THE EQUATOR: |+⟩, |−⟩, |I⟩, |−I⟩
Four special states that look identical under one measurement and totally different under another.
The equator of the Bloch sphere (θ = 90°, per Lesson 3) contains infinitely many points, each representing a different phase (φ) but the same 50/50 split between measuring 0 and measuring 1. So why do exactly four of these infinitely many points get special names and dedicated notation?
These four points are the ones that show up constantly in practice:
- They are the natural outputs of the most commonly used quantum gate, the Hadamard gate (Module 4)
- They are the states you get "definite" (100% certain) answers for when measuring in the X or Y basis
- They're the "poles" of those bases, the way
|0⟩/|1⟩are the poles of the Z basis
Think of them as the compass rose of the equator. A compass can point in any of 360 directions, but we still give special names to North, East, South, and West because they're the directions we reference constantly.
The Four Points, Precisely Located
Going around the equator (increasing φ, per Lesson 3's longitude angle):
| STATE | Φ (LONGITUDE) | PLAIN DESCRIPTION |
|---|---|---|
| |+⟩ | 0° | "Plus" — the reference starting point |
| |i⟩ | 90° | Quarter-turn around from |+⟩ |
| |−⟩ | 180° | "Minus" — directly opposite |+⟩ |
| |−i⟩ | 270° | Three-quarter turn around, opposite |i⟩ |
Same Probability, Different States — How Can That Be?
Here's the important, slightly mind-bending fact this lesson is built around: all four of these states give exactly 50% chance of measuring 0 and 50% chance of measuring 1 in the standard (Z-basis) measurement. If all you ever did was measure in the Z-basis, these four states would be statistically indistinguishable from each other — and from any other point on the equator.
So in what sense are they different states at all? Two ways:
1. They're different points on the sphere. Geometrically, they're obviously not the same arrow — they point in four different directions. Two things can be visually/geometrically distinct even if one particular type of question (a Z-measurement) can't tell them apart.
2. A different measurement can tell them apart — perfectly. Measuring |+⟩ in the X-basis gives one outcome with 100% certainty. Measuring |−⟩ in the X-basis gives the opposite outcome with 100% certainty. Similarly, measuring |i⟩ in the Y-basis gives one outcome with 100% certainty; |−i⟩ gives the opposite.
A Worked Comparison Table
| STATE | Z-BASIS MEASUREMENT | X-BASIS MEASUREMENT | Y-BASIS MEASUREMENT |
|---|---|---|---|
| |+⟩ | 50% / 50% | 100% "plus" outcome | 50% / 50% |
| |−⟩ | 50% / 50% | 100% "minus" outcome | 50% / 50% |
| |i⟩ | 50% / 50% | 50% / 50% | 100% "i" outcome |
| |−i⟩ | 50% / 50% | 50% / 50% | 100% "−i" outcome |
Each state is "certain" in exactly one basis — its own — and "50/50" in the other two. This is a preview of the deeper concept that what you can learn from a measurement depends on what basis you choose to measure in.
|+⟩ and |−⟩ are named for being the two states left completely unchanged in direction (only sign-flipped, in the math you'll see in Module 2) by the Pauli-X gate — the quantum equivalent of a classical NOT gate, covered in Module 4. |i⟩ and |−i⟩ play the same special role for the Pauli-Y gate.
Practice Questions
Test your understanding
KEY TAKEAWAYS
Remember these points
|+⟩, |i⟩, |−⟩, |−i⟩ are four specially-named points on the Bloch sphere's equator, spaced 90° apart in phase
All four give a 50/50 split when measured in the standard Z-basis — they are statistically indistinguishable by that specific measurement
They are, however, geometrically distinct states, and each is perfectly distinguishable using a different measurement basis (X-basis distinguishes plus/minus; Y-basis distinguishes i/minus-i)
What you can learn from a measurement depends on what basis you choose to measure in — a theme formalized fully in Module 8