It is well known that any operation on a quantum computer is described with a unitary matrix (a quantum gate) because quantum computing is reversible.

Only non-reversible operation is a measurement of a qubit. Therefore, I would expect that the measurement can be described by a non-unitary matrix (or even by matrix that is not invertible).

A result of the measurement should be a probability distribution of all possible states of measured qubits.

My question: How does a matrix representing measurement of qubit(s) look like?


2 Answers 2


There is no single matrix representing measurement. Projective measurements are represented by a set of orthogonal projectors. For example, measurement of a single qubit in the standard basis is represented by projectors $$\pi_0=\begin{pmatrix} 1 &0\\ 0 &0 \end{pmatrix}$$ and $$\pi_1=\begin{pmatrix} 0 &0\\ 0 &1 \end{pmatrix}$$

If a qubit was initially in state with density matrix $\rho$, then the post-measurement density matrix $\rho'$ is


Also, probability distribution of possible measurement outcomes can be obtained only by multiple measurements on an ensemble of identical systems; a single measurement gives no information about probabilities of possible measurement outcomes.


These are called observables, which are unitary hermitian matrices whose eigenvectors are the possible outcomes of the measurement. For example, the observable of the standard computational basis is the Pauli Z operator, and the observable of the sign basis is the Pauli X operator.


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