What is the difference between superpositions and mixed states?

[quantumadmin]

Superpositions and mixed states are both concepts in quantum mechanics that describe different aspects of the state of a quantum system. They represent different types of quantum uncertainty and information.

Superpositions:

A superposition is a property of a quantum state where the state can exist in a linear combination of two or more distinct states. Mathematically, if |ψ⟩ and |φ⟩ are two possible states of a quantum system, then a superposition of these states is represented as α|ψ⟩ + β|φ⟩, where α and β are complex probability amplitudes that satisfy the normalization condition |α|^2 + |β|^2 = 1. Superpositions represent a form of quantum coherence, where the quantum system simultaneously exhibits characteristics of both states.

Key points about superpositions:

Superpositions are a fundamental feature of quantum mechanics.

The concept of superpositions is closely related to the wave-like nature of quantum particles.

Superpositions can be used to encode and process information in quantum computation, allowing for parallel processing of different possibilities.

Mixed States:

A mixed state, also known as a statistical mixture, is a probabilistic combination of multiple pure states. It represents a scenario where the observer lacks knowledge about the exact quantum state of the system. Mathematically, a mixed state is described by a density matrix ρ that is a Hermitian, positive-semidefinite matrix with a trace equal to 1. A mixed state ρ can be expressed as a convex combination of pure states: ρ = ∑ pi |ψi⟩⟨ψi|, where pi are probabilities and |ψi⟩ are the associated pure states.

Key points about mixed states:

Mixed states describe the statistical distribution of possible pure states in a quantum ensemble.

Mixed states arise when there is classical uncertainty or decoherence affecting a quantum system.

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

Superpositions describe the coexistence of multiple states with specific probability amplitudes, allowing for the quantum interference of possibilities. Mixed states, on the other hand, describe the probabilistic distribution of possible states in situations of quantum uncertainty or decoherence. While superpositions represent a fundamental feature of quantum mechanics, mixed states account for our lack of complete knowledge about a quantum system's state.