Abstract The transition from unitary, reversible von Neumann-Everett quantum processes to non-unitary, irreversible processes and measurements is explored through infinite tensor products interpreted as nested, chained, or iterated Wigner’s friend scenarios. Infinite tensor products can disrupt unitary equivalence through sectorization and factorization, drawing parallels to concepts from real analysis, recursive mathematics, and statistical physics.
Abstract The transition from unitary, reversible von Neumann-Everett quantum processes to non-unitary, irreversible processes and measurements is explored through infinite tensor products interpreted as nested, chained, or iterated Wigner’s friend scenarios. Infinite tensor products can disrupt unitary equivalence through sectorization and factorization, drawing parallels to concepts from real analysis, recursive mathematics, and statistical physics.
Infinity is central to deriving macroscopic irreversibility from reversible microscopic laws across mathematics, theoretical computer science and physics. In analysis, infinite processes—such as Dedekind cuts and Cauchy sequences—construct real numbers as equivalence classes of rational approximations, bridging discrete rationals to the continuous real line. In quantum mechanics, infinite tensor products model nested measurements, where sectorization partitions the Hilbert space into equivalence classes, reconciling unitary evolution with wavefunction collapse. In statistical mechanics, macrostates emerge as equivalence classes of microstates sharing identical macroscopic properties, providing the statistical basis for thermodynamic irreversibility despite reversible dynamics. Equivalence relations formalize For-All-Practical-Purposes (FAPP) indistinguishability, reflecting operational limits on precision and observation. Together, these examples reveal a unified framework where infinity and equivalence underpin emergent macroscopic behavior from microscopic reversibility.
By assuming (i) a universal, observer-independent quantum state, by considering (ii) measurement processes with wave function collapse as part of quantum mechanical time evolution (also within isolated systems), and by (iii) clearly distinguishing the state of a quantum system from the knowledge of conscious observers about this state, we suggest a modified Copenhagen interpretation of quantum mechanics that resolves Wigner's Friend and extended Wigner's Friend paradoxes. The suggested interpretation leads to differences in some outcome probabilities compared to previous analyses which, in principle, make it possible to falsify the suggested modifications or the previous analyses (or both). The fundamental problem of the incompleteness of quantum mechanics in the Copenhagen interpretation regarding the lack of a precise definition and a mechanistic description of the measurement process (including the collapse of the wave function) is not addressed in this study. However, the argumentation regarding the resolution of Wigner's Friend paradoxes also applies to attempts for such a mechanistic description using collapse models with stochastic extensions to the Schrödinger equation or trying to model wave function collapse with unitary time evolution and irreversibility of the measurement process resulting from quantum statistical mechanics.
We revisit von Neumann's theory of infinite tensor products of Hilbert spaces. On the set Γ of equivalence classes of C0-sequences, which labels the incomplete tensor products inside the complete tensor product, we introduce a natural pseudo-ultrametric d: the distance between two classes is the convergence exponent of the series ∑j∣⟨φj,ψj⟩−1∣ formed from any pair of representatives. We show that d is well defined on equivalence classes, satisfies the strong triangle inequality, and is complete. Distinct classes may lie at distance zero, so d separates points only after passing to the quotient Γ of Γ by the relation d=0; the pair (Γ,d) is then a complete ultrametric space. As an application, we show that a product unitary ⨂jU whose factor U satisfies inf∥x∥=1∣⟨x,Ux⟩−1∣>0 (in particular, a unitary on a finite dimensional space with 1∈/σ(U)) displaces every class to the maximal distance 1. Guided by the intended application -- a caricature of Everettian branching, in which the sectors of the infinite tensor product play the role of worlds -- we also develop a gauge-invariant variant d~ of the metric, based on von Neumann's weak equivalence and matched to the quasi-equivalence of product states on the quasi-local algebra. The displacement of a class under a product unitary, measured by d~, is class dependent and realizes every value in [0,1]. We interpret d~ as a decoherence exponent: it measures the polynomial rate at which two branches of the universal state vector become operationally distinct as ever larger portions of the environment are monitored.
In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B , is associated with a tensor product space HA ? HB . Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems. As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement , which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing. Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al . [80] and Plenio et al . [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.
The role played by irreversible processes, at the microscopic scale, in several examples of quantum measurements is studied within a "minimally realist" interpretation of the quantum formalism introduced in a recent publication. The analysis extends and confirms the basic results of the earlier work: separation of the collapse and amplification phases of the measurement, and decoherence and collapse happening already as a consequence of irreversible microscopic processes at the level of the "probe" itself (typically an atom in the examples considered). Additionally, the proposed interpretation of the formalism is shown to be consistent with previous interpretations, and a case for true irreversibility in physics is argued in some detail.