Tensor products are the mathematical backbone of composite quantum systems. They allow us to combine separate C-algebras into more complex structures, crucial for describing systems with multiple parts. This operation preserves key properties and enables analysis of interactions between subsystems.

Entanglement is a uniquely quantum phenomenon where subsystems exhibit correlations that can't be explained classically. It's central to quantum information and computation, leading to non-local effects and violating Bell inequalities. Understanding entanglement is key to grasping quantum mechanics' weirdness.

Tensor Products and Composite Systems

Mathematical Foundations of Tensor Products

• Tensor product of C-algebras combines two separate algebras into a larger, more complex structure
• Operation denoted by $A \otimes B$ for C-algebras A and B
• Resulting tensor product preserves algebraic and topological properties of original algebras
• Tensor product space dimension equals product of individual space dimensions
• Crucial for describing composite quantum systems with multiple subsystems

Composite Quantum Systems and Their Properties

• Composite quantum systems consist of two or more subsystems combined using tensor products
• State space of composite system represented by tensor product of subsystem state spaces
• Total Hilbert space given by $H = H_1 \otimes H_2 \otimes ... \otimes H_n$ for n subsystems
• Allows description of complex quantum systems (atoms, molecules, quantum circuits)
• Enables analysis of interactions and correlations between subsystems

Partial Trace and Reduced Density Matrices

• Partial trace maps operators on composite system to operators on subsystem
• Mathematically expressed as $Tr_B(\rho_{AB}) = \sum_i \langle b_i | \rho_{AB} | b_i \rangle$ for basis $\{|b_i\rangle\}$ of subsystem B
• Reduced density matrix obtained by performing partial trace over unwanted subsystems
• Describes state of subsystem when information about other subsystems unavailable
• Crucial for analyzing entanglement and quantum information tasks

Entanglement

Fundamentals of Quantum Entanglement

• Entangled states exhibit quantum correlations between subsystems that cannot be described classically
• Cannot be written as tensor product of individual subsystem states
• Mathematically represented as $|\psi\rangle_{AB} \neq |\phi\rangle_A \otimes |\chi\rangle_B$ for any choice of $|\phi\rangle_A$ and $|\chi\rangle_B$
• Leads to non-local correlations and violation of Bell inequalities
• Central resource in quantum information and computation (teleportation, dense coding)

Separable States and Their Properties

• Separable states lack quantum entanglement between subsystems
• Can be written as convex combinations of product states
• Mathematically expressed as $\rho_{AB} = \sum_i p_i \rho_A^i \otimes \rho_B^i$ with $\sum_i p_i = 1$ and $p_i \geq 0$
• Serve as reference point for quantifying entanglement in quantum states
• Possess only classical correlations between subsystems

Bell States and Schmidt Decomposition

• Bell states represent maximally entangled two-qubit states
• Four Bell states: $|\Phi^+\rangle, |\Phi^-\rangle, |\Psi^+\rangle, |\Psi^-\rangle$
• $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$ serves as prototypical example
• Schmidt decomposition expresses bipartite pure states in special form
• Mathematically written as $|\psi\rangle_{AB} = \sum_i \sqrt{\lambda_i} |i\rangle_A \otimes |i\rangle_B$
• Schmidt coefficients $\lambda_i$ quantify entanglement in the state
• Provides powerful tool for analyzing and characterizing entanglement in quantum systems