Quantum Foundations

The Plus-Minus Basis

The X basis {|+>, |->} is the diagonal basis conjugate to the computational basis, essential for reading out phase information.

A complementary frame

Alongside the computational basis {|0>, |1>}, the most important alternative is the plus-minus basis {|+>, |->}, also called the X basis or diagonal basis. Its states are |+> = (|0>+|1>)/sqrt(2) and |-> = (|0>-|1>)/sqrt(2).

Relationship to Z basis

Kronos motion — quantum verdict

The two bases are related by the Hadamard gate: H|0> = |+> and H|1> = |->, and conversely H|+> = |0>, H|-> = |1>. They are maximally incompatible — a state definite in one is an equal superposition in the other. |0> measured in the X basis gives + or - with equal probability.

|+> = (|0> + |1>)/sqrt(2), the +1 eigenstate of X
|-> = (|0> - |1>)/sqrt(2), the -1 eigenstate of X
On the Bloch sphere: the +x and -x poles

Reading phase

The X basis is how relative phase becomes visible. The states (|0>+|1>)/sqrt(2) and (|0>-|1>)/sqrt(2) look identical in the computational basis (both 50/50) but are perfectly distinguished in the X basis. Measuring in X is done by applying H and then measuring in Z. This is the standard readout for phase-based algorithms.

Mutually unbiased bases

The Z and X bases are mutually unbiased: complete ignorance in one means complete certainty is impossible in the other. Adding the Y basis gives three mutually unbiased bases for a qubit, aligned with the three axes of the Bloch sphere. This structure underlies the BB84 quantum key distribution protocol, which encodes bits randomly in the Z or X basis.

Why it matters

Choosing between the Z and X bases for preparation and measurement is a routine part of circuit design. Phase kickback deposits information in the X basis, and many error-correction parity checks measure X-type operators, so fluency with the plus-minus basis is essential.