Why is a qubits rotation and position represented in only three-dimensional space? At minimum, should it not be represented in space time? Something seems odd in looking at this as a sphere. > [!ai]+ AI > > You’re right to question the seemingly simplistic 3D representation of a qubit’s state. While the Bloch sphere visualization is incredibly useful, it’s a simplification that hides some of the underlying complexity. Here’s a breakdown of why it works and its limitations: > **Why the Bloch Sphere Works (and what it represents):** > - **Qubit State:** A qubit’s state is a superposition of two basis states, often labeled |0⟩ and |1⟩. Mathematically, this is represented as: |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers. These complex amplitudes encode both the probability of measuring the qubit in each basis state (|α|² for |0⟩ and |β|² for |1⟩) and the *relative phase* between the two states. > - **Normalization and Global Phase:** Because the probabilities must sum to 1 (|α|² + |β|² = 1), we have a constraint. Furthermore, a global phase factor (a common complex multiplier to both α and β) doesn’t affect measurable probabilities. These two constraints allow us to reduce the four degrees of freedom (two complex numbers) to two real numbers. > - **Spherical Coordinates:** These two real numbers can be conveniently represented as angles on a sphere: > - **θ (Theta):** Relates to the probabilities of |0⟩ and |1⟩. At the north pole (θ=0), the qubit is in state |0⟩. At the south pole (θ=π), it’s in state |1⟩. > - **φ (Phi):** Represents the relative phase between |0⟩ and |1⟩. > - **Bloch Vector:** A vector pointing from the origin to a point on the Bloch sphere then represents the qubit’s state. The vector’s length is always 1 due to normalization. > **Why the Bloch Sphere *Doesn’t* Include Time (directly):** > - **Static Representation:** The Bloch sphere represents the state of a qubit *at a particular instant in time*. It doesn’t show how the state evolves over time. > - **Unitary Evolution:** The evolution of a qubit’s state is governed by unitary transformations. These transformations can be visualized as rotations of the Bloch vector. So, while time isn’t explicitly shown on the Bloch sphere, the *effect* of time (through unitary evolution) is represented by changes in the vector’s direction. > - **Separate Time Parameter:** If you want to represent the time evolution, you’d need to add time as a separate parameter. You could imagine a series of Bloch spheres, each representing the qubit’s state at a different time step, or a path traced by the Bloch vector through the sphere over time. > **Why Spacetime Isn’t Necessary (for single qubits):** > - **Internal Degree of Freedom:** The qubit’s state is an *internal* degree of freedom, analogous to the spin of an electron. It doesn’t directly relate to the qubit’s position in physical space. While the physical implementation of a qubit might occupy space, the Bloch sphere representation deals with the abstract quantum state, not its physical location. > - **Multi-Qubit Systems and Entanglement:** When dealing with multiple qubits, especially entangled ones, the Bloch sphere representation becomes inadequate. You can’t represent the joint state of entangled qubits on individual Bloch spheres. More complex mathematical tools, like density matrices and higher-dimensional Hilbert spaces, are needed. However, even then, spacetime isn’t directly involved in describing the entanglement itself. > In summary, the Bloch sphere is a powerful visualization tool for single qubit states, but it’s a simplification. It represents the state at a single point in time and doesn’t directly incorporate spatial information. Time evolution is represented by rotations of the Bloch vector, and more complex mathematical tools are needed for multi-qubit systems.