Table of Contents

1. Introduction
2. Conceptual Overview
3. The Wavefunction and Pure States
4. Statistical Mixtures and Mixed States
5. Density Matrix Representation
6. Mathematical Criteria
7. Visualization: Bloch Sphere
8. Physical Interpretation
9. Examples of Pure States
10. Examples of Mixed States
11. Quantum Ensembles
12. Decoherence and the Emergence of Mixed States
13. Experimental Distinction
14. Entropy and Purity Measures
15. Role in Quantum Information Theory
16. Quantum State Tomography
17. Conclusion

1. Introduction

In quantum mechanics, understanding the distinction between pure and mixed states is fundamental to interpreting the behavior of quantum systems. While pure states represent maximal information about a system, mixed states describe statistical uncertainty or entanglement-induced ignorance.

2. Conceptual Overview

• Pure states are coherent and deterministic descriptions of quantum systems.
• Mixed states represent an ensemble of pure states or the result of ignoring part of a larger entangled system.

3. The Wavefunction and Pure States

A pure quantum state is described by a single state vector \( |\psi\rangle \) in a Hilbert space. All measurable properties can be calculated directly from \( |\psi\rangle \). The density matrix is:

\[
\rho = |\psi\rangle \langle \psi|
\]

This state is said to be coherent and retains phase information.

4. Statistical Mixtures and Mixed States

A mixed state represents a statistical ensemble of pure states \( \{(p_i, |\psi_i\rangle)\} \). Its density matrix is:

\[
\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|
\]

where \( p_i \) are classical probabilities. The system is not in any one of the states \( |\psi_i\rangle \), but rather a probabilistic mixture.

5. Density Matrix Representation

Both pure and mixed states are represented using density matrices:

• Pure: \( \rho^2 = \rho \), \( \text{Tr}(\rho^2) = 1 \)
• Mixed: \( \rho^2 \ne \rho \), \( \text{Tr}(\rho^2) < 1 \)

This provides a unified framework for all quantum states.

6. Mathematical Criteria

Let \( \rho \) be a density operator:

• If \( \text{Tr}(\rho^2) = 1 \), the state is pure.
• If \( \text{Tr}(\rho^2) < 1 \), the state is mixed.

The von Neumann entropy,

\[
S(\rho) = -\text{Tr}(\rho \log \rho)
\]

is zero for pure states and positive for mixed ones.

7. Visualization: Bloch Sphere

For qubits, the state space is visualized as a Bloch sphere:

• Pure states: lie on the surface (\( |\vec{r}| = 1 \))
• Mixed states: lie inside the sphere (\( |\vec{r}| < 1 \))

This provides a geometric interpretation of coherence and statistical mixing.

8. Physical Interpretation

• A pure state reflects complete knowledge (e.g., prepared by a well-controlled experiment).
• A mixed state can arise from:
• Uncertainty in preparation.
• Tracing out part of an entangled system.
• Decoherence due to environmental interaction.

9. Examples of Pure States

• Electron in a definite spin state: \( |\psi\rangle = |\uparrow_z\rangle \)
• Photon in a definite polarization state: \( |\psi\rangle = \frac{1}{\sqrt{2}}(|H\rangle + |V\rangle) \)

Each of these has maximal coherence and minimal entropy.

10. Examples of Mixed States

• Unpolarized photon: 50% horizontal, 50% vertical.
• Thermal states at non-zero temperature.
• Qubits entangled with another system and then traced out.

11. Quantum Ensembles

A quantum ensemble is a collection of systems, each in a pure state \( |\psi_i\rangle \) with probability \( p_i \). The density matrix describes the average behavior of the ensemble, not the state of any individual system.

12. Decoherence and the Emergence of Mixed States

Decoherence causes pure states to evolve into mixed states due to entanglement with the environment. This results in suppression of interference and a transition to classical-like behavior.

13. Experimental Distinction

Although different mixtures can produce the same density matrix (ensemble equivalence), interference experiments or quantum tomography can reveal coherence and determine whether a state is pure.

14. Entropy and Purity Measures

• Purity: \( \gamma = \text{Tr}(\rho^2) \)
• Entropy: \( S(\rho) \)

These are essential for quantifying quantum information and decoherence. Pure states have maximal purity and zero entropy.

15. Role in Quantum Information Theory

• Pure states are the backbone of quantum algorithms.
• Mixed states model real-world noise and imperfections.
• Quantum channels, teleportation, and cryptography require careful control of both.

16. Quantum State Tomography

Tomography allows reconstruction of the density matrix from measurements. This reveals whether a state is pure or mixed and is a crucial tool in quantum computing and quantum optics.

17. Conclusion

The distinction between pure and mixed states is central to quantum theory. While pure states capture idealized behavior, mixed states reflect the statistical nature of quantum systems in practice. The density matrix formalism allows both to be treated on equal footing, enabling precise analysis of coherence, entanglement, and environmental effects.

Table of Contents

1. Introduction
2. The Measurement Problem and Superpositions
3. What Is Decoherence?
4. The Environment and Open Quantum Systems
5. Formal Definition and Mathematical Framework
6. Reduced Density Matrix and Tracing Out the Environment
7. Decoherence in the Position Basis
8. Pointer States and Einselection
9. Decoherence Time Scale
10. Examples: Schrödinger’s Cat and Interference Loss
11. Decoherence vs Wavefunction Collapse
12. Role of Entanglement in Decoherence
13. Experimental Evidence for Decoherence
14. Decoherence in Quantum Computing
15. Philosophical Implications and Interpretations
16. Limitations of the Decoherence Program
17. Conclusion

1. Introduction

Quantum decoherence is the process by which a quantum system loses its ability to exhibit coherent superposition due to interactions with its environment. It explains why quantum systems appear classical under everyday conditions, resolving part of the measurement problem without invoking collapse.

2. The Measurement Problem and Superpositions

In quantum mechanics, particles can exist in superpositions of states, such as:

\[
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle
\]

Upon measurement, we observe only one outcome. The mystery is: why do we never observe macroscopic superpositions like Schrödinger’s cat being alive and dead simultaneously?

3. What Is Decoherence?

Decoherence is the disappearance of quantum coherence in a system due to entanglement with the environment. It causes interference terms (off-diagonal elements of the density matrix) to vanish, leading to classical probabilistic behavior.

4. The Environment and Open Quantum Systems

No system is perfectly isolated. Every quantum system interacts with its environment (e.g., air, photons, thermal fluctuations), making it an open quantum system. These interactions induce entanglement and lead to decoherence.

5. Formal Definition and Mathematical Framework

Given a system \( S \) and an environment \( E \), the total state evolves unitarily:

\[
|\Psi_{SE}\rangle = \sum_i c_i |s_i\rangle \otimes |e_i\rangle
\]

The reduced density matrix for the system is obtained by tracing out the environment:

\[
\rho_S = \text{Tr}E (|\Psi{SE}\rangle \langle \Psi_{SE}|)
\]

6. Reduced Density Matrix and Tracing Out the Environment

For a pure entangled state, tracing out the environment yields a mixed state:

\[
\rho_S = \sum_{i,j} c_i c_j^* \langle e_j | e_i \rangle |s_i\rangle \langle s_j|
\]

If \( \langle e_j | e_i \rangle \rightarrow \delta_{ij} \), then:

\[
\rho_S \rightarrow \sum_i |c_i|^2 |s_i\rangle \langle s_i|
\]

This resembles a classical probability distribution over outcomes.

7. Decoherence in the Position Basis

In many physical cases, decoherence is strongest in the position basis due to spatially localized environmental interactions. The interference between spatial wave packets vanishes, giving rise to classical trajectories.

8. Pointer States and Einselection

Certain states remain stable under environmental interaction — these are pointer states. The environment “selects” these states as classical-like, a process known as environment-induced superselection or einselection.

9. Decoherence Time Scale

Decoherence is extremely fast for macroscopic systems. For example:

• A dust particle in air decoheres in \( \sim 10^{-31} \, \text{seconds} \)
• The timescale depends on system-environment coupling, temperature, and spatial resolution.

10. Examples: Schrödinger’s Cat and Interference Loss

The Schrödinger’s cat paradox illustrates decoherence. The cat becomes entangled with a quantum state (e.g., a radioactive atom). Decoherence rapidly transforms the state into an apparent classical mixture:

\[
\rho_{\text{cat}} = |\alpha|^2 | \text{alive} \rangle \langle \text{alive} | + |\beta|^2 | \text{dead} \rangle \langle \text{dead} |
\]

This suppresses quantum interference.

11. Decoherence vs Wavefunction Collapse

• Decoherence explains why we don’t observe interference but does not specify why one outcome is realized.
• Collapse (as in Copenhagen) assumes one outcome is randomly chosen.
• Decoherence turns a pure superposition into a mixed state, but the observer’s knowledge is not updated.

12. Role of Entanglement in Decoherence

Entanglement with the environment is essential. It’s not the disturbance of the system that causes decoherence, but the information leakage into the environment, which becomes correlated with the system.

13. Experimental Evidence for Decoherence

• Loss of interference in double-slit experiments with massive molecules.
• Superconducting qubits and decoherence times in quantum computers.
• Interference suppression in photon and atom interferometry.

These experiments match theoretical predictions of decoherence.

14. Decoherence in Quantum Computing

Decoherence is a major challenge:

• It leads to loss of quantum information.
• Requires quantum error correction and decoherence-free subspaces.
• Dictates qubit coherence times and operational limits.

Understanding and mitigating decoherence is key to building stable quantum devices.

15. Philosophical Implications and Interpretations

Decoherence supports interpretations like:

• Many-worlds, where all branches persist without collapse.
• Relational quantum mechanics, where the observer-environment relation determines outcomes.

However, decoherence alone doesn’t explain why we observe definite results.

16. Limitations of the Decoherence Program

• Does not solve the measurement problem completely.
• Does not choose a single outcome.
• Only explains emergence of classicality, not the subjective experience of an observer.

17. Conclusion

Quantum decoherence provides a powerful and natural explanation for the apparent transition from quantum to classical worlds. By accounting for entanglement with the environment, it explains the loss of interference and stability of classical states. Though not a full resolution of the measurement problem, decoherence is indispensable for understanding open quantum systems and for advancing quantum technology.