This note is part of [[quantum/Practical Quantum Information System]]. > [!info] Course code > Use the companion repository for this lecture's runnable lab, helper functions, and regression checks: > - [notebooks/by_concept/ghz_states_entanglement_swapping_and_monogamy.ipynb](https://github.com/montekkundan/quantum-code/blob/main/notebooks/by_concept/ghz_states_entanglement_swapping_and_monogamy.ipynb) > - [qcourse/protocols.py](https://github.com/montekkundan/quantum-code/blob/main/qcourse/protocols.py) > - [tests/test_protocols_algorithms_qec.py](https://github.com/montekkundan/quantum-code/blob/main/tests/test_protocols_algorithms_qec.py) [TODO: add video - GHZ States, Entanglement Swapping, and Monogamy] ## What This Concept Is Bell pairs are the beginning of entanglement, not the whole story. Once you move to three or more systems, entanglement can have structures that do not reduce to "just more Bell states." GHZ states, entanglement swapping, and monogamy of entanglement are three of the best early ways to see that richer structure. This note is where multipartite entanglement becomes concrete. You stop asking only whether two systems are entangled and start asking how entanglement is distributed across a network or across several subsystems at once. ## Foundation Terms You Need First A [[quantum/Glossary#GHZ state|GHZ state]] is a standard multipartite entangled state. [[quantum/Glossary#Monogamy of entanglement|Monogamy of entanglement]] is the principle that strong entanglement between two systems limits how strongly either can be entangled with a third. [[quantum/Glossary#Reduced density matrix|Reduced density matrices]] remain the main tool for checking what subsystems individually look like. The key habit here is to name the partition each time. In multipartite settings, "entangled" without a partition often hides the real structure. ## How The Idea Actually Works A GHZ state such as $ \frac{|000\rangle + |111\rangle}{\sqrt{2}} $ is not just three Bell pairs glued together. It carries genuinely multipartite correlation structure. Measuring one subsystem changes what you can say about the others in a way that is not captured by pairwise correlation language alone. Entanglement swapping is the opposite style of lesson. Start with two Bell pairs that do not directly connect the outer qubits. By performing an appropriate joint measurement on the middle qubits, you can leave the outer qubits entangled even though they never interacted directly. This is one of the most important conceptual steps toward quantum networking and repeater-style thinking. Monogamy then tells you entanglement is not a freely shareable commodity. If one qubit is maximally entangled with another, it cannot also be maximally entangled with a third. This prevents many naive but incorrect pictures of "all the qubits are equally entangled with all the others in the same way." ## Why It Matters - It broadens entanglement intuition beyond two-qubit textbook examples. - It prepares you for nonlocality, networks, and subsystem reasoning. - It shows that entanglement distribution is structured and constrained, not generic. ## Practical Lab Use a multipartite notebook here so you stop treating all entanglement as a Bell pair. - Prepare a GHZ state and measure correlations in more than one basis. - Implement entanglement swapping with two Bell pairs and a Bell measurement on the middle qubits. - Inspect reduced states so you can see which subsystems look mixed and why. ## Homework Use the homework to make the network picture precise. - Explain what entanglement swapping achieves operationally. - Describe one way GHZ correlations differ from pairwise Bell correlations. - Explain monogamy of entanglement in subsystem language rather than as a slogan only. ## References - Scott Aaronson, [Introduction to Quantum Information Science](https://www.scottaaronson.com/qclec.pdf), Lecture 10. - IBM Quantum Learning, [learning portal](https://quantum.cloud.ibm.com/learning/en). - Qiskit documentation, [main docs](https://qiskit.org/documentation/).