nLab deferred measurement principle



Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

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quantum algorithms:



In quantum physics and specifically in quantum information theory, the principle of deferred measurement is a theorem which says that

is equivalent (equal as a function from given input to output data types) as

The principle can be useful in practice for optimizing quantum circuits. It also clearly relates to the issue of interpretations of quantum mechanics: Since it is the collapse of the wavefunction upon quantum measurement which makes the interpretation of quantum mechanics subtle, it is interesting to note that this collapse may be (arbitrarily) deferred, in a precise sense.


See Gurevich & Blass 2021 for a general formalization and proof.

As an axiom for syntax of a quantum programming language: Staton (2015), Axiom B (p. 6 of 12).

Alternatively: In terms of the discussion at quantum circuits via dependent linear types, the deferred measurement principle is essentially the Kleisli equivalence for the necessity comonad B\Box_B on dependent linear types, like this:


While the principle of deferred measurement is a classical statement in quantum information theory, it was not defined or proven in generality and with precision (hence has remained folklore) until Gurevich & Blass 2021 (see the critical discussion of the literature provided there).

Accounts of the informal statement:

As an axiom for quantum programming languages:

  • Sam Staton, Axiom B (p. 6) in: Algebraic Effects, Linearity, and Quantum Programming Languages, POPL ‘15: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (2015) [doi:10.1145/2676726.2676999, pdf]

The above discussion and graphics follows:

See also:

General precise statement and proof:

Last revised on November 13, 2022 at 09:05:15. See the history of this page for a list of all contributions to it.