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.

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).

Bra-ket notation for two-qubit operations: inner product, outer product, and tensor product.

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 state is 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 state is 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 |10> and |11> 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 |00> and |10> 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 |0000>, |0100>, |0110>, 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.