No information without disturbance

If a quantum instrument leaves every input state unchanged even after conditioning on its classical output, then its output distribution is independent of the input state. Thus an ordinary quantum process cannot learn arbitrary state information while preserving every state.

A quantum instrument associates each classical outcome with a completely positive trace-nonincreasing map; summing the outcome maps gives a quantum channel. It specifies both outcome probabilities and conditional output states.

For Kraus operators satisfying ${\sum }_i{A}_i^{†}{A}_i=I$, the map $V|\varphi ⟩={\sum }_i{A}_i|\varphi ⟩\otimes |i⟩$ is an isometry. Implement it by a unitary operator on the system and a quantum ancilla, initially in an extra unused state $|0⟩$. Projecting the output quantum ancilla onto $|i⟩$ gives joint unnormalized state $A_i\rho A_i^{†}\otimes |i⟩⟨i|$. The outcome probability is $\mathrm{Tr}(A_i\rho A_i^{†})$, and the conditional system state is $A_i\rho A_i^{†}$ divided by this probability. The unused level can be merged into the first orthogonal projection without changing any outcome on the prepared state, giving a complete projective measurement with exactly the original outcome labels.

// Target: quantum-theory.bigb

A POVM fixes outcome probabilities, but a quantum instrument additionally fixes the conditional output state. For computational-basis effects $E_y=|y⟩⟨y|$, the operators $M_y=|y⟩⟨y|$ leave the measured eigenstate as output, whereas $N_y=|0⟩⟨y|$ reset every outcome to $|0⟩$. Both have $M_y^{†}{M}_y=N_y^{†}{N}_y=E_y$. Thus a specification of effects alone cannot select a unique joint state that retains the measured quantum system.

A local quantum operation on one tensor-factor subsystem acts as ${\Phi }_A\otimes {\mathrm{id}}_B$, with ${\Phi }_A$ completely positive. An individual outcome branch can decrease trace, while the sum over outcomes of a normalized instrument preserves trace. The unselected operation consequently leaves the remote reduced state unchanged. In a local field algebra, Kraus operators inside one region and commuting with a distant observable give an analogous no-signalling statement. Locality of the operation must be established; an arbitrary globally defined projection is not automatically a physically local measurement.