stand by that statement, but every now and then I want to poke further down a different rabbit hole.

Most recently, I’ve used the image of little wave pulses overlapping in multiple directions and dimensions to represent qubits, as shown below:

The story that a qubit simultaneously holds values 1, 0, and everything in between, again, is a metaphor to help us grasp something that few people really understand and even fewer can explain. We use the “1 and 0” language because humans don’t really think in the mathematical terms that more accurately describe quantum states, but it’s time to bridge that gap.

In 1939, physicist Paul Dirac wrote his paper: "A New Notation for Quantum Mechanics" in which he gave us a standard for mathematical notation in quantum mechanics. He called this notation is called “bra-ket” (pronounced like bracket) notation. It’s sometimes also referred to as Dirac notation. It describes symbols in two forms: bra and ket.

The brackets used in bra-ket notation are a combination of a vertical line | and angle brackets ⟨ ⟩. The use of those symbols makes math easier and gives the ability to show that a lot of things can go into the definition of “superposition.”

A ket has the vertical bar first and the angle bracket second. For example, at the end of issue 13, I said that the symbol name for a qubit is the Greek alphabet character: Ψ, called psi (pronounced like “sigh”). A qubit with an unknown state written in bra-ket notation would be: |Ψ⟩. If the vertical line is first, it is called a ket. When the angle bracket is first, it is called a bra.

By definition (Wikipedia under “Bra-ket notation”), a ket is: “a vector, v, in an abstract (complex) vector space V, and physically it represents a state of some quantum system.” A ket represents a vector.

While a ket is a vector, a bra is a function that acts on a vector. A bra, ⟨f|, has the italicized “f” to indicate that it is a function. Combine the two and you would have: ⟨f|Ψ⟩ which is a function that would act on a qubit.

Wait. What?

Yes. A qubit is a vector. Though we talk about qubits as 0 and 1, qubits are more than just a simple number.

Most numbers we talk about are pretty simple, but not all are. Two of the terms we will be using here are “scalar” and “vector.” A scalar is a number that just means one thing. Like 65, as in I’m driving 65 miles per hour. It’s just a number and all it represents is the speed that I’m traveling.

We don’t normally run around saying that I’m driving 65 scalar miles per hour. We really only need to use “scalar” as a label when we are also talking about vectors, so we can differentiate between the two.

A vector is a number that means two things. Like, I’m driving 65 miles per hour heading south. That sentence has speed and direction so it represents a vector. You could write this as: 𝑣⃗ = (65,180). The parentheses group 65 and 180 as being both part of the same vector.

𝑣⃗ is the symbol that means “vector”

65 is the speed in miles per hour

, separates the two parts of the vector

180 is the direction (south) in degrees on a compass

( and ) group the two parts into the same vector

Scalar numbers are great when you know all of the other information. Like, I’m going 65 miles per hour on the freeway from Seattle to Tacoma. Funny thing here. While “65” is still a scalar because it just means our speed, the whole sentence turned into a vector because it has both speed (65) and direction (from Seattle to Tacoma).

𝑣⃗ = (65,from Seattle to Tacoma)

So, a vector is a set of numbers that contains more than one piece of information. A scalar is just about one piece of information. 65 is a scalar. The phrase “from Seattle to Tacoma“ just indicates the direction so it is also a scalar. Combine the two and you have a vector.

That above example is a two-dimensional vector. the speed is one dimension, and the direction is another dimension. Vectors can be more complex than that too. A multi-dimensional vector might say that I’m right now at 70 miles per hour and accelerating from 65 to 75 miles per hour while traveling from Seattle to Tacoma and spinning in circles at 10 rotations per minute because I hit ice. Speed, acceleration, direction, and rotation - that’s a four-dimensional vector.

Qubits are vectors. That means there is more than just one bit of information in a qubit. Even when not in superposition, qubits are still vectors and have two pieces of information in them. A non-superposition qubit may be in the state (1,0) or in the state (0,1). In bra-ket notation, these would be |0⟩ and |1⟩. When the qubit is in superposition, the vector would be written as (1,1), which in bra-ket notation is |+⟩.

That is all for today. Next week we will put qubit vectors into perspective with more metaphor and maybe look at quantum entanglement.

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I’ve wrapped the first ten issues of The Quantum Edge newsletter into book form. The collection, called “The Quantum Computing Anthology, Volume 1”, is now available in Kindle and paperback on Amazon. The book collects newsletter issues 1 through 10 and has some additional material and edits for continuity and clarity. I will add another volume to the series every ten newsletter issues, so look for Volume 2 (newsletter issues 11 - 20) in early 2026.

You can order the Kindle or paperback editions on Amazon today: The Quantum Computing Anthology, Volume 1

Check your email box Thursday - probably. (Okay, some of these weekly issues have come out on Friday, or not at all. But, in a quantum world, how can you tell?)

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Positive Edge is the consulting arm of Duane Benson, Tech journalist, Futurist, Entrepreneur. Positive Edge is your conduit to decades of leading-edge technology development, management and communications expertise.