Quantum Instruments
A quantum instrument records a measurement outcome and the conditional state together, the most complete description of measurement.
Outcome and state at once
A POVM gives outcome probabilities; a channel gives state evolution. A quantum instrument unifies both. It is a collection of completely positive maps {E_k}, one per outcome, whose sum E = sum_k E_k is trace-preserving. The probability of outcome k is Tr(E_k(rho)), and the conditional post-measurement state is E_k(rho)/Tr(E_k(rho)). The instrument records what happened and what the system became.
Relation to other formalisms
- Extract only probabilities: E_k(rho) = M_k rho M_k-dagger gives the POVM E_k = M_k-dagger M_k
- Forget the outcome: summing gives the channel E, an ordinary CPTP map
- Keep the outcome, discard the state: recover a plain measurement
Thus the instrument sits above both the POVM and the channel; each is a partial view obtained by discarding one kind of information.
Selective vs non-selective
A selective use of the instrument keeps the outcome and the conditional state, updating based on what was seen. A non-selective use averages over outcomes, giving the unconditioned channel E(rho) = sum_k E_k(rho). Feedback and error correction depend on selective use: the recorded syndrome determines the corrective operation applied to the conditional state.
Why the full object is needed
Each E_k can involve several Kraus operators, so an instrument is strictly more general than a set of single measurement operators. This matters when a single outcome can be reached by physically distinct processes, as in coarse-grained or noisy measurements. Modeling real detectors, quantum non-demolition schemes, and continuous monitoring accurately requires the instrument, since only it captures both the readout statistics and the back-action on the system in one consistent structure.