The Measurement Problem

The second argument put forward to demonstrate the incompleteness of quan-tum mechanics revolves around the difficulty raised by the question of “measure-ment” within its theoretical framework. As we have seen, two types of time evo-lution occur within it: the first, before the measurement, (4), is deterministic and unitary; the second, occurring during or immediately after the measurement, (6), is random and non-unitary. These two modes of evolution effectively account for the experimental results. However, the successes of quantum mechanics over the past hundred years, its universal character, as well as the limits of its applicability, which have been constantly pushed further, have led to a new perspective on the act of measurement and prompted physicists to ask the following questions: is the measuring apparatus not a physical system like any other? And is the act of measurement not simply an interaction between a “large” system and a “small” sys-tem46, which could be described in terms of this fundamental and universal theory, quantum mechanics? And if we accept that two methods of treatment can be use-ful, even indispensable, in practice—because considering the “large” system within the framework of quantum mechanics could prove extremely complex—should we not seek to understand the transition from unitary evolution (4) to non-unitary evolution (6) in physical terms, rather than simply postulating them?

Let us again quote Bell, who expresses these ideas with unparalleled intellectual clarity: “It would seem that the theory is exclusively concerned with ‘measure-ment results’ and has nothing to say about anything else. What exactly qualifies certain physical systems to play the role of ‘measurer’? Did the wave function of the universe wait for billions of years before a single-celled creature appeared? Or did it have to wait a little longer, for a slightly better-qualified system... holding a PhD? If the theory is to apply to anything other than highly idealized laboratory operations, aren’t we forced to admit that processes more or less ‘resembling measurements’ are occurring more or less all the time, more or less everywhere? And doesn’t this imply that wave function jumps happen all the time?”

These considerations, like the EPR paradox, raise a new conceptual difficulty, now known as the measurement problem. To better understand its nature, let’s examine the measurement process in more detail by considering a quantum system S described by a state vector of the form (1). Let’s rewrite this state, ignoring the time dependence, which is not essential here:

(13)

We then introduce a measurement device M specifically designed for its ability to measure the quantity A, and we treat it, this time, within the framework of quan-tum mechanics. Like any quantum system, the device M is described by a state vector |M0⟩, representing its initial, neutral state. Just before the measurement, the composed system S + M is thus described by the state vector |Φi⟩ = |Ψ⟩ ⊗ |M0⟩. The interaction between the two systems leads the composed system from the state |Φi⟩ to the correlated state |Φc⟩, defined as follows:

(14)

The state |Φc⟩ describes the existence of correlations—even an entanglement, anal-ogous to that encountered in the EPR paradox—between the eigenstates {|an⟩} of the operator  and the corresponding states {|Mn⟩} of the measurement device M. However, if we follow the formalism of quantum mechanics, the expression (14) must be understood as a superposition of macroscopic states {|Mn⟩} of the meas-urement device.

To highlight the difficulty, Schrödinger proposed illustrating this situation with the famous cat paradox47. In this thought experiment, the macroscopic