Quantum nondemolition (QND) measurement is a special type of measurement of a quantum system in which the uncertainty of the measured observable does not increase from its measured value during the subsequent normal evolution of the system. This necessarily requires that the measurement process preserves the physical integrity of the measured system, and moreover places requirements on the relationship between the measured observable and the self-Hamiltonian of the system. In a sense, QND measurements are the "most classical" and least disturbing type of measurement in quantum mechanics. Most devices capable of detecting a single particle and measuring its position strongly modify the particle's state in the measurement process, e.g. photons are destroyed when striking a screen. Less dramatically, the measurement may simply perturb the particle in an unpredictable way; a second measurement, no matter how quickly after the first, is then not guaranteed to find the particle in the same location. Even for ideal, "first-kind" projective measurements in which the particle is in the measured eigenstate immediately after the measurement, the subsequent free evolution of the particle will cause uncertainty in position to quickly grow. In contrast, a momentum (rather than position) measurement of a free particle can be QND because the momentum distribution is preserved by the particle's self-Hamiltonian p2/2m. Because the Hamiltonian of the free particle commutes with the momentum operator, a momentum eigenstate is also an energy eigenstate, so once momentum is measured its uncertainty does not increase due to free evolution. Note that the term "nondemolition" does not imply that the wave function fails to collapse. QND measurements are extremely difficult to carry out experimentally. Much of the investigation into QND measurements was motivated by the desire to avoid the standard quantum limit in the experimental detection of gravitational waves. The general theory of QND measurements was laid out by Braginsky, Vorontsov, and Thorne following much theoretical work by Braginsky, Caves, Drever, Hollenhorts, Khalili, Sandberg, Thorne, Unruh, Vorontsov, and Zimmermann.
Technical definition Let A {\displaystyle A} be an observable for some system S {\displaystyle {\mathcal {S}}} with self-Hamiltonian H S {\displaystyle H_{\mathcal {S}}} . The system S {\displaystyle {\mathcal {S}}} is measured by an apparatus R {\displaystyle {\mathcal {R}}} which is coupled to S {\displaystyle {\mathcal {S}}} through interactions Hamiltonian H R S {\displaystyle H_{\mathcal {RS}}} for only brief moments. Otherwise, S {\displaystyle {\mathcal {S}}} evolves freely according to H S {\displaystyle H_{\mathcal {S}}} . A precise measurement of A {\displaystyle A} is one which brings the global state of S {\displaystyle {\mathcal {S}}} and R {\displaystyle {\mathcal {R}}} into the approximate form
| ψ ⟩ ≈ ∑ i | A i ⟩ S | R i ⟩ R {\displaystyle \vert \psi \rangle \approx \sum _{i}\vert A_{i}\rangle _{\mathcal {S}}\vert R_{i}\rangle _{\mathcal {R}}}
where | A i ⟩ S {\displaystyle \vert A_{i}\rangle _{\mathcal {S}}} are the eigenvectors of A {\displaystyle A} corresponding to the possible outcomes of the measurement, and | R i ⟩ R {\displaystyle \vert R_{i}\rangle _{\mathcal {R}}} are the corresponding states of the apparatus which record them. Allow time-dependence to denote the Heisenberg picture observables:
A ( t ) = e i t H S A e − i t H S . {\displaystyle A(t)=e^{itH_{\mathcal {S}}}Ae^{-itH_{\mathcal {S}}}.}
A sequence of measurements of A {\displaystyle A} are said to be QND measurements if and only if
[ A ( t n ) , A ( t m ) ] = 0 {\displaystyle [A(t_{n}),A(t_{m})]=0}
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