Lesson: Tensor Products

Introduction

In quantum mechanics, the tensor product is a mathematical operation that combines two or more vectors or matrices to form a new vector or matrix. It is a fundamental concept in quantum computing and physics, used to describe the state of multiple qubits or other quantum systems.

Definition

The tensor product of two vectors A and B, denoted as A ⊗ B, is a new vector with components that are the products of the components of A and B. For example, if:

A = (a1, a2, a3) B = (b1, b2)

Then the tensor product A ⊗ B is:

A ⊗ B = (a1b1, a1b2, a2b1, a2b2, a3b1, a3b2)

Properties

The tensor product has several important properties:

• Associativity: (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
• Commutativity: A ⊗ B = B ⊗ A
• Distributivity: A ⊗ (B + C) = A ⊗ B + A ⊗ C
• Identity element: A ⊗ (1) = A
• Zero element: A ⊗ (0) = 0

Applications in Quantum Computing

In quantum computing, the tensor product is used to represent the state of multiple qubits. A single qubit can be in the state |0⟩ or |1⟩. Two qubits can be represented by the tensor product |0⟩ ⊗ |0⟩, |0⟩ ⊗ |1⟩, |1⟩ ⊗ |0⟩, or |1⟩ ⊗ |1⟩. These states represent the four possible combinations of the two qubits.

Applications in Quantum Physics

In quantum physics, the tensor product is used to describe the state of multiple particles. For example, the state of two electrons can be represented by the tensor product of their individual states. The tensor product can also be used to describe the entanglement between particles.

Learning Resources

• Tensor Products in Quantum Mechanics
• Tensor Product in Quantum Computing
• Quantum Entanglement and Tensor Products

Exercises

1. Calculate the tensor product of the following vectors:
2. A = (1, 2, 3)
3. B = (4, 5)
4. Describe how the tensor product is used to represent the state of multiple qubits.
5. Explain how the tensor product can be used to describe the entanglement between particles.