# Qubit, An Intuition #3 — Quantum Measurement, Full and Partial Qubits

*The beauty of Quantum Mechanics for Quantum Computation, featuring IBM Quantum*

**Andi Sama**** **— *CIO, **Sinergi Wahana Gemilang** with **Cahyati S. Sangaji*

TL;DR;

Quantum measurement for full and partial qubits, with examples. Quantum state. Pure state.Please refer to the previous articles:

“Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World” for a discussion on a single qubit as a computing unit for quantum computation.

- "Qubit, An Intuition #2 - Inner Product, Outer Product, and Tensor Product" for a discussion on two-qubits operations.

For an introductory helicopter view of the overall six articles in the series, please visit this link “Embarking on a Journey to Quantum Computing — Without Physics Degree.”

In the last two articles on the “Qubit, An Intuition” series, we have discussed the basics of a quantum bit (qubit) and two-qubits operations: inner product, outer product, as well as the tensor product. In this article, we discuss quantum measurement, the observation of quantum states that change the wave function *Psi *|Ψ> from a superposition state to a pure quantum state.

# Qubit and Two-Qubits operations

A single quantum bit (qubit) is exciting with its nature of being in superposition. Hence |Ψ> = α|0> + β|1>. The quantum state *Psi* is in the linear combination of |0> and |1> with the probability of being in state |0> is |α|², and the probability of being in state|1> is |β|².

α and β are the probability amplitudes while α* and β* are the complex conjugate of α and β, respectively. It fulfills the normalization constraint, such that |α|²+|β|²=1 or α*α+β*β=1, where α, β ∈ ℂ².

An **inner product **is a product of two quantum states bra *Psi *<Ψ| and ket *Phi* |Φ>, producing a **value**. An inner product is also called an* overlap, *the overlap between quantum states.

An **outer product** is a product of two quantum states, ket *Psi*|Ψ> and bra *Phi* <Φ|, producing a **matrix**. The outer product is also called a* projection*.

A **tensor product **is a product of two quantum states, ket *Psi*|Ψ> and ket *Phi* |Φ>, producing a **column vector **with length 2ⁿ (where n is the number of qubits).

# The Quantum States & Pure States

A few rules in quantum measurements:

- First, following a measurement, a qubit (or qubits) in a
**superposition state**switches to a**pure state**. It means the qubits lost their quantumness.

A

superposition stateis a linear combination of multiple states such as |Ψ> = α|0> and β|1> for a qubit, with probabilities to be in that state are defined by its probability amplitudes α and β as |α|², β|², respectively. For two qubits, quantum state|ΨΦ> = αγ|00>, αδ|01>, βγ|10>, and βδ|11> are defined by its probability amplitudes α, β, γ, and δ as |αγ|², |αδ|², |βγ|², and |βδ|², respectively.A

pure stateis a state, in which if we re-measure, we will always get that state again at 100% probability.

- Second, the new resultant state must always preserve the normalization constraint.

# Full Qubits Quantum Measurement

Let’s take an example of a two-qubits state |Ψ>. Note that there are only three linear combinations of states |00>, |10>, and |11>.

The probabilities and resultant states to measure 00, 10, and 11 are 25%, 25%, and 50%, respectively.

Let’s take another example of a two-qubits state |Ψ> as follows:

The probabilities and resultant states to measure 00, 01, 10, and 11 are 16%, 48%, 9%, and 27%, respectively.

# Partial Qubits Quantum Measurement

Let’s use the same two-qubits state |Ψ> to illustrate partial qubits measurement.

## Example 1 — Partial measurement of 2 Qubits State

The following is the measurement of the first qubit to get the first qubit = 1. To do this, we only need to pay attention to the |10> and |11> states, which have the first qubit = 1 (highlighted below).

The probability’s rule of A union B is as follows. In this case, A intersection B is 0, so it can be omitted.

In this case, the probability of measuring |**1**0> and |**1**1> is 75%. The resultant state depends on the 2 quantum measurement rules stated above.

- After measurement, the measured qubits changed from superposition state to pure state.
- The new resultant state must always preserve the normalization constraint. Thus, it must be renormalized.

The following is the resultant state that has been normalized by dividing with its norm. The norm is the square root of probability, which is sqrt(3/4).

Then, we can verify the normalization constraint as follows:

## Example 2 —Partial measurement of 2 Qubits State

Let’s use another example for the same two-qubits state |Ψ> to illustrate a different partial qubits measurement.

The example is a measurement of the second qubit to get the second qubit = 0. To do this, we only need to pay attention to the |0**0**> and |1**0**> states, which have the second qubit = 0 (highlighted below).

In this case, the probability of measuring |00> and |10> is 50%. The following is the resultant state that has been normalized by dividing with its norm. The norm is the square root of probability, which is sqrt(1/2).

We can again verify the normalization constraint (total should be 1) as follows:

## Example 3 — Partial measurement of 4 Qubits State

Let’s use one more example for the four-qubits state |Ψ> to illustrate a different partial qubits measurement.

The example is a measurement of the first qubit and fourth qubit to get the first qubit =0 and the fourth qubit= 0. To do this, we only need to pay attention to the |**0**00**0**>, |**0**10**0**>, |**0**11**0**>, and |1111> states, which have the second qubit = 0 (highlighted below).

In this case, the probability of measuring |0000> and |0100>and |0110> is 80%. The following is the resultant state that has been normalized by dividing with its norm. The norm is the square root of probability, which is sqrt(4/5).

As usual, we verify the normalization constraint (total should be 1) as follows:

# Moving Forward

Next in the “*Qubit, An Intuition #4*” article, we will discuss unitary matrices, with examples and their implementation in IBM quantum.

# References

- Advanced Maths, 2020–2021, “
*Quantum Computing: #3 Examples of Full and Partial Measurements of Qubits.*” - Andi Sama, Cahyati S. Sangaji, 2021a, “
*Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World**.*” - Andi Sama, Cahyati S. Sangaji, 2021b, “
*Qubit, An Intuition #2 — Inner Product, Outer Product, and Tensor Product**.*” - Andi Sama, 2021, “
*My Journey to Quantum Computing**.*” - IBM, 2021a, “
*IBM Quantum on IBM Cloud**.*” - IBM, 2021b, “
*Open-Source Quantum Development*.” - IBM, 2021c, “C
*lassical and quantum operations**.*” - Yongshan Ding, 2021, “
*Lecture2b: Measurement, Composition, Entanglement | Quantum Computer Systems@UChicago*,” University of Chicago.