What it means for mixed state qubits going below the Bloch sphere surface?

by EITCA Academy / Saturday, 25 October 2025 / Published in Quantum Information, EITC/QI/QIF Quantum Information Fundamentals, Introduction to Quantum Information, Geometric representation

The geometric representation of qubits via the Bloch sphere constitutes a powerful intuitive aid in quantum information science. The Bloch sphere provides a visualization framework for understanding both pure and mixed quantum states of a two-level system (qubit). Analyzing what occurs when mixed state qubits are represented by points inside, as opposed to on, the surface of the Bloch sphere, requires a detailed understanding of density operators, their relation to physical states, and the geometry of state space.

1. The Bloch Sphere and Qubit States

The state space of a single qubit is described by 2-dimensional complex Hilbert space. Any state of a qubit can be characterized by a density matrix $\rho$ , which is a 2×2 Hermitian, positive semi-definite matrix with unit trace. There are two categories of qubit states:

– Pure states, described by a state vector $|\psi\rangle$ , with $\rho = |\psi\rangle\langle\psi|$ .
– Mixed states, which are probabilistic mixtures of pure states and cannot be described by a single state vector.

The general form of a qubit density matrix is:

$\rho = \frac{1}{2}(I + \vec{r} \cdot \vec{\sigma})$

where:
– $I$ is the 2×2 identity matrix,
– $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ are the Pauli matrices,
– $\vec{r} = (r_x, r_y, r_z)$ is the Bloch vector, a real three-dimensional vector.

The Bloch vector provides a geometric representation of the state: for any physical state, $|\vec{r}| \leq 1$ .

– Pure states are those where $|\vec{r}| = 1$ ; these lie on the surface of the Bloch sphere.
– Mixed states have $|\vec{r}| < 1$ ; these are found inside the Bloch sphere.

2. Physical Meaning of Points Inside the Bloch Sphere

When the Bloch vector has norm strictly less than one ( $|\vec{r}| < 1$ ), the corresponding density matrix does not represent a pure quantum state but rather a mixed state. This mixedness can be interpreted as the state representing a statistical ensemble of pure states, rather than a completely definite quantum state.

Mathematically, the purity of a quantum state is quantified as:

$\text{Purity} = \mathrm{Tr}[\rho^2]$

For qubits, this can be related to the Bloch vector as:

$\mathrm{Tr}[\rho^2] = \frac{1}{2}(1 + |\vec{r}|^2)$

– For $|\vec{r}| = 1$ , $\mathrm{Tr}[\rho^2] = 1$ : the state is pure.
– For $|\vec{r}| < 1$ , $\mathrm{Tr}[\rho^2] < 1$ : the state is mixed.
– The maximally mixed state has $|\vec{r}| = 0$ , corresponding to the center of the Bloch sphere, with density matrix $\rho = \frac{I}{2}$ .

3. Significance of Mixed States

Mixed states arise in practice when a system is not completely isolated or suffers from decoherence. In physical terms, a mixed state reflects partial knowledge about the system: it could, for example, result from tracing out the environment from a larger entangled state, or from classical statistical uncertainty about the preparation of the state.

The movement from the surface toward the center of the Bloch sphere corresponds to increasing mixedness (or decreasing purity). The radius of the Bloch vector can thus be viewed as a measure of how "quantum" or "definite" a state is. Points nearer the center represent states that are more uncertain or more disordered. The extreme case at the center ( $\vec{r}=0$ ) represents the maximally mixed state, which is completely random and carries no information about any preferred basis.

4. Examples of Mixed States and Their Representation

– Pure State Example: $|0\rangle$ corresponds to the Bloch vector $(0, 0, 1)$ : on the north pole of the sphere.
– Maximally Mixed State: The density matrix is $\frac{1}{2}I$ , with $\vec{r} = (0,0,0)$ : at the center.
– Partially Mixed State: Consider a state that is a classical mixture of $|0\rangle$ with probability $p$ and $|1\rangle$ with probability $1-p$ :

$\rho = p|0\rangle\langle0| + (1-p)|1\rangle\langle1| = \frac{1}{2}\left(I + (2p-1)\sigma_z\right)$

The Bloch vector is $(0, 0, 2p-1)$ , which lies along the $z$ -axis but at a distance $|2p-1|$ from the origin. For $p=0.7$ , the Bloch vector is at $z=0.4$ , inside the sphere.

– Depolarized Pure State: If a pure state is sent through a depolarizing channel that leaves it unchanged with probability $1-p$ and replaces it by the maximally mixed state with probability $p$ , the resulting state is closer to the center. This is a typical example of physical noise leading to a reduced Bloch vector length.

5. Physical Interpretation and Relevance

The length of the Bloch vector not only describes the purity of the state but also affects observable quantities. For a given observable, the expectation value is:

$\langle O \rangle = \mathrm{Tr}[\rho O]$

For a spin- $\frac{1}{2}$ particle, the expectation value of spin along an axis $\vec{n}$ is $\vec{r} \cdot \vec{n}$ . For mixed states ( $|\vec{r}| < 1$ ), the maximum possible expectation value is reduced. This reflects the inherent uncertainty or entropy in the system, caused by noise, loss of coherence, or incomplete knowledge.

6. Entropy and Mixedness

The degree of mixedness can be quantified using the von Neumann entropy:

$S(\rho) = -\mathrm{Tr}[\rho \log_2 \rho]$

– Pure states ( $|\vec{r}|=1$ ) have zero entropy.
– The maximally mixed state ( $\vec{r}=0$ ) achieves the maximum entropy ( $S(\rho)=1$ ) for a qubit.

This entropy is directly connected to the Bloch vector length: as the state moves inside the sphere, entropy increases.

7. Mathematical Constraints on the Bloch Vector

Not every vector inside the sphere corresponds to a physical state: the constraints are
– The density matrix must be positive semi-definite.
– This requirement is equivalent to $|\vec{r}| \leq 1$ .

Any point inside the sphere (including on the surface) corresponds to a valid qubit state, with the surface corresponding to pure states and the interior to mixed states.

8. Non-uniqueness of Mixed State Decomposition

A key feature of mixed states is that they can be decomposed into ensembles of pure states in infinitely many ways. For example, the maximally mixed state can be written as an equal mixture of any pair of orthogonal states, or as a uniform mixture over all pure states. Geometrically, this is reflected in the fact that all points on the surface are equally "mixed" into the center.

9. Experimental Manifestations

In laboratory practice, pure states are difficult to maintain due to coupling with the environment (decoherence). As a result, real qubits frequently correspond to points inside the Bloch sphere. Quantum error correction and decoherence suppression techniques aim to keep qubit states as close as possible to the surface.

10. Extension: General Quantum Systems

The concept generalizes beyond qubits. For higher-dimensional quantum systems (qudits), the analogy of the Bloch sphere becomes more complicated, but the core idea persists: pure states form the "surface" of the allowed state space, and mixed states fill the interior.

Summary Paragraph

States represented by points inside the Bloch sphere correspond to mixed quantum states, characterized by a loss of purity and an increase in entropy. This geometric representation provides both a visual and quantitative understanding of state purity, the impact of decoherence, and the statistical nature of quantum mechanics. The movement from the surface towards the center of the Bloch sphere directly visualizes the process of quantum state mixing, with the center representing maximal mixing and total lack of information about the system's state. This interpretation is central to quantum information theory, providing the basis for understanding quantum noise, decoherence, and the challenges in maintaining quantum coherence in practical systems.

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