Among the axioms of quantum mechanics there is one axiom that says transformations of a quantum state need to be continuous, linear, and reversible (and this together with the other axioms results in unitary operators).

What is the simplest experiment that confirms the need for this axiom, in particular reversibility of the operations?

Perhaps another way to state this is: what experiment definitively shows that we can't simply use classical probability (and probabilistic operations) to model quantum mechanics?

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    $\begingroup$ Don't you ask yourself the same for classical transformations? Why focus on the "quantum case" here? $\endgroup$
    – ACuriousMind
    Sep 7, 2015 at 17:55
  • $\begingroup$ There are no axioms in physics. That's a term that only applies to mathematics. That we use linear reversible transformations in quantum mechanics is simply the result of there being no evidence that non-linear, irreversible ones would be needed. $\endgroup$
    – CuriousOne
    Sep 7, 2015 at 18:01
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    $\begingroup$ I am not entirely sure about your question? Are you asking about "quantum operations" in the sense used in quantum computing? The axiom you mention is not usually included in the (admittedly not very rigorous) axioms of quantum mechanics used by physicists. $\endgroup$ Sep 7, 2015 at 18:04
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    $\begingroup$ I like this question, although I think use of the word "axiom" is throwing people off a bit. One might say in other words, why don't we use a different theory (instead of QM) which includes irreversible evolution? I'm not sure you're going to get a satisfying answer, because I don't think there's a single experiment anyone can point to which singlehandedly demonstrates the need for time evolution to be reversible. $\endgroup$
    – David Z
    Sep 7, 2015 at 18:27

3 Answers 3


One place to look for relatively direct evidence is in the cross sections of time-reversed nuclear and particle physics reactions. For instance comparing $$A + n \to B + \alpha $$ with $$B + \alpha \to A + n$$ consistently shows the same (energy dependent) cross-sections for both directions where these reaction can be done between ground states of the nuclei (or including long-lived meta-stable excited states).

Likewise with proton-alpha and proton-neutron reaction pairs.


Time evolution is governed by a one to one relationship. The Schrödinger equation says $$i\hbar \frac{\partial}{\partial t} \Psi=\hat H \Psi.$$

The fact that $\hat H$ is self adjoint and there is an $i$ on the left hand side means that norm doesn't change so two orthonormal states states stay orthonormal as they evolve. Thus it can't send two different states to the same state and so time evolution is reversible.

Every single experiment is about testing the predictions of the Schrödinger equation. The double slit has a free particle $\hat H$ so the phase changes at a fixed rate depending on the path distance. Which leads to interference, which is what we observe when we do the experiment. Other situations have different Hamiltonians, but each one shows the Schrödinger equation working.

In fact every aspect of how we use it shows it doesn't change the norm. For instance when we get frequencies of different results, the fact the frequencies sum to one at two different times is because the evolution was norm preserving.

So we would need a completely different theory if it wasn't reversible, since so many aspects are based on it being reversible.


Quantum mechanics conceives 'physical' transformations on physical systems represented in unitary operators. Such transformations range from simple operations like a global rotation, to complex microscopic operations like the Schrödinger operator that gives time evolution

But unitary transformations do not exhaust the list of available transformations that you can perform on a physical quantum system. Measurements, for instance, are neither unitary nor deterministic, and despite some special cases like the delayed quantum-eraser experiments, they are in general irreversible

So, to answer your question, any axiom that pretends to impose only unitary operations on quantum system, is just flagrantly ignoring measurements

Certain textbooks will try to avoid the conclusion of measurements being intrinsically non-unitary operations by confusing them with decoherence. The problem with equating measurement with decoherence is that decoherence can only remove the interference terms from a quantum density matrix, but the distribution of probabilities are still there. When you make measurements, you don't see the probability distribution, but individual eigenvalues, and each such measurement is intrinsically irreversible

  • $\begingroup$ But this doesn't involve a closed system. If the laws of physics are fundamentally reversible, then having what looks like a non-reversible change somewhere due to interactions with another system that is only going to be treated effectively (observers consisting of atoms and molecules are magically supposed to not be subject to the same laws), isn't a good counterexample. You could just as well claim that momentum isn't conserved when you bounce a ball of the ground when treating the Earth effectively in the infinite mass limit (and ignore that it is non-rigid). $\endgroup$ Sep 8, 2015 at 18:58
  • $\begingroup$ third-party observers interacting with quantum systems do interact unitarily from your perspective: in fact you (the first-party observer) sees the third-party observer become entangled with the quantum system and become a superposition of tensor product of eigenvalues and eigenobservers, but from their point of view the information is irreversibly lost, and communicating with you will not give any useful information to reverse their measurements, unless all eigenobservers do it consistently in the same way, which is the case of the Quantum-Eraser experiment $\endgroup$
    – lurscher
    Sep 8, 2015 at 20:55

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