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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?

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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

$$\rho'=\sum_{i=0}^1\pi_i\rho\pi_i$$

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.

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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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