How are the zero and one states represented on the Bloch sphere and why do they become antipodal states?
The Bloch sphere is a geometric representation of the quantum state of a two-level quantum system, such as a qubit. It provides a clear visualization of the quantum states and their properties. In the context of the Bloch sphere, the zero and one states are represented by specific points on the sphere's surface. These points are known as antipodal states due to their positioning on opposite sides of the sphere.
To understand why the zero and one states become antipodal states on the Bloch sphere, we need to consider the mathematical foundations of quantum mechanics. In quantum mechanics, a qubit can be described by a superposition of basis states. The basis states for a qubit are typically denoted as |0⟩ and |1⟩.
The Bloch sphere provides a convenient way to visualize the state of a qubit. The north pole of the sphere represents the state |0⟩, while the south pole represents the state |1⟩. The equator of the sphere represents a superposition of the two states, where the relative position along the equator corresponds to the relative weights of the |0⟩ and |1⟩ states in the superposition.
The reason why the zero and one states become antipodal states on the Bloch sphere can be understood by considering the concept of orthogonal states. In quantum mechanics, two states are said to be orthogonal if their inner product is zero. In the case of a qubit, the states |0⟩ and |1⟩ are orthogonal to each other.
On the Bloch sphere, the antipodal points at the north and south poles represent the orthogonal states |0⟩ and |1⟩, respectively. This means that the inner product between these two states is zero. The antipodal nature of these states is a consequence of their orthogonality.
To illustrate this concept, let's consider an example. Suppose we have a qubit in the state (|0⟩ + |1⟩)/√2, which represents an equal superposition of the zero and one states. On the Bloch sphere, this state is represented by a point on the equator. If we measure the qubit, we will obtain either the |0⟩ state or the |1⟩ state with equal probability.
Now, let's consider the measurement outcome where we obtain the |0⟩ state. On the Bloch sphere, this corresponds to a measurement result along the north pole. If we perform the same measurement on a qubit in the state (|0⟩ – |1⟩)/√2, which represents another equal superposition of the zero and one states, we will obtain the |0⟩ state again. However, on the Bloch sphere, this measurement result corresponds to a measurement along the south pole.
This example demonstrates that the zero and one states are antipodal on the Bloch sphere. When a qubit is in the zero state, it is represented by a point on the north pole, and when it is in the one state, it is represented by a point on the south pole. This antipodal relationship arises due to the orthogonality of the zero and one states.
The zero and one states are represented as antipodal points on the Bloch sphere because they are orthogonal to each other. The north and south poles of the sphere correspond to the zero and one states, respectively, while the equator represents superpositions of these states.
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