Pure state

Definition

A pure state in quantum mechanics refers to a quantum system that is fully described by a single wave function or a single point on the Bloch sphere. This means it has a precise set of properties, such as specific values for observables like position and momentum. Pure states contrast with mixed states, which represent statistical mixtures of different states and lack complete information about the system's properties.

5 Must Know Facts For Your Next Test

1. Pure states can be represented mathematically by vectors in Hilbert space or as points on the Bloch sphere for two-level systems.
2. Measurements performed on a pure state yield definite outcomes based on the probabilities determined by the wave function.
3. The evolution of pure states follows deterministic laws governed by the Schrödinger equation, allowing for precise predictions of future states.
4. Pure states have maximum knowledge about the system, meaning they represent situations where all relevant information about the quantum system is known.
5. Entangled pure states can exist in multi-partite systems, where individual subsystems cannot be described independently but are instead linked in a way that affects measurements.

Review Questions

• How does the representation of a pure state differ from that of a mixed state, particularly in terms of observable properties?
  • A pure state is represented by a single wave function or point on the Bloch sphere and reflects maximum knowledge about the system, meaning it has specific values for its observable properties. In contrast, a mixed state represents a statistical mixture of different pure states and reflects uncertainty in the measurement outcomes. The key difference lies in the completeness of information: pure states provide exact details about a quantum system's characteristics, while mixed states indicate some degree of lack of knowledge.
• Discuss how pure states can be represented on the Bloch sphere and what this representation signifies for two-level quantum systems.
  • On the Bloch sphere, pure states are represented as points on its surface. Each point corresponds to a specific quantum state of a two-level system (like a qubit), where the poles typically represent the computational basis states (|0⟩ and |1⟩). The angles on the sphere determine the relative phase and probability amplitudes between these basis states. This geometric representation allows for visualizing operations like rotations, which correspond to quantum gates acting on pure states in quantum computing.
• Evaluate the implications of pure states on measurement outcomes in quantum mechanics and how they influence our understanding of quantum information.
  • Pure states have significant implications for measurement outcomes because they lead to definitive results based on their wave functions, allowing us to predict probabilities of various observables accurately. When a measurement is performed, it collapses the pure state into one of its eigenstates with corresponding probabilities dictated by the wave function. This deterministic nature is essential for understanding quantum information since algorithms rely on manipulating pure states to achieve reliable computation. Furthermore, entangled pure states challenge our classical intuitions about locality and reality, fundamentally altering our grasp of information transfer and correlations between distant particles.

Related terms

Wave Function:

A mathematical function that describes the quantum state of a particle or system, containing all the information about its position, momentum, and other observable properties.

Mixed State: A statistical ensemble of different quantum states that reflects uncertainty or lack of complete knowledge about a system's properties.

Density Matrix:

A mathematical representation used to describe the statistical state of a quantum system, particularly useful for mixed states, summarizing probabilities of various pure states.