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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum Buridan’s ass

Jaroslav Novotný, Goce Chadzitaskos, Igor Jex, Stephen Mark Barnett, T. Kiss

Published Sep 1, 2025• 1 citation
Publisher

Abstract

The interplay between symmetry and dynamics is central to physics, as well as to other branches of science. An interesting situation arises in decision-making when you are offered several equally viable solutions and you are forced to select one. The ensuing delay is generally known as Buridan’s paradox. We investigate a sketch of this situation in the context of optical multi-ports and comment on the role of symmetry and how physics deals with the dilemma of many options. It is noteworthy that in its simplest form, the evolution away from its initial state is sped up by quantum interference. However, more fully connected networks can display the frustration familiar from the classical paradox. The role of symmetry breaking is analysed for several cases and implications for the paradox are discussed. Our results have implications for quantum communications networks and other distributed quantum systems.

DOI
10.1098/rspa.2025.0023

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The interplay between symmetry and dynamics is central to physics, as well as to other branches of science. An interesting situation arises in decision-making when you are offered several equally viable solutions and you are forced to select one. The ensuing delay is generally known as Buridan’s paradox. We investigate a sketch of this situation in the context of optical multi-ports and comment on the role of symmetry and how physics deals with the dilemma of many options. It is noteworthy that in its simplest form, the evolution away from its initial state is sped up by quantum interference. However, more fully connected networks can display the frustration familiar from the classical paradox. The role of symmetry breaking is analysed for several cases and implications for the paradox are discussed. Our results have implications for quantum communications networks and other distributed quantum systems.
It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. \n \nIn part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. \n \nIn part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.
The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical consequences that go beyond the simple mathematical idea of linearly combining vectors in a Hilbert space. Specifically, the discussion turns around the connection between symmetries characterizing the wave function and the behavior in time displayed by the quantum flux when the latter is analyzed in terms of streamlines (Bohmian trajectories). This is illustrated with a series of analytical results and numerical simulations, which include Young's two-slit experiment, counter-propagating wave packets, grating diffiraction and quantum carpets (e.g., Talbot carpets), and diffiraction under confinement conditions. From the analysis presented it follows that quantum paradoxes appear whenever symmetries related to interference are neglected in the interpretation and understanding of the corresponding phenomena.
The quantum Buridan's ass problem concerns a network of interconnected nodes in which transitions between the nodes are governed by the laws of quantum mechanics. In such situations interference between probability amplitudes plays a key role in determining whether the probability of leaving the initial node is large or small and also the speed at which the associated transitions occur. In this paper we extend this idea to consider transitions between one given set of coupled nodes and the remainder of the network. We have in mind a quantum analogy to changing an opinion, in which the initially occupied set of nodes is one opinion bubble, with the remaining parts of the network corresponding to a different opinion. We find that quantum interference again plays a crucial role in the dynamics. A further key component is the role of symmetry between the nodes and their interconnections, as it is symmetry that lies at the heart of the Buridan's ass puzzle. In the classical Buridan's ass paradox it is symmetry between competing alternatives that freezes the ass into indecision. In the quantum case, however, interference results in a far richer dynamics.
In a quantum measurement, a coupling g between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling g behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of N interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off‐diagonal elements of S + A, rapid due to the large size of N. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states. If g is too weak, the measurement fails due to a “Buridan’s ass” effect. The probability distribution for the magnetization then develops not one but two narrow peaks at the ferromagnetic values. During its evolution it goes through wide shapes extending between these values.
Quantum information is scrambled via a chaotic time evolution in many-body systems. Clarifying how and to what extent we can recover the quantum information from the scrambled state is a central issue of today's physics, since it commonly appears as an essential factor in various topics including irreversibility of quantum chaotic dynamics, fault-tolerant quantum computation, and black hole information paradox. In realistic settings, symmetry can ubiquitously exist in scrambling dynamics. Here we establish fundamental limitations on the information recovery arising from the Lie group symmetries in the scrambling dynamics. The results are applicable to many situations, including the Hayden-Preskill black hole model with symmetry where the success rate of the information recovery qualitatively changes from the original no-symmetry case. Our findings show a close relationship between information recovery, symmetry, and quantum coherence, and that symmetry hinders the quantum information recovery.