Measuring Multi-Qubit Circuits

Multi-Qubit circuits are great, however, until they are measured, we will never know their true outcome. Looking back at the ket of a multi-qubit state, it’s easy to see the possible measurements.

In this example, the two possible outcomes from measurement would be the 00 state and the 01 state. This is because the wave function has to collapse on only one of these two states.

Through Born’s rule, it’s also possible to see the probability of each of these states happening. Firstly, the ket needs to be converted into vector notation.

In the vector notation of the state, we see the contribution to each state, starting from 00 at the top to 11 at the bottom. Born’s rule shows us that to find the probability of each state happening, the contribution needs to be squared.

After squaring the contributions, the resulting probabilities are 50% for the outcome of the circuit to be in the 00 state and 50% for the 01 state.

Entanglement

Entanglement is where one object’s state depends on another object’s state. For qubits, this means that when two qubits are entangled and one is measured, information about the other is immediately known. However, when they aren’t entangled, the measurements must be taken independently.

To know if a quantum state is entangled or not, you can try to separate the qubits to find their individual states. If this is possible, then the qubits are not entangled. However, if it isn’t then the qubits are entangled. We will use the below ket as the first example to demonstrate this behavior. Take a moment to find the separate qubit states of this ket if possible, then read on.

After looking at this two-qubit state, we see that qubit 0 is in the 0 state in both of the two-qubit states. From this, we know that qubit 0 actually is in the 0 state. This leaves qubit 1 in the superposition plus state. In this example, we were able to separate the states of the qubits. This means that they are not entangled. Try this next example.

When we look at this two-qubit state, we notice something interesting — whenever qubit 0 is in the 0 state, qubit 1 is also in the 0 state. This goes the same way for when qubit 0 is in the 1 state because, coincidentally, qubit 1 is also in the 1 state. If we try to separate these qubits into the plus states, we wouldn’t get the same ket. Because we can’t find the separate qubit states, we know these two qubits are entangled. This ket state is also known as one of the four Bell states. The three other Bell states are as follows: