quantumphysicslearning

Qubits: A Light Switch That Isn't Just On or Off

October 11, 202616 min read

I have been dabbing into quantum physics lately. The surprise is not how weird it is, it is how neatly it sits next to the physics I already knew. It blends into it the way one colour of light fades into the next on a rainbow, with no line where one stops and the other starts.

The book that got me going is Why Nobody Understands Quantum Physics. In one of teh chapters, it sorts all of physics with two questions: how big is the thing, and how fast is it moving?

Four questions, four rulebooks

size: small → big

speed: slow → fast

Size: an atomSpeed: an atomic electron’s speed

An atom at an atomic electron’s speed follows Schrödinger & Heisenberg. Particles act like waves, and exact positions give way to probabilities. This is where qubits live.

Drag the sliders, or click a corner for an example. The borders are rough: physics fades from one rulebook into the next, like colours in a rainbow, instead of switching at a line.

	Slow	Fast (near light speed)
Big	Newton. Ants, cars, planets. Three laws about forces, speed and momentum. F = ma	Einstein's relativity. We sit on a planet moving around a sun that moves around a black hole, and energy and mass turn out to be the same thing. E = mc²
Small	Schrödinger and Heisenberg. Electrons and atoms. Particles behave like waves, and light is both a wave and a particle. iħ ∂ψ/∂t = Ĥψ	Dirac. Small things moving very fast. (iħγ^μ∂_μ − mc)ψ = 0

You do not need to read those equations. The point is only that every cell has its own rulebook, and each one is a better description than the one before it once you enter its territory.

The bottom-right cell was the one I knew least about. Dirac wrote his equation to agree with both Schrödinger (when things are slow) and Einstein (when they are fast). Two things fell out of it that he did not put in: the electron's spin, and a mirror-image particle, the positron, which turned up in experiments a few years later. That is antimatter, predicted by an equation before anyone saw it.

Spin is the one that matters today, because spin is where the qubit comes from.

From a light switch to a qubit

Take the switch on your wall. Up, the light is on. Down, it is off. That on/off choice is called a bit, and every photo, song and game on your computer is a gigantic pile of them, flipped by billions of tiny transistor switches.

A bit is a 0 or a 1. Nothing in between, ever. Click the switch to flip it yourself.

Now swap the wall switch for something smaller: an electron. Besides its charge, an electron has a property called spin.

Physicists draw this needle inside a ball, and it is the standard picture of a qubit as shown below. The north pole is "up", the south pole is "down", and every other spot on the surface is a blend (superposition) of the two. The two points on the equator are the perfectly even blends, 50% up and 50% down. (The |0⟩ and |1⟩ labels are just names for "up" and "down", and I explain them in a moment.)

Try it in three steps. Drag the arrow to any spot, choose which question to ask, then measure. A normal compass would give a smooth reading that depends on the angle. A real spin never does: it snaps to one of two answers, and the angle only decides the odds (probabilty). That what is the probablity that the the position is up or down and the sign in between gives the direction of the spin. Spinning left or spinning right (which is the momentum). And here you can see if you know the position of the electron top or bottom you sort of eliminated the superposition state.

1 · Set the state

Drag the arrow tip, or use the slider: 35°

|ψ⟩ = 0.95|0⟩ + 0.30|1⟩

2 · Choose what to ask

Chance of up: 91%. Chance of down: 9%.

3 · Measure

Drag the arrow tip, choose what to ask, then measure.

This ball is the standard picture of one qubit, called the Bloch sphere. Only the slice facing you is drawn, so the arrow stays in the page. A simulation of P(+) = cos²(θ/2), not lab data.

So an electron's spin looks like a switch too. We call the two answers 0 (up) and 1 (down). Here is where it gets odd: before you measure, the spin does not have to be either one. It can be a blend of both, and a qubit is the name for that.

|ψ⟩ = α|0⟩ + β|1⟩

ψ (psi) is just the name of the state, like a variable name. |0⟩ and |1⟩ are the two answers you can get, ‘up’ and ‘down’. α and β (alpha and beta) say how much of each is in the mix. The angle-bracket thing, | ⟩, is called a ket, and all it means here is ‘this is a quantum state’. It is called bra-ket notation, and I will go into it properly in another post.

That is the whole definition of a qubit. The two numbers α and β are called amplitudes, and they decide what happens when you finally look.

P(0) = |α|² P(1) = |β|² |α|² + |β|² = 1

Square an amplitude to get the chance of that answer. The two chances must add up to 100%, because you will get one of the answers.

A quick example so the symbols feel real. Say α = 0.6 and β = 0.8.

The chance of "up" is 0.6² = 0.36, which is 36%.
The chance of "down" is 0.8² = 0.64, which is 64%.
Check: 36% + 64% = 100%. Good.

A perfectly even blend has α = β = 1/√2, which gives 50% each:

|ψ⟩ = |0⟩ + |1⟩√2

Half the squared amplitude each way: (1/√2)² = ½. A coin flip, but a coin flip that has not happened yet.

Written as plain numbers, a qubit is just a list of two of them. This are called as eigen vectors (I know I should have paid more attention to matrices in school):

|0⟩ = 10 |1⟩ = 01 |ψ⟩ = αβ

‘Up’ is the list (1, 0), ‘down’ is the list (0, 1), and any blend is the list (α, β). Everything a quantum computer does is arithmetic on lists like this.

The catch: you cannot ask every question at once

Heisenberg's famous rule says the more precisely you know where something is, the less precisely you can know how fast it is moving, and the other way around. My first draft listed this as two rules. It is one rule, read from either side:

Δx · Δp ≥ ħ2

Δx is how fuzzy the position is and Δp is how fuzzy the momentum (mass times velocity) is. Multiply them and you can never get below ħ/2, a very small number. Squeeze one fuzziness down and the other has to grow.

The number is tiny, so it only bites for tiny things. Trap an electron in a space the width of an atom (0.1 nanometres) and its speed is uncertain by about 580 kilometres per second. Do the same to a baseball (0.145 kg) in a 1 nanometre space and its speed is uncertain by about 4 × 10⁻²⁵ metres per second. That is why you never see a baseball blur.

Spin has its own version of the same idea. "Is the spin up or down along the vertical axis?" and "Is it left or right along the sideways axis?" are two different questions, and they cannot both have a sharp answer at the same time. If you are sure about one, the other is a coin flip. In the language from above:

|→⟩ = |0⟩ + |1⟩√2 |←⟩ = |0⟩ − |1⟩√2

An atom that is definitely ‘right’ is an even blend of ‘up’ and ‘down’, so asking up-or-down gives 50/50. An atom that is definitely ‘left’ is the same blend with a minus sign. Notice the minus sign. We will need it.

And the same thing the other way around, an "up" atom is an even blend of "right" and "left":

|0⟩ = |→⟩ + |←⟩√2 |1⟩ = |→⟩ − |←⟩√2

Being sure about up means being unsure about left versus right.

Stern–Gerlach: three gates, one surprise

In 1922 Otto Stern and Walther Gerlach fired silver atoms through a magnet whose field changes from place to place. Atoms with spin pointing one way get pushed up, the other way down. A classical needle could point any direction, so they expected a smear on the detector. They got two separate spots. Spin really does come in just two values.

So a Stern–Gerlach magnet works as a measuring gate for spin. Now chain three of them together, which is a staple of quantum textbooks:

Gate 1 measures up/down (the Z direction) and we keep only the "up" atoms.
Gate 2 is turned sideways. It measures right/left (the X direction).
Gate 3 measures up/down again, and a screen records where the atoms land.

Three runs, changing only what gate 2 lets through:

Block the "left" path. At the end you find two clouds on the screen.
Block the "right" path instead. The same result, two clouds.
Let both paths through. Now there is only one cloud.

Read that again. Opening a second path made one of the spots disappear. Play with it. The labels on the wires say what state the atom is in at each stage, using the same kets as before, and you will meet them again in the arithmetic below:

0 sent0 stopped at gate 10 stopped at the X gate0 landed up0 landed down

Pick a run, then send the atoms from the oven.

Why: Nobody looks at which arm the atom took, so it takes both at once (the two faint dots) and recombines. Each arm carries half an amplitude to each spot. The two routes to "down" have opposite signs and cancel to exactly zero, while the two routes to "up" add up. Every atom lands on "up". One cloud.

The labels on the wires say what state the atom is in at that stage, and the one with an atom in it lights up. |↑⟩ is the |0⟩ from earlier and |↓⟩ is |1⟩. A simulation drawn from the quantum probabilities, not lab data. The "let both through" run is the idealised textbook version, where the two arms recombine and nothing records which one the atom took. The faint pair of dots is one atom on both arms.

Why one cloud? Do the arithmetic

Every atom leaving gate 1 is in the state |0⟩ ("up"). Using the blend from above, that is also an even mix of right and left:

|0⟩ = |→⟩ + |←⟩√2

Same atom, written in the sideways language. Each path has amplitude 1/√2.

Runs 1 and 2: one path blocked. Gate 2 acts as a measurement. You find out the atom went "right" (say), so it is now exactly |→⟩. Gate 3 then asks up-or-down of an even blend: a coin flip, so two clouds. Half the atoms were stopped at gate 2, and the survivors land 50/50.

Run 3: both paths open. Nobody finds out which path the atom took, so both stay alive. Now each way of arriving at "down" has its own amplitude, and amplitudes get added, not their probabilities. Go via the "right" path, where the amplitude of "down" is 1/√2 × 1/√2. Go via the "left" path, where the sign flips (remember the minus!) and it is 1/√2 × (−1/√2).

amplitude(↓) = 12 − 12 = 0
amplitude(↑) = 12 + 12 = 1

Two routes to ‘down’ with opposite signs add up to exactly zero: they cancel, like a wave crest meeting a trough. Two routes to ‘up’ have the same sign and add up to 1, which means a 100% chance. Every atom lands on ‘up’. One cloud.

If those had been plain probabilities, ¼ + ¼ = ½ would have given "down" a 50% chance in both cases and nothing would be strange. Probabilities can only add up. Amplitudes can cancel, and that cancellation is what you see in the experiment. It is also the trick quantum computers are built around: arrange things so the wrong answers cancel out and the right answer adds up.

You have probably lain inside a quantum experiment

The part of the book that really got me: an MRI scanner is spin physics at work. You are mostly water, and water is H₂O, so you are full of hydrogen. A hydrogen nucleus is a single proton, and a proton has spin, just like the electron above. Each one is a tiny compass needle that can sit in a state we can call 0 or 1.

(One honest caveat: a scanner is not a computer handling one neat qubit at a time. It measures a huge crowd of spins at once, and "qubit" is an analogy that helps me think about it.)

Here is what happens, step by step:

1. The big magnet. In a strong magnetic field, each proton's needle wobbles around the field direction, like a spinning top around the vertical. How fast it wobbles depends only on the field strength, and it is called the Larmor frequency:

ω = γ B

ω is the wobble rate, B is the strength of the magnet in tesla, and γ (gamma) is a constant that is different for every kind of nucleus. For hydrogen, γ/2π is about 42.58 million wobbles per second for every tesla.

A typical 1.5 tesla hospital scanner makes hydrogen wobble at about 64 million times a second. That is radio frequency, a bit below the FM band. Hold that thought, it is why a radio wave is the tool for the next step.

2. Two energy levels. The field splits the proton's two spin states into a lower-energy "0" and a higher-energy "1", separated by a small gap:

ΔE = ħ ω

The gap between the two states is the wobble rate times ħ. It is the same ħ from Heisenberg's rule, the quantum yardstick.

I assumed from the book that the protons fall into state 0. They do, but only just. The gap is so small that heat nearly drowns it out, and only a thin sliver more protons end up in 0 than 1:

excess ≈ ħ ω2 k T ≈ 5 × 10⁻⁶

k is Boltzmann’s constant and T is body temperature. Put in 1.5 tesla and 37 °C and the answer is about 5 in a million. That tiny excess is the signal. It is enough only because a single cubic millimetre of water holds around 7 × 10¹⁹ hydrogen nuclei.

3. The radio pulse. A short burst of radio waves at exactly the wobble frequency nudges the spins out of their neat 0 state into a blend of 0 and 1, like the |ψ⟩ = α|0⟩ + β|1⟩ from earlier. Only the right frequency works, which is why the number from step 1 matters.

4. The echo. When the pulse stops, the spins drift back to their resting state and give off a faint radio signal that the scanner's coils pick up. Fat, water and tumour tissue settle back at different speeds, and that difference is what makes the image contrast. Extra magnetic fields that vary across your body shift the frequency slightly by position, which is how the scanner knows where each signal came from.

I kept filing quantum mechanics under teleportation and supercomputers. I had been inside one the whole time, or at least been scanned by one. Knowing that blew my mind.

Cheat sheet

Bit: a switch that is 0 or 1, never both.
Spin: a tiny built-in compass-needle property of particles like electrons and protons. Measured along any direction, it gives one of two answers.
Qubit: a spin (or anything like it) described as a blend of 0 and 1: |ψ⟩ = α|0⟩ + β|1⟩.
Amplitude (α, β): the number in front of each answer. Square it to get a probability. It can be negative, and amplitudes can cancel each other.
Superposition: being in a blend of answers before you measure.
Measurement: asking the question. You get one answer, and the blend is gone.
Uncertainty: some pairs of questions, such as position and speed, or up/down and left/right spin, cannot both have sharp answers at once.

Phew, that was a lot. It took me months to get a grip on this, and I hope laying it out step by step made it easier for you than it was for me. Yes, it is a bit math heavy, so I drew as much of it as I could. The part that still gives me goosebumps: at the tiny scale, physics stops handing out answers and starts handing out odds like as if we are gambling. Remember the map at the top? We live in the big-and-slow corner, where Newton is enough. The other three corners were always there. We just never meet them, because of how big and how slow we are. Scale is strange in general. An atom is tens of millions of times smaller than an ant, which is about a thousand times smaller than an elephant, which is millions of times smaller than the Earth. And an atom is almost entirely empty space: scale it up to a 100-metre stadium and the nucleus is a grain of sand in the middle. Now a pure daydream, with no physics behind it. We only ever see the world from our own size. Is something far bigger looking at us the way we look at atoms? Are blackholes the atom to them and we are the particle that is spining around? Or could they even detect us while we are so tiny to them? Exciting times!

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