How Does Readout Depend on the Measurement Basis?

• One box with two faces
• Seen from the front: the teal face
• Seen from the side: the amber face
• The two images obtained from the two views

What you will learn

• Explain what a measurement basis is.
• Give an example where one state gives two different readouts in two bases.
• Describe how a different basis is measured with a Hadamard gate.

Before you start

Know that a qubit is a quantum bit and that measuring it gives 0 or 1 with some probability.

What is a measurement basis?

A measurement basis is the set of directions the state is compared against. The AWS Quantum Technologies article says quantum measurement determines the degree of projection of the state onto the measurement basis. Like light casting a shadow of an object, the readout shows how much of the state lies along each of those directions.

According to AWS, a measurement produces an outcome with a probability given by the squared magnitude of its corresponding amplitude; it does not directly read the complex amplitude. For a qubit measured in the computational basis, the outcome is |0⟩ or |1⟩. Superposition describes the state before measurement; it does not mean a single readout reveals every component.

• Tilted stick (like the state)
• Shadow on the ground (the projection)
• Projection lines
• Shadow on the wall (projection another way)

How can one state give two different readouts in two bases?

The clearest example is the two states |+⟩ and |−⟩. They differ only in the relative phase between their |0⟩ and |1⟩ components. According to the AWS article, states that differ only in relative phase give identical measurement statistics in that basis, although they are distinct quantum states.

Specifically, in computational-basis measurements both |+⟩ and |−⟩ give 50% |0⟩ and 50% |1⟩. But in the {|+⟩, |−⟩} basis, |+⟩ always gives |+⟩ and |−⟩ always gives |−⟩, both with probability one. The same two states go from indistinguishable to fully distinguishable by changing basis.

• Computational basis: an even split for both states
• The |+⟩, |−⟩ basis: one certain result per state
• Divider between the two states

How is a different basis measured?

By rotating the state before measuring. AWS writes that most quantum hardware can only measure in a single fixed computational basis, the Z basis. Measuring in a different basis is achieved by an equivalent unitary transformation. For example, a Pauli X measurement is implemented by applying a Hadamard gate before a Z-basis measurement.

• The qubit state
• The rotation (like a Hadamard gate)
• Fixed measuring device in the Z basis

Why do we need several measurement bases?

Because each basis shows only part of the picture. The AWS article says different measurement bases reveal complementary aspects of quantum states. What appears as phase in the computational basis shows up as amplitude in a different basis. Each measurement basis accesses specific quantum information while leaving other aspects unobservable, and we cannot get all perspectives simultaneously.

Real-world example

Compare it with a box whose front face is blue and whose side face is yellow. Looking straight from the front you only see blue. Looking from the side you only see yellow. The same box gives different information from each direction, but this analogy does not capture randomness or the change of state caused by quantum measurement. This is only a comparison to help picture it, not an exact description of quantum measurement.

Common misunderstandings

Measurement reads out a value the qubit already holds. Measurement determines the degree of projection of the state onto the measurement basis, so the readout depends on the basis chosen.

Two states with the same readout are the same state. States that differ only in relative phase are still different states; they only share statistics in that basis.

One measurement tells you everything about the state. Each measurement basis accesses specific information and leaves other aspects unobservable.

Frequently asked questions

What is a measurement basis?

The set of directions chosen for a measurement; the readout shows how the state projects onto them.

Why does one state give different readouts in two bases?

Because measurement determines the projection onto the chosen basis, so changing the basis changes the statistics.

How do you measure in another basis on hardware that only measures in the Z basis?

Apply an equivalent unitary transformation before the measurement, such as a Hadamard gate for a Pauli X measurement.

What does relative phase change?

Two states differing only in relative phase give the same statistics in the computational basis but can be told apart in another basis.

Summary

• The readout depends on the measurement basis, because measurement determines the projection onto that basis.
• Two states with identical measurement statistics in the computational basis can still be distinguished in another basis.
• Another basis is measured by a unitary transformation, such as a Hadamard gate, before a Z-basis measurement.