Introduction to Multipartite Quantum Entanglement and Quantum Nonlocality

lecture for Warsaw-4-PhD School, Summer Term 2025

Wojciech Bruzda (Center for Theoretical Physics Polish Academy of Sciences, Warszawa)

syllabus

introduction

• Hilbert space formalism
• postulates of quantum mechanics
• tensor structure and partial operations (partial: trace, transpose)
• qudit (register)
• Bloch sphere representation

quantum algebra

• pure states, mixed states and their purification
• no-cloning theorem
• Schmidt decomposition (SVD)
• von Neumann entropy

quantum measurement

• von Neumann's measurement postulate
• projective measurement (PVM)
• generalized measurement (POVM)
• Born rule
• Neumark's dilation theorem
• partial measurement

quantum entanglement

• quantum correlations vs. classical correlations
• von Neumann entropy and Schmidt decomposition revisited
• (maximally entangled) Bell state(s)
• quantum entanglement vs. tensor structure

(S)LOCC paradigm

• (S)LOCC – idea and consequences (classification of entangement)
• quantum circuit (one- and two-qubit quantum gates)
• quantum circuit for Bell state
• quantum "teleportation" protocol (quantum entanglement as a resource)
• state distinguishability

multipartite quantum entanglement

• different multipartite entanglement scenarios
• $k$-separability and $k$-producibility
• genuine multipartite entanglement

entanglement measure

• entanglement of formation
• entanglement cost
• distillation of entanglement (bound entanglement)
• concurrence / tangle (monogamy of entanglement)
• geometric measure of entanglement
• negativity

entanglement criteria

• entanglement witness
• positive partial transpose (PPT) criterion
• computable cross-norm realignment (CCNR) criterion → cf. reshuflling operation...

special classes of multiqudit (highly) entangled states

\(|{\rm GHZ}\rangle\) and \(|{\rm W}\rangle\) states

• Borromean rings and "topological" entanglement
• monogamy of entanglement revisited

absolutely maximally entangled (AME) states

• reshuffling operation
• (orthogonal) Latin squares
• multi-unitary matrix
• local unitary equivalence and classification of AME states

graph states

• generalized Pauli group, Weyl-Heisenberg group and Clifford algebra
• stabilizer formalism
• AME states revisited

quantum nonlocality

• quantum correlations revisited
• Bell inequalities
• sum-of-square decomposition

certification and self-testing protocols

genuine multipartite nonlocality