What's the difference between a pure and mixed quantum state?

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In quantum mechanics, a quantum state is a mathematical description that represents the physical properties of a quantum system. Quantum states can be categorized into two main types: pure states and mixed states. These states have distinct characteristics and properties.

Pure Quantum State:

A pure quantum state is a state that is completely described by a single, normalized vector in a complex Hilbert space. Mathematically, a pure state |ψ⟩ is represented by a vector that satisfies the normalization condition ⟨ψ|ψ⟩ = 1. Pure states have a well-defined quantum phase and exhibit wave-like interference patterns. They represent the simplest and most idealized form of a quantum state.
Properties of Pure States:

Pure states are deterministic and have a definite outcome when measured in a compatible basis.

Quantum operations on pure states are described by unitary transformations.

Entanglement between multiple qubits is a characteristic feature of pure states.

Pure states can be represented using the Dirac notation, such as |0⟩, |1⟩, |+⟩, or |ψ⟩.

Mixed Quantum State:

A mixed quantum state, also known as a statistical mixture or ensemble, is a state that cannot be described by a single, normalized vector. Instead, it is a probabilistic combination of multiple pure states, each with associated probabilities. A mixed state ρ is described by a density matrix that is a positive-semidefinite, Hermitian matrix with unit trace.

Properties of Mixed States:

Mixed states represent uncertainty and lack of knowledge about the exact quantum state of a system.

Mixed states can arise due to classical uncertainty, measurement outcomes, or decoherence.

Quantum operations on mixed states are described by completely positive, trace-preserving maps (CPTP maps), which can include both unitary
operations and non-unitary processes (e.g., measurement or noise).

In all, the key distinction between pure and mixed quantum states is that pure states are fully specified by a single vector and represent the most idealized form of a quantum state, while mixed states are probabilistic combinations of pure states and describe uncertainty or lack of knowledge about a quantum system. The choice between pure and mixed states depends on the context and level of information available about the quantum system under consideration.