POVM

POVM means positive-operator-valued measurement. A POVM is the collection of positive effects that determines the outcome probabilities of a generalized measurement.

Formula Hook

For outcomes labeled by $i$, a POVM is a set $\{E_i\}$ satisfying

$$E_i\ge0, \qquad \sum_i E_i=I.$$

For a density operator $\rho$,

$$p(i)=\operatorname{Tr}(\rho E_i).$$

Canonical Home

See POVMs: First Encounter for the teaching page and Generalized Measurements Overview for the broader measurement framework.

Common Confusions

• A POVM determines probabilities, not the post-measurement state by itself.
• Effects are positive operators; they need not be orthogonal projectors.
• Kraus operators, measurement instruments, and POVM effects are related but distinct objects.

References

• P. Busch, P. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, 2nd ed., Springer, 1996.
• A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, 1982.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.