A Mathematical Lesson On How Measurements Affect A Qubit

Use Linear Algebra on a Qubit to learn how it transforms upon being measured

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We have learned all the mathematics needed to understand quantum systems in a series of 10 lessons.

In this lesson, it’s time to track a qubit, learn to measure its state on different orthonormal bases, and mathematically learn how measurements affect it.

In case you missed the previous lessons, here they are:

• Lesson 1: Imaginary and Complex Numbers

• Lesson 2: Trigonometry

• Lesson 3: Differential Calculus

• Lesson 4: Probability

• Lesson 5: Linear Algebra (1)

• Lesson 6: Linear Algebra (2)

• Lesson 7: Linear Algebra (3)

• Lesson 8: Linear Algebra (4)

• Lesson 9: Linear Algebra (5)

• Lesson 10: Linear Algebra (6)

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An Electron As A Qubit

An electron has an intrinsic property called the Spin that can represent a Qubit in quantum computing.

Note that this does not literally mean that the electron is physically spinning in space. Instead, it just represents a fundamental aspect of its quantum state.

When measured along a specific axis (typically the Z-axis), the spin of an electron can be found in one of two states:

• Spin-Up (∣↑⟩): Aligned with the positive direction of the measurement axis (e.g., pointing up along the Z-axis on the standard orthonormal basis).

• Spin-Down (∣↓⟩): Aligned against the positive direction, pointing down along the negative Z-axis on the standard orthonormal basis.

These two states form the computational basis for a single qubit system.

|0⟩ or ∣↑⟩ and |1⟩ or ∣↓⟩ are the quantum analogues of the classical binary states 0 and 1.

(In the complex vector space C², the standard orthonormal basis and the computational basis are the same.)

But there’s a little nuance here.

A qubit can exist in a superposition of these states.

This is represented using the ket ∣ψ⟩.

When measured, the qubit collapses to either |0⟩ or |1⟩ with probabilities ∣α∣² and ∣β∣², respectively.

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The Experiment

Let’s run an experiment on an electron initially in the Spin-up state.

Step 1: Measuring In Vertical Axis

When we measure it using an apparatus (which can be a magnetic field gradient) aligned along the vertical direction, ∣α∣² will be 1, or the measurement result will always be Spin-up with probability 1.

Also ∣β∣² will be 0, or the probability that the qubit is in the Spin-down state will be 0.

Step 2: Measuring In Horizontal Axis

Next, we turn our measuring apparatus, aligning it along the horizontal/ X-axis.

The orthonormal basis for measuring spin in the horizontal direction is represented using ∣→⟩ and |←⟩, as shown below.

Or,

On measuring along this direction, the Spin-up qubit gets in a superposition state of the horizontal basis as follows:

Our task now is to find how much of this Spin-up electron is in the positive horizontal state and how much is in the negative.

This can be done by finding the coefficients a and b.

To express a given ket, let’s say, ∣v⟩ as a linear combination of an orthonormal basis { ∣b(1)⟩, ∣b(2)⟩, … , ∣b(n)⟩ }, the following formula can be used:

In the above equation, ⟨b(i)​∣v⟩ represents the inner product (or projection) of ∣v⟩ onto the basis vector ∣b(i)​⟩.

Using this, let’s reformulate the representation of the Spin-up electron on the horizontal axis.

Now, our task is to calculate a and b, where:

We calculate ‘a’ as follows:

We calculate ‘b’ as follows:

This leads to the following superposition state for the Spin-up electron when measured along the horizontal axis:

In this state, the square of the coefficients a and b gives the probabilities of finding the electron in the positive and negative horizontal/ X-axis, respectively.

In either case, the probability of ending up with either of the horizontal states is 50%.

Step 3: Measuring Again In The Vertical Axis

Let’s again measure using the vertical basis {∣↑⟩, ∣↓⟩}.

In the last step, we obtained ∣→⟩ and |←⟩, with equal probabilities.