Using the example of a single qubit state and the observable X, describe the process of measuring the state and determining the outcome.
In the field of quantum information, the measurement of a quantum state is a fundamental process that allows us to extract information about the system under study. In this context, let us consider the example of a single qubit state and the observable X. We will describe the process of measuring the state and determining the outcome.
A qubit is the basic unit of quantum information, analogous to a classical bit. It can exist in a superposition of two orthogonal states, conventionally denoted as |0⟩ and |1⟩. These states correspond to the eigenstates of the Pauli X operator, which is often referred to as the X observable. The X operator is represented by the matrix:
X = |0⟩⟨1| + |1⟩⟨0|.
To measure the state of a qubit, we need to perform a measurement in a specific basis. In this case, we will consider the computational basis, which consists of the eigenstates of the X operator. These eigenstates are |+⟩ and |−⟩, defined as:
|+⟩ = (|0⟩ + |1⟩)/√2,
|−⟩ = (|0⟩ – |1⟩)/√2.
To carry out the measurement, we prepare an ancillary qubit in the state |+rangle, which serves as the measurement device. We then apply a controlled-X gate, also known as a CNOT gate, with the target qubit being the state we wish to measure and the control qubit being the ancillary qubit. The CNOT gate performs a conditional operation on the target qubit based on the state of the control qubit. In this case, if the control qubit is in the state |+rangle, the CNOT gate leaves the target qubit unchanged. However, if the control qubit is in the state |-rangle, the CNOT gate applies the X operator to the target qubit.
After applying the CNOT gate, we measure the ancillary qubit in the computational basis. This measurement collapses the combined state of the target and ancillary qubits into one of the four possible outcomes: |0+rangle, |1+rangle, |0-rangle, or |1-rangle. The probability of obtaining each outcome depends on the initial state of the target qubit.
For example, if the initial state of the target qubit is |0⟩, the combined state before measurement is |0+rangle. In this case, the CNOT gate leaves the target qubit unchanged, and the measurement of the ancillary qubit will always yield the outcome |+rangle. Similarly, if the initial state of the target qubit is |1⟩, the combined state before measurement is |1-rangle. In this case, the CNOT gate applies the X operator to the target qubit, and the measurement of the ancillary qubit will always yield the outcome |-rangle.
In general, the measurement outcome provides information about the state of the target qubit. If the measurement outcome is |+rangle, we can conclude that the target qubit was in the state |0⟩. If the measurement outcome is |-rangle, we can conclude that the target qubit was in the state |1⟩.
To summarize, the process of measuring the state of a qubit involves preparing an ancillary qubit in a specific state, applying a controlled-X gate to the target and ancillary qubits, and measuring the ancillary qubit in the computational basis. The measurement outcome corresponds to the state of the target qubit, providing information about its initial state.
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