How is the state of a qubit represented using the Bloch sphere representation?
The Bloch sphere representation is a powerful tool in the field of quantum information for visualizing and understanding the state of a qubit. In this representation, the state of a qubit is represented as a point on the surface of a unit sphere known as the Bloch sphere. The Bloch sphere provides a geometric interpretation of the state of a qubit, allowing us to easily visualize and analyze its properties.
To understand how the state of a qubit is represented using the Bloch sphere, let's first consider the general state of a qubit. A qubit is a two-level quantum system, and its state can be described by a superposition of two basis states, conventionally denoted as |0⟩ and |1⟩. These basis states correspond to the two orthogonal states of the qubit, often referred to as the computational basis.
The Bloch sphere representation allows us to express any state of a qubit as a linear combination of the basis states |0⟩ and |1⟩. Mathematically, any qubit state can be written as:
|ψ⟩ = α|0⟩ + β|1⟩
where α and β are complex numbers that satisfy the normalization condition |α|^2 + |β|^2 = 1. The coefficients α and β are often referred to as probability amplitudes, and they determine the probabilities of measuring the qubit in the basis states |0⟩ and |1⟩, respectively.
Now, let's see how this state |ψ⟩ is represented on the Bloch sphere. The Bloch sphere is a unit sphere where the north and south poles represent the basis states |0⟩ and |1⟩, respectively. The equator of the Bloch sphere represents a mixture of the two basis states, with the north and south poles representing pure states and points on the equator representing superposition states.
To find the point on the Bloch sphere that represents the state |ψ⟩, we can use the following formula:
x = 2Re(αβ^*)
y = 2Im(αβ^*)
z = |α|^2 – |β|^2
where Re(αβ^*) and Im(αβ^*) denote the real and imaginary parts of αβ^*, respectively.
Once we have the values of x, y, and z, we can locate the point (x, y, z) on the Bloch sphere. This point represents the state |ψ⟩ in the Bloch sphere representation.
For example, let's consider a qubit in the state |ψ⟩ = (1/√2)|0⟩ + (1/√2)|1⟩. Using the formula above, we can calculate the values of x, y, and z:
x = 2Re((1/√2)(1/√2)^*) = 0
y = 2Im((1/√2)(1/√2)^*) = 0
z = |1/√2|^2 – |1/√2|^2 = 0
Therefore, the state |ψ⟩ = (1/√2)|0⟩ + (1/√2)|1⟩ is represented by the point (0, 0, 0) on the Bloch sphere. This point is located at the center of the Bloch sphere, indicating that the state is a pure state.
The Bloch sphere representation provides a geometric visualization of the state of a qubit. It allows us to represent any qubit state as a point on the surface of a unit sphere, with the north and south poles representing the basis states |0⟩ and |1⟩, respectively. The Bloch sphere representation is a valuable tool for understanding and analyzing the properties of qubits in quantum information.
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