The Equator: |+⟩, |−⟩, |i⟩, |−i⟩

The Bloch sphere has two special poles: $|0\rangle$ at the north and $|1\rangle$ at the south. Every other point on the sphere is a superposition, and the equator — the great circle lying halfway between the poles — is where things get interesting. The four states that sit squarely on that equator are $|+\rangle$, $|-\rangle$, $|i\rangle$, and $|-i\rangle$.

The X-basis states: $|+\rangle$ and $|-\rangle$

The Hadamard gate takes $|0\rangle$ to the $+x$ axis and $|1\rangle$ to the $-x$ axis:

These two states form the X basis (also called the Hadamard basis or $\pm x$ basis). The state $|+\rangle$ sits on the $+x$ side of the equator; $|-\rangle$ sits directly opposite on the $-x$ side. Because they are orthogonal ($\langle +|-\rangle = 0$), they can serve as an alternative measurement basis — you measure in the X basis by applying $H$ before a standard readout.

The Y-basis states: $|i\rangle$ and $|-i\rangle$

The $+y$ and $-y$ points on the equator are reached by adding a complex phase. Applying the S gate (phase gate) after Hadamard multiplies the $|1\rangle$ amplitude by $i$:

The $-y$ counterpart is

These two states form the Y basis. On the Bloch sphere, $|i\rangle$ sits at the $+y$ axis of the equator and $|-i\rangle$ at the $-y$ axis. Together with the X-basis states, they tile the equator into four equally-spaced points separated by $90°$.

Why these four states matter

The four equator states are the eigenstates of the $X$ and $Y$ Pauli operators:

• $X|+\rangle = +|+\rangle$, $\; X|-\rangle = -|-\rangle$
• $Y|i\rangle = +|i\rangle$, $\; Y|-i\rangle = -|-i\rangle$

They appear throughout quantum computing: $|+\rangle$ is the starting state for interference in the Hadamard transform, and the Y-basis states arise naturally in quantum phase estimation and in characterizing qubit noise.

Try it

Prepare the state $|i\rangle = \tfrac{1}{\sqrt{2}}\bigl(|0\rangle + i|1\rangle\bigr)$, which sits on the $+y$ axis of the Bloch sphere equator. The grader checks the full statevector (up to global phase).

After running your circuit, inspect the amplitudes: qubit $|0\rangle$ should have amplitude $\approx 0.707$ (real) and qubit $|1\rangle$ should have amplitude $\approx 0.707i$ (purely imaginary). That is the signature of the $+y$ equator point.

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