There are many accounts of the roots of quantum indeterminism. Take, for instance, Vaidman’s recent review [544], leaning toward deterministic ontology.

In any case, in 1926, Born [570, p. 54] suggested that “from the standpoint of our quantum mechanics, there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which condition a definite outcome for the collision. Ought we to hope later to discover such properties \(\ldots \) and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment – as to the impossibility of prescribing conditions? \(\ldots \) I myself am inclined to give up determinism in the world of atoms.” Footnote 1

A quantum mechanical gap of casuality can be realized by a half-silvered mirror [292, 484, 498], with a 50:50 chance of transmission and reflection, as depicted in Fig. 13.1. A gap certified by quantum value indefiniteness necessarily has to operate with more than two exclusive outcomes [5]. Reference [3] presents such a qutrit configuration.

Fig. 13.1
figure 1

(Colour online) A gap created by a quantum coin toss. A single quantum (symbolized by a black circle from a source (left crossed circle) impinges on a semi-transparent mirror (dashed line), where it is reflected and transmitted with a 50:50 chance. The two final states are indicated by grey circles. The exit ports of the mirror can be coded by 0 and 1, respectively

Full size image

One may object to the orthodox view [589] of quantum indeterminism by pointing out that it is merely based on a belief without proof. It is not at all clear where exactly the randomness generated by a half-silvered mirror resides; that is, where the stochasticity comes from, and what are its origin. (Often vacuum fluctuations originating from the second, empty, input port are mentioned, but, pointedly stated [236, p. 249], these “mysterious vacuum fluctuations \(\ldots \) may be regarded as sugar coating for the bitter pill of quantum theory.”)

More generally, any irreversible measurement process, and, in particular, any associated ‘collapse,’ or, by another denomination, ‘reduction’ of the quantum state (or the wave function) to the post-measurement state is a postulate which appears to be means relative in the following sense.

The beam splitter setup is not irreversible at all because a 50:50 mirror has a quantum mechanical representation as a permutation of the state, such as a unitary Hadamard transformation; that is, with regard to the quantum state evolution the beam splitter acts totally deterministic; it can be represented by a one-to-one function, a permutation. (Experimentally, this can be demonstrated by serially concatenating two such 50:50 mirrors so that the output ports of the first mirror are the input ports of the second mirror. The result (modulo an overall phase) is a Mach–Zehnder interferometer reconstructing the original quantum state.)

Formally – that is, within quantum theory proper, augmented by the prevalent orthodox ‘Copenhagen-type’ interpretation – it is not too difficult to locate the origin of randomness at the beam splitter configuration: it is

(i) the possibility that a quantum state can be in a coherent superposition of classically distinct and mutually exclusive (outcome or scattering) states of a single quantum; and

(ii) the possibility that an irreversible measurement ad hoc and ex nihilo stochastically ‘chooses’ or ‘selects’ one of these classically mutually exclusive properties, associated with a measurement outcome. This, according to the orthodox interpretation of quantum mechanics, is an irreducible indeterministic many-to-one process – it transforms the coherent superposition of a multitude of (classically distinct) properties into a single, classical property . This latter assumption (ii) is sometimes referred to as the reduction postulate.

Already Schrödinger has expressed his dissatisfaction with both assumptions (i) and (ii), and, in particular, with the quantum mechanical concept of ontological existence of coherent superposition, in various forms at various stages of his life: he polemicized against the uncritical perception of the quantum formalism (i) by quoting the burlesque situation of a cat which is supposed to be in a coherent superposition between death and life [452]. He also noted the curious fact that, as a consequence of (i) and in the absence of measurement and state reduction (ii), according to quantum mechanics we all (as well as the physical universe in general), would become quantum jelly [457].

What in the orthodox scriptures of quantum mechanics often is referred to as “irreversible measurement” remains conceptually unclear, and is inconsistent with other parts of quantum theory. Indeed, it is not even clear that, ontologically, an irreversible measurement exists! Wigner [571] and, in particular, Everett [206, 208] put forward ontologic arguments against irreversible measurements by extending the cut between a quantum object and the classical measurement apparatus to include both object as well as the measurement apparatus in a uniform quantum description. As this latter situation is described by a permutation (i.e. by a unitary transformation), irreversibility, and what constitutes ‘measurement’ is lost. Indeed, the reduction postulate (ii) and the uniform unitarity of the quantum evolution cannot both be true, because the former essentially yields a many-to-one mapping of states, whereas uniform unitarity merely amounts to a one-to-one mapping, that is, a permutation, of states. In no way can a many-to-one mapping ‘emerge’ from any sort of concatenation of one-to-one mappings! Stated differently, according to the reduction postulate (ii), information is lost; whereas, according to the unitary state evolution, no information is ever lost. So, one of these postulates must be ontologically wrong (they may be epistemically justified for all practical purposes [44], though). In view of this situation, I am (to use Born’s dictum [68, p. 866]) inclined to give up the reduction postulate disrupting permutativity, and, in particular, unitarity, in the world of single quantum phenomena, in favour of the latter; that is, in favour of permutativity, and, in particular, unitarity.

The effort to do so may be high, as detailed beam recombination analysis of a Stern–Gerlach device (the spin analogue of a beam splitter in the Mach–Zehnder interferometer) shows [202, 461]. Nonetheless, experiments (and proposals) to “undo” quantum measurements abound [137, 252, 275, 323, 389, 394, 462, 463, 585]. Thus we could say that for all practical purposes [43], that is, relative to the physical means [375] available to resolve the huge number of degrees of freedom involving a macroscopic measurement apparatus, measurements appear to be irreversible, but a close inspection reveals that they are not. So, irreversibility of quantum measurements appears to be epistemic and means relative, subjective and conventional; but not ontic. (As already argued by Maxwell, this is just the same for the second law of thermodynamics [375].)