© European Physical Society, EDP Sciences, 2025

Route to interpretation

Despite the extraordinary success of quantum theory, consensus about interpretation has proven hard to reach. Many proposals can be considered as educated guesses or rely on philosophical prejudices; they have to be analyzed for consistency and meaning. The Copenhagen interpretation emerged as the most reasonable and practical one. It involves Born’s rule for the probability of outcomes and Heisenberg’s reduction of the quantum state after the measurement. Most theorists eventually left the subject aside and focused on calculations.

However, although the quantum formalism allows to evaluate results trustfully and efficiently, understanding its meaning requires an indisputable interpretation. According to Bohr and recent authors like Darrigol [4], an interpretation of a physical theory arises by working out its formalism for some idealized physical experiments. A concrete physical meaning can then be given to some mathematical quantities – connecting the abstract formalism to events in a laboratory.

In this prospect, interpretation of quantum mechanics can emerge from the theoretical analysis of an ideal quantum measurement, supposedly performed by letting a tested system interact with a model apparatus. One then faces the “measurement problem”, an apparent contradiction between the unitarity of such dynamical processes and the physical properties of measurements. Solvable models thus appear as gems to capture the meaning of quantum theory.

Setup for an ideal measurement

In our 2013 review [2] of the many existing dynamical models of ideal quantum measurements, we have focused on a detailed quantitative analysis of one of them, including discussion of approximations and non-idealities. The pedagogical interest of this approach led us to produce in 2024 a paper [1] intended to help teachers to include measurement theory in their courses for improving the understanding of quantum mechanics by students at various levels (high school, introductory, advanced).

The task is to choose a quantity to be measured, to build a model for a suitable apparatus and then to solve the dynamics of the compound system: object plus apparatus. We consider the apparatus as a macroscopic system described by quantum statistical mechanics. It includes a mechanism involving a pointer that indicates the outcome of the experiment. For reading it off, the pointer must be macroscopic, which brings a classical-like element into the description.

We limit ourselves here to the Curie-Weiss model for quantum measurement [3]. As shown in [1] and [2], its main features can be generalized to more complicated models. A schematic setup is presented in figure 1. The system S is a spin ½. Its z-component s_z = ±1 (in units of ħ/2) is to be measured with an apparatus A, which consists of a magnet M and a thermal bath B. We suppose that the Hamiltonian of S vanishes; hence s_z does not evolve, “waiting to be measured”. M consists of N >>1 spins ½, while B is a thermal bath. M is an Ising magnet with spin-spin couplings of the z components with strength J/N. The pointer variable is the magnetization along z. The interaction Hamiltonian between S and M involves similar spin-spin couplings, of strength g, between the measured spin S and each spin of M. B is a phonon bath, which consists of many harmonic oscillators in equilibrium at a temperature T (with a Debye cutoff Γ), and interacting with the x,y,z components of the apparatus spins through a coupling γ.

Fig. 1 Schematic setup of the measurement of system S by an apparatus A, consisting of a magnet M and a thermal bath B. The parameters of the model are indicated.

The measurement involves a thermodynamic phase transition in M. Keeping aside its use as a measurement device, consider this thermal magnet alone. As depicted in figure 2, at low enough T it has two stable ferromagnetic states with magnetization upwards or downwards in the z-direction. M starts in the paramagnetic state, which is metastable with an extremely long lifetime if M is left untouched. However, application on M of a sufficiently strong magnetic field h suppresses one of the free energy barriers, see figure 3. A rapid relaxation will then lead to the corresponding ferromagnetic state, depending on the sign of h.

Fig. 2 The free energy of the magnet as function of m, the magnetization per spin, for various temperatures. It has a metastable minimum at m=0. At low enough temperature it has two absolute minima, one near m=+1 and the other one near m= –1.

Fig. 3 The free energy for various magnetic fields h, at a given, sufficiently low T. For h=0, the minima near m=+1 and m= -1 describe stable states. When a large enough magnetic field is applied, one of the barriers of the metastable m=0 state is suppressed, and a quick transition to the thermodynamically stable state occurs. In the measurement setup, h is replaced by +g in the spin-up sector, by -g in the spin-down sector.

Dynamics

After S gets coupled to A, initially in the paramagnetic state, the subsequent evolution of S+A amounts to relaxation towards thermodynamic equilibrium [1,3]. The upwards or downwards direction of the magnetization is used as pointer and will be correlated with the sign of s_z = ±1. In an ideal measurement, s_z commutes with the total Hamiltonian H, whence the dynamical equation governing the density matrix of S+A splits up into the diagonal ↑↑ and ↓↓ sectors and the off-diagonal ↑↓ and ↓↑ sectors, related to the spin S.

The ↑↓ and ↓↑ sectors of the density matrix of S+A, the “Schrödinger cat terms”, decay quickly due to dephasing induced by the large size of the magnet, followed by decoherence. Initial information about s_x and s_y, which do not commute with H, are thus dynamically lost, a price to pay for getting information about s_z. This disappearance is essential for von Neumann’s reduction of the quantum state of S by the measurement process.

The evolution of the diagonal blocks ↑↑ and ↓↓ describe the registration of the measurement, that is, the transition of the magnet to one of its stable states. In the ↑↑ sector with s_z = +1, S acts on M just as an applied magnetic field h equal to g, see figure 3, so that a rapid transition leads M to its ferromagnetic state with positive magnetization. In the ↓↓ sector with s_z = -1, S acts on M as a negative magnetic field h = - g, so that M ends up in the other ferromagnetic state. The sign of the final magnetization therefore indicates whether the spin S has s_z = 1 or s_z = -1. Due to the conservation of s_z, the weights of the blocks ↑↑ and ↓↓ remain constant during the evolution, so that the dynamics automatically produces the Born probabilities.

Connection to individual events

Quantum physics is a statistical theory which describes ensembles, but unlike classical statistical mechanics, a connection to individual events like measurements is not ensured, see the Box. However, this can be done rigorously by a postulate about the macroscopic quantum apparatus, beyond the quantum formalism, but compatible with it [1].

After an individual measurement, the pointer lies in a stable state, magnetized either upwards or downwards. Being macroscopic, it can be read off by a person or automatically by a device. The value of s_z = ±1 is correlated with the sign of the pointer, so that each individual measurement will leave S, after achievement, in a state with either s_z = +1 or s s_z = -1. Resetting the apparatus for a new measurement costs a macroscopic amount of energy.

Ensuing interpretation of some quantum properties

The above analysis, and that of the class of ideal measurements generalizing the above one [1], [2], justifies the features usually stated in textbooks, and provides them with a physical interpretation. The ordinary probabilities readily found for the pointer indications can be interpreted as probabilities for the post-measurement state of the system S. They are expressed by Born’s rule in terms of the initial state of S owing to the conservation of s_z; this provides some information about the initial state – which however remains globally uninterpreted.

By selecting, after measurement, the events with magnetization of M upwards, we gather spins S produced for these events a final state “up”, mathematically obtained as a projection of the initial state. (This property is confirmed by a subsequent measurement of s_z.) Thus, von Neumann’s “collapse “or “reduction” is the result of post-measurement selection — not of quantum dynamics for individual measurement processes, that quantum theory does not describe.

These properties highlight the fundamental role of the apparatus in the interpretation of quantum theory.

About the Authors

Armen E. Allahverdyan (Yerevan) has worked on statistical physics and quantum mechanics, including quantum thermodynamics, measurements, physics of quantum technologies, hydrodynamics, and complex systems.

Roger Balian (Saclay; Academie des Sciences) has worked on various topics of quantum or statistical physics: superfluid ³He, strong uncertainty principle, relation between waves and trajectories, Casimir effect, critical phenomena, lattice gauge fields, instantons, distribution of galaxies, nuclear structure and reactions…

Theo M. Nieuwenhuizen

(Amsterdam) has worked on disordered systems, light scattering, spin glasses, thermodynamics of glasses, transport of molecular motors, quantum measurement, quantum thermodynamics, stochastic electrodynamics, the Lorentz electron, black holes, dark matter and cosmology.

References

All Figures

	Fig. 1 Schematic setup of the measurement of system S by an apparatus A, consisting of a magnet M and a thermal bath B. The parameters of the model are indicated.
In the text

	Fig. 2 The free energy of the magnet as function of m, the magnetization per spin, for various temperatures. It has a metastable minimum at m=0. At low enough temperature it has two absolute minima, one near m=+1 and the other one near m= –1.
In the text

Fig. 3 The free energy for various magnetic fields h, at a given, sufficiently low T. For h=0, the minima near m=+1 and m= -1 describe stable states. When a large enough magnetic field is applied, one of the barriers of the metastable m=0 state is suppressed, and a quick transition to the thermodynamically stable state occurs. In the measurement setup, h is replaced by +g in the spin-up sector, by -g in the spin-down sector.

In the text