Entanglement and Correlations of Measurement Outcomes

We saw earlier that entangled quantum states have unusual properties. First, none of the qubits whose system state is entangled has its own state. Second, measurements on one part of the multi-qubit entangled system affect the state we assign to the other parts. At a fundamental level, these issues are all tied to the concept of correlation. For an entangled state, outcomes of measurements carried out on different parts of the system are correlated.

Correlations are quantitative relationships between (or among) sets of data—an issue that is important in statistical analysis. However, for our purposes, we will need only a few simple ideas about correlations. In this section, we will show how correlations among measurement outcomes are related to entanglement. Those results will give us yet another way of determining if a quantum state is entangled. They also tell us that if we want to develop a quantum algorithm that includes correlations, we need to build in entangled states.