So far we have seen quantum circuits with qubits, quantum wires and quantum gates. But we have left out an essential part of the quantum circuit - the measurement instrument.
What are we measuring again?
Well of course, the quantum state of the qubit, but, in reality, we are measuring the property that exhibits quantum behavior and that we have chosen to represent the quantum state of our qubit. So if our qubit is an electron, we might be measuring its spin; or if our qubit is a photon, we might be measuring its polarization.
So depending on the type of quantum mechanical system (electron or photon etc.), our measurement instrument may change. But the general representation of a measurement instrument (and naturally the act of measurement), is given below:
Measurement Device - Quantum Circuit Symbol
To the left is an incoming quantum-wire, drawn as a single line that is capable of transmitting qubits, whereas to the right is an outgoing normal wire (electrical, optical, etc.,) drawn as a double line, that is capable of transmitting classical bits.
Case 1 : Measurement of a Single Qubit
We have already seen that when a qubit's quantum state is measured, the superposition collapses to one of the basis states as per the respective probabilities.
The observed result of measurement on a single qubit is thus as given below:
Single Qubit Measurement Results
Case 2 : Measurement of a Multi Qubit System
Making measurements on a multi-qubit system is quite similar to measurement of single qubits. The result is based on the probabilities associated with the basis states in the joint state of the system.
Consider below two qubits coming out of a quantum circuit, and each is measured simultaneously using measuring devices.
The observed result of measurement is thus as given below:
Measurement of a Multi Qubit System
Case 3 : Partial Measurement of a Multi Qubit System
Let's look at a very interesting case.
What happens if we measure just one qubit in a two-qubit system? Does it have any effect on the measurement of the second qubit later on?
Well first of all, measuring the first qubit, does have an effect on the measurement of the second qubit later on. It basically changes the probabilities of observing a 0 or 1 for the second qubit.
Shown below is the setup for this arrangement.
Partial Measurement of a 2-Qubit System
Working out how the probabilities play out is just some basic conditional probability mathematics. Let's name the event of measuring the first qubit - event 'E', and the event of measuring the second qubit - event 'F'. On the same lines, let's use the following notations for the observed measurements:
Now let's make a small probability chart for different outcomes:
Probability Chart of different outcomes
Conditional probability has a simple formula for "Probability of A given B(already occurred)":
Once the measurement of the first qubit has been made, its result (0 or 1) becomes a given for the measurement of the second qubit. Therefore, the probability of the second qubit turning out to be 0 or 1, is given as below :