Quantum States¶
What is a qubit actually holding, and what do we see when we look?
From Bits to States¶
A classical bit is simple. Whatever value it holds is the value you read.
Module 01 gave you the qualitative picture of a qubit. This module makes it precise, and it starts with one question.
For a qubit, is what it holds the same as what you see?
This page separates the two. What a qubit holds is its quantum state. What you see when you measure it is an outcome. They are not the same thing, and keeping them apart is the foundation for everything that follows.
Preparing a State¶
To work with a qubit, you first set it up. That setup is a preparation: a recipe. The state is what the recipe produces.
|0⟩ and |1⟩ are names for the two basis states. The third recipe prepares a superposition, and "balanced" means it treats 0 and 1 equally. Prepare a qubit the same way again and you get the same state again.
If you prepare a qubit the same way twice, do you get the same result twice?
You get the same state. What you see when you measure it can still differ.
Prepare, Predict, Observe¶
Choose a preparation. Predict what repeated measurements will show. Then measure and compare. The results are simulated: ideal and noiseless, not from real hardware.
1. Choose a preparation
2. Predict what repeated measurements will show
3. Measure
Outcomes
Notice the three separate places. The preparation is what you chose. The state is what that preparation produces. The outcomes are what measurements returned. Each outcome token is one result. The bars show how the results add up across many shots.
State vs. Outcome¶
| State | Outcome | |
|---|---|---|
| What it is | What the preparation produced | What one measurement returned |
| How you get it | By preparing the qubit | By measuring it |
| One run vs. many | The same every time you prepare the same way | One result per run; a pattern across many runs |
A state is what was prepared. An outcome is what one measurement returned. Check your understanding:
And one more, applying the idea to a new case:
A Superposition Is Not a Hidden Bit¶
Picture a coin flipped and covered with a cup. It is already heads or tails; you just have not looked. That is ordinary uncertainty about a value that exists.
A balanced superposition is not described that way. But look at the counts from the demo above: they look just like a fair coin. Counts from this one kind of measurement cannot tell the two apart.
If the counts look like a fair coin, how could a superposition be anything more?
Other tests tell them apart, and Module 01 already showed two. In the Superposition demo, mixing a qubit twice brings it back to a certain outcome, while mixing a random bit twice leaves it 50/50. In the CHSH game, an entangled pair wins more often than any pre-arranged strategy can. Later pages in this module, on gates and on measurement, develop the first of these. For now, the point is only that these counts alone do not settle it.
This page does not claim that a qubit is "both 0 and 1 at once" in any literal sense, and it does not claim that measuring reveals a value that was already there. It claims only what the demo shows: a state is one thing, and the outcomes you see are another.
The Numbers Behind the Probabilities¶
A state comes with numbers, called amplitudes, that decide the probabilities of the outcomes. A balanced superposition has amplitudes that make 0 and 1 equally likely. A prepared |0⟩ has amplitudes that make 0 certain.
A state can also carry something that these outcome probabilities do not show. It is called phase. We will not calculate with either idea yet. The next page shows how to write a state down so that you can.
Beyond the Core¶
Same counts, different states¶
Two different states can give identical counts when measured this way. For example, a state can differ from the balanced superposition only in its phase, which this measurement cannot see. The counts alone therefore do not determine the state.
The next pages show how to describe that difference and how to picture it.
Pure and mixed states¶
Every state on this page is a pure state: a complete description of a qubit that was prepared in one definite way.
A mixed state describes a qubit whose preparation is itself uncertain, for example one that was prepared as |0⟩ or as |1⟩ with some probability. Such a qubit also gives 0 and 1 about equally often, yet it is a different thing from the balanced superposition. A later treatment makes the difference precise.
From States to Writing Them Down¶
Compare what you can do now with what is still missing.
You can describe the balanced superposition in words, and you can run it and watch the counts. You cannot yet predict what it will do in a new situation. For that you need a precise way to write a state down.
Takeaway¶
A quantum state is what a preparation produces. What you see is an outcome drawn from it, and one kind of measurement does not show you the whole state.
How can we write a state down so we can calculate what it will do?
Next: Qubit Representation → (coming soon)