Quantum mechanical weak values of projection operators have been used to answer which-way questions, e . g . to trace which arms in a multiple Mach–Zehnder setup a particle may have traversed from a given initial to a prescribed final state. I show that this procedure might lead to logical inconsistencies in the sense that different methods used to answer composite questions, like “Has the particle traversed the way X or the way Y ?”, may result in different answers depending on which methods are used to find the answer. I illustrate the problem by considering some examples: the “quantum pigeonhole” framework of Aharonov et al. , the three-box problem, and Hardy’s paradox. To prepare the ground for my main conclusion on the incompatibility in certain cases of weak values and logic, I study the corresponding situation for strong/projective measurements. In this case, no logical inconsistencies occur provided one is always careful in specifying exactly to which ensemble or sample space one refers. My results cast doubts on the utility of quantum weak values in treating cases like the examples mentioned.
Quantum mechanical weak values of projection operators have been used to answer which-way questions, e . g . to trace which arms in a multiple Mach–Zehnder setup a particle may have traversed from a given initial to a prescribed final state. I show that this procedure might lead to logical inconsistencies in the sense that different methods used to answer composite questions, like “Has the particle traversed the way X or the way Y ?”, may result in different answers depending on which methods are used to find the answer. I illustrate the problem by considering some examples: the “quantum pigeonhole” framework of Aharonov et al. , the three-box problem, and Hardy’s paradox. To prepare the ground for my main conclusion on the incompatibility in certain cases of weak values and logic, I study the corresponding situation for strong/projective measurements. In this case, no logical inconsistencies occur provided one is always careful in specifying exactly to which ensemble or sample space one refers. My results cast doubts on the utility of quantum weak values in treating cases like the examples mentioned.
A so called “weak value” of an observable in quantum mechanics (QM) may be obtained in a weak measurement + post-selection procedure on the QM system under study. Applied to number operators, it has been invoked in revisiting some QM paradoxes ( e.g. , the so called Three-Box Paradox and Hardy’s Paradox). This requires the weak value to be interpreted as a bona fide property of the system considered, a par with entities like operator mean values and eigenvalues. I question such an interpretation; it has no support in the basic axioms of quantum mechanics and it leads to unreasonable results in concrete situations.
I show that the application of the quantum-mechanical (QM) which-way weak measurement scheme of Vaidman may lead to logical inconsistencies. To this end, I study weak values of projection operators. Weak values are (normalized) amplitudes, operationally defined by a weak measurement followed by postselection. Projector weak values have a direct physical significance. This allows formulating an inconsistency in classical terms, viz., the contradiction in the readings of several measuring devices. To arrive at the contradiction, I also invoke the property of weak measurements not to collapse the wave function but to leave the state of the system unchanged (to lowest order in the weak measurement strength). My arguments rely entirely on basic QM rules plus commonly accepted weak QM measurement approximations. Therefore, the inconsistency challenges QM as such.
Many quantum paradoxes based on a realistic view of weak values were discussed in the last decades. They lead to astonishing conclusions such as the measurement of a spin component of a spin-1/2 particle resulting in $100\ensuremath{\hbar}$, the separation of a photon from its polarization, and the possibility of having three particles in two boxes without any two particles being in the same box, among others. Here, we show that the realistic view of the weak values present in these (and other) works is equivalent to a realistic (and highly controversial) view of quantum measurements, where a measurement reveals the underlying reality of the measured quantity. We discuss that all quantum paradoxes based on weak values simply disappear if we deny these realistic views of quantum measurements and weak values. Our work thus aims to demonstrate the strong assumptions and the corresponding problems present in the interpretation of these quantum paradoxes.
In their paper (arXiv:2402.09879), Aredes and Saldanha analyze several paradoxes related to weak values and present a "general argument" that aims to show that "realistic interpretations ...of weak values lead to inconsistencies". Although we agree with the identified inconsistencies for the specific weak values analyzed there, in this Comment we demonstrate that the origin of these inconsistencies is not their general argument, which is formally incorrect. We use Bohmian mechanics as a counterexample to confirm that their conclusions are not valid for all weak values and quantum theories. In particular, we show that weak values postselected in position can in fact be interpreted within Bohmian mechanics as properties of quantum systems, detached from any measuring devices, in a consistent and meaningful way.
Interest in weak values has grown steadily since their introduction in 1988, driven by developments in quantum measurement theory and increasing attention to the foundational issues at the heart of quantum mechanics. This thesis undertakes a philosophical investigation into the nature and implications of weak values, with a particular focus on their interpretation and role in creating and resolving quantum paradoxes. Overall, we argue that sequential weak values offer novel conceptual tools for assessing logical consistency in an atemporal fashion, ultimately motivating a shift toward a fundamentally relation-based ontology. Chapter 1 begins by establishing the conceptual and philosophical foundations of the thesis. We argue that performing this conceptual analysis of weak values requires a philosophical methodology informed by the structure of quantum theory and the foundational questions it poses. In Chapter 2, we provide a critical exploration of weak measurement theory and weak values, outlining their historical foundations and development, the experimental context, and interpretative challenges. Through a detailed analysis of paradoxes related to weak values such as the nested interferometer and the quantum Cheshire cat, the chapter argues that weak values reveal spatiotemporally discontinuous trajectories which conflict with classical intuition about particles and their trajectories. We contend that weak values are inherently contextual, as their significance arises in the context of the pre- and post- selections of the system. Chapter 3 develops an original account of ‘particle presence' across time, building on metaphysical notions of persistence. This chapter proposes a way of recovering continuity in paradoxes such as the three-box gendankenexperiment. Our framework offers a means of assessing the persistence of particles through spacetime via sequential weak values which we argue test the logical consistency of propositions across time. We then explore the implications for temporality, developing the key argument of this chapter: presence is defined across durations of time, rather than in discrete spacetime. This raises the possibility of presence without a well-defined causal order. In Chapter 4, the insights developed in the previous chapters are extended to a consideration of Relational Quantum Mechanics (RQM), addressing the longstanding issue of a lack of intersubjective agreement in relational interpretations. Given RQM's rejection of absolutism, this chapter critiques existing attempts at bridging perspectives and proposes a new route to empirical confirmation grounded in the persistence and reliability of observer's memory records. Our two- step model, consisting of internal confirmation via sequential weak values and then a consistency check from a second observer, offers a new approach to confirmation in RQM without rejecting the non-absolutist core of the interpretation. Sequential weak values are re-interpreted as atemporal probes available to the informational observer, distinct from its physical counterpart. This atemporal nature allows them to establish logical consistency across the observer's records and assign credence to them without presupposing a classical temporal order. These results suggest an ontology grounded in primitive relations, rather than one built on temporally ordered events. Chapter 4 thus builds on the contextuality arguments of earlier chapters: just as weak values reveal measurement outcomes to be context-dependent, their sequential and time-symmetric structure exposes temporal contextuality, challenging classical assumptions about fixed causal order and reinforcing the case for a non-classical, relation-based ontology. Finally, in Chapter 5, we conclude by summarising our key results and giving a list of open questions.
Entanglement and nonlocality are studied in the framework of pre-/postselected ensembles with the aid of weak measurements and the two-state-vector formalism. In addition to the EPR–Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose entangled pre- or postselected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting “emerging correlations” between independent preand postselected ensembles of particles. This can be viewed as a quantum violation of the classical “pigeonhole principle.” Introduction It is seldom acknowledged that seven years before the celebrated Bell paper [1], Bohm and Aharonov [2] published an analysis of the EPR paradox [3]. They suggested an experimental setup, based on Compton scattering, for testing nonlocal correlations between the polarizations of two annihilation photons. In 1964, Bell proposed his general inequality, thereby excluding local realism. During the same time, Aharonov et al. constructed the foundations of a time-symmetric formalism of quantum mechanics [4]. While Bell's proof utilizes entanglement to demonstrate nonlocal correlations,wewill describe inwhat follows the emergence of nonlocal correlations between product states. For this purpose, however, we shall invoke weak measurements of pre- and postselected ensembles. In classical mechanics, initial conditions of position and velocity for every particle fully determine the time evolution of the system. Therefore, trying to impose an additional final condition would lead either to redundancy or to inconsistency with the initial conditions. This is radically different in quantum mechanics. Because of the uncertainty principle, an initial state vector does not fully determine, in general, the outcome of a future measurement. However, adding a final (backward-evolving) state vector results in a more complete description of the quantum system in between these two boundary conditions, which has a bearing on the determination of measurement outcomes. The basis for this time-symmetric formulation of quantum mechanics was laid down by Aharonov, Bergman, and Lebowitz (ABL), who derived a symmetric probability rule concerning measurements performed on systems, while taking into account the final state of the system, in addition to the usual initial state [4].