Partial Measurement of Multi-Qubit States

Measuring one qubit of an entangled register collapses the whole state consistently, updating the rest without fully determining it.

Measuring one part of many

In a multi-qubit register you often measure only some qubits. A partial measurement collapses the measured qubits to definite values and updates the state of the unmeasured ones accordingly. When the register is entangled, measuring one qubit can dramatically change the others.

How the update works

Suppose a two-qubit state is a|00> + b|01> + c|10> + d|11> and you measure the first qubit. Getting 0 keeps only the terms starting with 0: the state becomes (a|00> + b|01>) renormalised. Getting 1 keeps the |10> and |11> terms. The outcome probability is the total squared amplitude of the surviving branch, and the remaining qubit is left in the corresponding conditional state.

Entangled example

For the Bell state (|00>+|11>)/sqrt(2), measuring the first qubit as 0 forces the second to 0, and measuring it as 1 forces the second to 1. A single partial measurement fully determines the partner — the essence of entangled correlation. For a product state, by contrast, measuring one qubit tells you nothing about the other.

Deferred measurement

A useful principle is that measurements can often be deferred: a mid-circuit measurement followed by a classically controlled gate is equivalent to a coherent controlled gate followed by measurement at the end. This deferred-measurement principle lets analysts move all measurements to the end without changing the result, simplifying reasoning about circuits.

Uses

Partial measurement is central to many protocols: it is how teleportation extracts its two classical bits, how measurement-based computation drives the register forward, and how error correction reads syndromes without collapsing the protected logical information. In each case the skill is measuring just enough to learn what is needed while leaving the useful part of the state intact.