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A projective measurement is specified by mutually orthogonal projections $P_i$ with ${\sum }_i{P}_i=I$. On a state $\rho$, outcome $i$ has probability $\mathrm{Tr}(P_i\rho )$.

Under preparation independence, an ontological model of a quantum system reproducing exact quantum quantum measurement probabilities cannot assign overlapping preparation measures to distinct pure states. An exclusion quantum measurement on independent copies makes every outcome impossible for one possible preparation; a common ontic region would require every response to vanish there. Thus the conclusion is a psi-ontic model, not a blanket exclusion of hidden variables or all informational interpretations.

A unitary operation on a fixed number of qubits used as a component of a quantum circuit. Examples include the Hadamard gate and Controlled-Z gate. Measurements are described separately by quantum measurement; a gate is a reversible operation on its full input state.

The nonselective channel of a hypothetical spacelike nonlocal quantum measurement must not allow a distant input change to alter local output statistics. For a rank-one measurement in the rotated even and odd two-qubit bases with angle $\theta$, Alice's output satisfiesBob can change the last correlation using only his local Pauli Z gate while Alice keeps the input $|+⟩$. The resulting signal excludes $0<\theta <\pi /4$. Degenerate observables that do not distinguish this basis are not covered by the rank-one argument. Locally recording a jointly encoded result, which is decoded later by local operations and classical communication, is compatible with quantum no-signalling.

Remote state preparation uses a shared entangled state, local quantum measurements and classical communication to prepare a state known to the sender at the receiver. The sender can tailor the measurement to the desired state, so restricted state families can require less communication than quantum teleportation. A real great-circle family of qubit states admits one-bit remote preparation of real qubit states.

For two density operators $\rho ,\sigma$, let $P_{+}$ project onto the positive spectral subspace of $\rho -\sigma$. The binary POVM $\{P_{+},I-P_{+}\}$ has output probability difference $(D,-D)$, where $D=D(\rho ,\sigma )$. Its measurement channel therefore preserves this pair's trace distance exactly. The optimizing quantum measurement depends on the pair; a single fixed measurement need not preserve every pair's distance.