Understanding Quantum Information and Computation - Lesson 2 Notes

Introduction

• John Watrous, Technical Director of Education at IBM Quantum.
• Lesson 2 of a series on Quantum Information and Computation.
• Review of Lesson 1 is encouraged as it lays the foundation for this lesson.
• The series involves textbook content along with videos.

Unit 1 Overview

• Unit 1 covers the basics of quantum information.
• Previous lesson focused on single systems, describing their operation in isolation.
• Key topics from Lesson 1:
  • Comparison between classical and quantum information.
  • Representation of quantum states as vectors with complex numbers.
  • Standard basis measurements and unitary operations.

Focus of Lesson 2

• Expanding to multiple systems capable of storing quantum information.
• Concept that the whole can be greater than the sum of its parts.
• Introduction of the concept of entanglement, a key aspect of quantum information.

Classical Information Review

• Similarities between quantum and classical information are explored.
• Introduction of the tensor product concept, important in both classical and quantum contexts.
• Classical states of multiple systems are defined:
  • Classical state of a system is a configuration without uncertainty, defined mathematically as a finite non-empty set.
  • When combining two systems X (set sigma) and Y (set gamma), the classical state set is the Cartesian product of sigma and gamma.

Example of Classical States

• Example: X stores a binary value and Y stores one of four card suits, yielding 8 possible classical states.
• For n systems, the classical state set is represented by the Cartesian product of classical state sets of individual systems.

Cartesian Product and Notation

• Defined as the set of ordered pairs from two sets.
• Notation: n-tuple of classical states can be expressed as strings.
• Example with 10 bits shows 1024 total states.

Probabilistic States

• Probabilistic states assign probabilities to classical states.
• Example with bits X and Y showing correlation in outcomes.
• Independence vs. correlation defined in terms of probabilistic states.

Tensor Product

• Explanation of tensor products for vectors and matrices, applied to quantum states.
• Tensor product of quantum state vectors results in quantum state vectors.
• The tensor product represents independence between systems.

Measurements and Probabilistic Operations

• When measuring multiple systems, the operation is treated as a single system's measurement.
• If not all systems are measured, conditional probabilities are discussed.
• Operations on probabilistic states modeled by stochastic matrices.

Quantum Information for Multiple Systems

• Quantum states for multiple systems represented as vectors with complex entries.
• Tensor products represent independence between systems.

Quantum States and Entanglement

• Product states vs. entangled states explained.
• Entangled states show correlations that cannot be described by product states.
• Bell states and GHZ states discussed as examples of entangled states.

Measurements of Quantum States

• Measurement outcomes are determined by absolute values squared of entries in quantum state vectors.
• Conditional changes to the quantum state post-measurement are explored.

Unitary Operations on Multiple Systems

• Operations represented by unitary matrices corresponding to classical state sets.
• Independence between operations described by tensor products.
• Examples of controlled operations (e.g., controlled-NOT, controlled swap).

Conclusion

• Lesson 2 wraps up with discussions on measurements, operations, and the significance of quantum states for multiple systems.
• Next lesson previewed: Quantum circuits, protocols, and games.