Summary: Multi-Qubit Systems, Entanglement, and Quantum Weirdness

Here are the key takeaways:

• Most quantum applications involve systems of more than one qubit. Multi-qubit states can be built up as superpositions of products of single-qubit states. For $n$ qubits, we need $2^n$ basis states.

• The product of two state vectors is called a tensor product and is often denoted with the symbol $\otimes$. For example, $\ket \psi = \ket A \otimes \ket B$, or without the symbol $\ket \psi = \ket A \ket B$.

• Entangled states cannot be factored into simple product states because there is a correlation between the states.

• We can use quantum gates acting on multi-qubit systems to produce entangled states. Qubit interactions with their environment can wipe out the entangled state we produced with quantum gates.

• All entangled states are superposition states, but not all superposition states are entangled states.

• We can distinguish entangled from non-entangled states mathematically by a relationship among the state amplitudes.

• Entangled states imply correlations among the probabilities of the measurements carried out on the individual qubits. It is important to note that the correlations in the measurement outcomes are not due to an interaction that occurs between the qubits during the measurement.

• Measurements kill off entanglement. After an observation of one of the qubits, that qubit’s state is no longer entangled with the state of the other qubits.

• Entangled states do not assign separate states to each qubit. Only the system can be said to have a state.

• We can exchange basis states for multi-qubit states by using the methods developed in the previous chapter on each of the qubit states individually.