Lesson 3 of 9
Amplitudes, not probabilities
Explain why amplitudes can cancel each other and probabilities never can.
01 · Learn · the idea
Open a second slit and parts of the screen go dark. You gave the light another way to arrive and it stopped arriving.
How it works
- Every route a quantum system could take carries an amplitude
- An amplitude can be NEGATIVE this is the whole resource
- Add the amplitudes of every route to the same outcome
- THEN square the total to get the chance add first, square second
- Opposite signs cancel to nothing — an outcome that cannot happen
In numbers
Chance of landing at one spot on the screen
Same two slits, same light. Only a minus sign separates certainty from impossibility.
Side by side
Two ways of combining two routes
The common misread
That the two slits “share out” the light, so opening both spreads it thinner. Opening both makes some spots brighter than either slit alone and other spots completely dark — that is not sharing, it is adding and subtracting.
The words
- amplitude
- the number carried by each route; square it to get a chance
- cancellation
- two routes with opposite signs adding to nothing
Carry this
How two things combine is a separate question from how big each one is, and it is usually the one nobody asks.
Full lesson
Open a second door and the room gets darker
Shine light at a wall with one narrow slit in it, and a screen behind catches a bright band. Close that slit, open a second one a little to the side, and you get a bright band again — shifted over slightly, but bright.
Now open both slits at once.
There are places on that screen which were lit by the left slit alone, lit by the right slit alone, and are completely dark with both open. Not dimmer. Dark. You gave the light a second way to arrive, and it stopped arriving.
Nothing in ordinary chance behaves like this. Two routes to a place can only make it more likely, never less. If a second door to a room can leave the room empty, the thing walking through the doors is not being counted by probabilities.
What is being counted instead
Every route a quantum system could take carries a number. That number is called an amplitude, and the crucial thing about it — the thing this whole item exists for — is that it can be negative.
To get the probability of an outcome you do two steps, in this order:
- Add the amplitudes of every route that leads to that outcome.
- Square the total.
Adding first is what makes cancellation possible. Squaring second is what makes the answer a real chance, never negative.
Work the two-slit case with actual numbers. Say each slit on its own gives an amplitude of one half at some point on the screen:
- Left slit only: amplitude 0.5 → probability 0.5 × 0.5 = 0.25
- Right slit only: amplitude 0.5 → probability = 0.25
- Both open, same sign: amplitudes 0.5 + 0.5 = 1.0 → probability 1.00
- Both open, opposite signs: amplitudes 0.5 + (−0.5) = 0 → probability 0
Look at the last two lines. Same two slits, same two routes, same amount of light. The only difference is a minus sign, and the answer swings from certainty to impossibility.
If you had squared first and then added — the way probabilities work — you would have got 0.25 + 0.25 = 0.5 in both cases, and never seen a dark band at all.
Why negatives are the resource
An ordinary computer that wanted to check a million answers would have to look at a million answers. There is no arrangement of probabilities that helps, because probabilities only ever pile up. Everything is positive; nothing subtracts.
Amplitudes subtract. That means a quantum computer can set up a calculation where the routes leading to wrong answers carry amplitudes that add to nearly zero, while the routes leading to the right answer reinforce. The wrong answers do not get checked and rejected. They get cancelled before anyone looks.
This is also why the last item mattered so much. A qubit’s state is a direction, and which way round the equator it points is exactly the sign of its amplitudes. “Both at once” throws away the minus sign. The minus sign is the machine.
A rule the arithmetic will not let you break
There is a constraint that keeps all of this honest: the squares of the amplitudes over all possible outcomes must add to exactly 1. Something always happens.
So amplitude cannot simply be manufactured. Pushing amplitude towards one answer means pulling it away from others. A quantum algorithm is not a machine for creating certainty out of nothing; it is a machine for moving it, from the answers you do not want into the one you do.
Where else this shape shows up
The habit worth taking from this is not about physics. It is that the rule for combining things is a fact about the world, and it is not always addition.
We assume, almost always without noticing, that more routes means more likely, that more effort means more output, that adding a second person to a job speeds it up. Sometimes the quantities really do just add. Sometimes they cancel — a second messenger contradicting the first, a second policy undoing the first. The question of how these combine is separate from the question of how big each one is, and it is usually the one nobody asks.
02 · Try · the lab
03 · Check · quick quiz
1. Two routes lead to the same spot, with amplitudes 0.5 and −0.5. What is the chance of landing there?
- Zero — the amplitudes add to 0 first, and 0 squared is 0
- 0.5 — you square each one and add: 0.25 + 0.25
- 0.25 — the two cancel down to one route
- It is undefined, because a negative chance is impossible
Answer
Zero — the amplitudes add to 0 first, and 0 squared is 0 — Add the amplitudes, then square. Adding gives 0, so the outcome never happens. Squaring first and then adding would give 0.5 and would miss the whole effect.
2. Why can't an ordinary probability calculation ever produce a dark band?
- Probabilities are never negative, so extra routes can only pile up
- Probabilities are too small to cancel each other out
- Ordinary calculations round the numbers before adding them
- Light is not random, so probability does not apply to it
Answer
Probabilities are never negative, so extra routes can only pile up — Everything is positive, so nothing can subtract. Cancellation needs a minus sign, and only amplitudes have one.
3. A quantum algorithm makes the right answer far more likely. Where did that extra likelihood come from?
- From the wrong answers — the squares must always add to 1, so certainty is moved, not created
- From running many copies of the calculation in parallel universes
- From the machine checking each wrong answer faster and discarding it
- From measuring the qubits repeatedly until the right answer appears
Answer
From the wrong answers — the squares must always add to 1, so certainty is moved, not created — Something always happens, so the squared amplitudes total exactly 1. Pushing amplitude into one answer means pulling it out of the others — the wrong answers are cancelled, never checked.