Can a qubit be in the inside of the Bloch Sphere?

Question

Can qubit be inside Bloch Sphere, i.e. its length is less than 1?
If yes, how we represent that state since we have only parameter for angles and not the length (norm) of the vector?

Answer 1

Mixed states are represented by vectors inside Bloch sphere.

Suppose you have mixed state with density matrix

$$\rho=q|\psi\rangle\langle\psi| + (1-q)|\phi\rangle\langle\phi|$$ where $|\psi\rangle$ and $|\phi\rangle$ are pure states with vectors $\overset{\rightarrow}{r_{\psi}}$ and $\overset{\rightarrow}{r_{\phi}}$ on Bloch sphere, and $0<q<1$; then the vector $$\overset{\rightarrow}{r_{\rho}}=q\overset{\rightarrow}{r_{\psi}}+(1-q)\overset{\rightarrow}{r_{\phi}}$$ inside Bloch sphere represents mixed state with density matrix $\rho$. Geometrically, $\overset{\rightarrow}{r_{\rho}}$ is a point on the line connecting points $\overset{\rightarrow}{r_{\psi}}$ and $\overset{\rightarrow}{r_{\phi}}$

Answer 2

Yes, the state of a qubit can be described by a point inside the Bloch sphere. However, you cannot use the state vector formalism to describe it, you have to generalise to the concept of mixed states. For a qubit, these have three parameters: the two angles and the length of the vector.