Journal of Physics A: Mathematical and Theoretical

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.

Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential, a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.

The motion of run-and-tumble particles in one-dimensional finite domains are analyzed in the presence of generic boundary conditions. These describe accumulation at walls, where particles can either be absorbed at a given rate, or tumble, with a rate that may be, in general, different from that in the bulk. This formulation allows us to treat in a unified way very different boundary conditions (fully and partially absorbing/reflecting, sticky, sticky-reactive and sticky-absorbing boundaries) which can be recovered as appropriate limits of the general case. We report the general expression of the mean exit time, valid for generic boundaries, discussing many case studies, from equal boundaries to more interesting cases of different boundary conditions at the two ends of the domain, resulting in nontrivial expressions of mean exit times.

This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world.

The curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes.

These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states.

The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

In this topical review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

One of the most widely known building blocks of modern physics is Heisenberg’s indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach, where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.

These are exciting times for quantum physics as new quantum technologies are expected to soon transform computing at an unprecedented level. Simultaneously network science is flourishing proving an ideal mathematical and computational framework to capture the complexity of large interacting systems. Here we provide a comprehensive and timely review of the rising field of complex quantum networks. On one side, this subject is key to harness the potential of complex networks in order to provide design principles to boost and enhance quantum algorithms and quantum technologies. On the other side this subject can provide a new generation of quantum algorithms to infer significant complex network properties. The field features fundamental research questions as diverse as designing networks to shape Hamiltonians and their corresponding phase diagram, taming the complexity of many-body quantum systems with network theory, revealing how quantum physics and quantum algorithms can predict novel network properties and phase transitions, and studying the interplay between architecture, topology and performance in quantum communication networks. Our review covers all of these multifaceted aspects in a self-contained presentation aimed both at network-curious quantum physicists and at quantum-curious network theorists. We provide a framework that unifies the field of quantum complex networks along four main research lines: network-generalized, quantum-applied, quantum-generalized and quantum-enhanced. Finally we draw attention to the connections between these research lines, which can lead to new opportunities and new discoveries at the interface between quantum physics and network science.

We study a discrete variant of the Airy equation and show that discretization produces a more intricate Stokes structure than in the continuous case, inducing the higher-order Stokes phenomenon and infinite accumulations of Stokes and anti-Stokes curves. These features are absent in the continuous Airy equation and are typically seen only in solutions of at least third-order linear homogeneous, second-order or higher linear inhomogeneous, or nonlinear differential equations. Remarkably, this behavior is seen here to arise in a second-order homogeneous linear difference equation. Using exponential asymptotic methods, we derive the asymptotic solutions and the corresponding Stokes structure, with numerical simulations confirming our predictions. We conjecture that the higher order Stokes phenomenon is able to be present in other second order linear difference equations.

Quantum pseudo-telepathy games, such as the Mermin–Peres magic square and the doily game (DG), theoretically allow players to win with unit probability when using entangled quantum strategies. We quantitatively characterize the quantum advantage in these games and compare it with violations of two Bell inequalities: the Clauser–Horne–Shimony–Holt and the Collins–Gisin inequalities. The analysis is restricted to two families of two-qubit states: modified Werner states and Bell-diagonal states. For each case, we identify and quantify the regions of quantum state space that exhibit either a quantum advantage or a Bell inequality violation, relative to the set of all entangled states. Within these families, the DG encompasses a larger fraction of entangled states than the Mermin–Peres magic square game, although both cover significantly smaller regions of entangled states compared to those where Bell inequalities are violated. Although both approaches are fundamentally linked to quantum contextuality, our analysis of the examined two-qubit state families suggests that Bell inequalities are more effective at revealing entanglement, even if pseudo-telepathy games offer a more intuitive and conceptually appealing perspective.

We unveil a quantum Pontus–Mpemba effect (PME) enabled by the Liouvillian skin effect (LSE) in a dissipative tight-binding chain with asymmetric incoherent hopping and coherent boundary coupling. The skin effect, induced by non-reciprocal dissipation, localizes relaxation modes near the system boundaries and gives rise to non-orthogonal spectral geometry. While such non-normality is often linked to slow relaxation, we show that it can instead accelerate relaxation through a two-step protocol—realizing a quantum PME. Specifically, we consider a one-dimensional open chain with coherent hopping J, asymmetric incoherent hoppings , and a controllable end-to-end coupling ε. For ε = 0, the system exhibits the LSE, with left and right eigenmodes localized at opposite edges. We compare two relaxation protocols toward the same stationary state: (i) a direct relaxation with ε = 0, and (ii) a two-step (Pontus) protocol where a brief coherent evolution transfers the excitation across the lattice before relaxation. Although both share the same asymptotic decay rate, the two-step protocol relaxes significantly faster due to its reduced overlap with the slow boundary-localized Liouvillian mode. The effect disappears when , i.e. when the skin effect vanishes. Our results reveal a clear connection between boundary-induced non-normality and protocol-dependent relaxation acceleration, suggesting new routes for controlling dissipation and transient dynamics in open quantum systems.

Classical background fields are a foundational technique in quantum field theory, playing a central role in developments such as the Higgs mechanism. Independently, supersymmetric twisting à la Witten has emerged as a key tool underlying phenomena such as supersymmetric localisation. Although these constructions are traditionally treated as distinct, they arise on an equal footing within the homotopy-algebraic approach to quantum field theory. In this work, we formalise this connection by interpreting both supersymmetric twisting and classical backgrounds as instances of twisting curved quantum -superalgebras. Using the language of homotopy algebras and the Batalin–Vilkovisky formalism, we provide a unified algebraic framework that encompasses topological/holomorphic twists, spontaneous symmetry breaking, computation of anomalies, and supersymmetric localisation à la Festuccia–Seiberg. We examine a variety of applications and examples illustrating this perspective in supersymmetric and general quantum fields theories alike. As a byproduct, we introduce a notion of twisting for quantum -algebras and a homotopy-algebraic reformulation of the one-particle-irreducible effective action.

The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra and a -module V, one may ask about the structure of the sub-ring of -invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.

We investigate the role of energy-invariant assistants in energy extraction from quantum batteries. To this end, for energy extraction, we restrict to unitaries that jointly act on the battery and the assistant but keep the assistant’s energy preserved. We demonstrate that in the presence of an energy-invariant assistant, having the same dimension as the battery, all stored energy of the battery can always be extracted, transforming the battery into its ground state, when an appropriate joint unitary and assistant’s state are employed. Additionally, we offer a necessary and sufficient condition for a battery to be unable to provide any energy, i.e. to be inactive, even when an energy-invariant assistant is present, prepared in an arbitrary but fixed state.

These lecture notes present an introduction to classical Affine Gaudin models, which provide a general framework for the systematic construction and study of a large class of integrable two-dimensional field theories. A key role is played by Kac–Moody currents, which are fields satisfying a particular Poisson bracket. After reviewing this notion, we discuss in detail the construction of Affine Gaudin models in the language of Hamiltonian field theories. Special emphasis is placed on their symmetries and conserved quantities, including the construction of infinite families of local and non-local Poisson-commuting charges in terms of Kac–Moody currents. Moreover, we study explicit examples of affine Gaudin models, making the link with the realm of integrable sigma-models. Finally, we mention briefly various perspectives concerning these theories, including the question of their quantisation. Minimal prerequisites on classical Hamiltonian field theories and integrability are required to follow the presentation and a brief reminder of these notions is given at the beginning of the notes. Moreover, various exercises are included throughout the document. These notes were prepared for the Young Researchers Integrability School and Workshop held in Durham from 17 to 21 July 2023.

Santosh Kumar was an active researcher on the topic of exact results in random matrix theory and their various applications, particularly to quantum chaos and information theory. Barely entering his mid-career, he died unexpectedly on the 18th October 2024. The present article gives an account of some of his research directions and findings. As well as serving as a tribute to his work, this is done also for the purpose of providing a resource for those who may continue along related lines in the future.

Stability theory plays a central role in the analysis of the behaviour of a solution of a real-order differential equations. In the literature, various stability concepts were introduced from the application point of view. The most popular ones are the Lyapunov and the Ulam-Hyers stability. Here, we first we establish a relation between the Lyapunov and Ulam-Hyers stability concepts for a dynamical system and prove that the concept of Ulam-Hyers is more general than that of Lyapunov. Second, we present a brief overview of recent developments in the Ulam-Hyers stability analysis of fractional-order differential equations (FDEs). These equations include linear FDEs, non-linear FDEs, delay FDEs, fractional-order boundary value problems and impulsive FDEs.

This paper is a review of the theory of abelian anyons in planar systems at an introductory level and with focus on the formalism of quantum field theory, but with the aim of clarify the connections between the mathematical structure and the theoretical and experimental physical aspects of these particle excitations.

Active matter has played a pivotal role in advancing understanding of non-equilibrium systems, leading to a fundamental shift in the study of biophysical phenomena. The foundation of active matter research is built on assumptions regarding the symmetry of microscopic constituents. While these assumptions have been validated extensively, instances of mixed or joint symmetries are prevalent in biological systems. This review explores the coexistence of polar and nematic order in active matter, emphasizing the theoretical and experimental challenges associated with these systems. By integrating insights from recent studies, we highlight the importance of considering mixed symmetries to accurately describe biological processes. This exploration not only benefits the field of biology but could also open new horizons for non-equilibrium physics, offering a comprehensive framework for understanding complex behavior in active matter.

Four-dimensional conformal fishnet theory is an integrable scalar theory which arises as a double scaling limit of -deformed maximally supersymmetric Yang–Mills. We give a perturbative reformulation of -deformed super-Yang–Mills theory in twistor space, and implement the double scaling limit to obtain a twistor description of conformal fishnet theory. The conformal fishnet theory retains an abelian gauge symmetry on twistor space which is absent in space-time, allowing us to obtain cohomological formulae for scattering amplitudes that manifest conformal invariance. We study various classes of scattering amplitudes in twistor space with this formalism.

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves. Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.

We consider anomaly cancellation for gauge theories where the left-handed chiral multiplets are in higher representations. In particular, if the left-handed quarks and leptons transform under the triplet representation of and if the gauge group is compact then up to an overall scaling there is only one possible nontrivial assignment for the hypercharges if N  =  3, and two if N  =  9. Otherwise there are infinitely many. We use the Mordell–Weil theorem, Mazur’s theorem and the Cremona elliptic curve database which uses Kolyvagin’s theorem on the Birch Swinnerton-Dyer conjecture to prove these statements.

Carrollian manifolds offer an intrinsic geometric framework for the physics in the ultrarelativistic limit. The recently introduced Carrollian Lie algebroids are generalised to the setting of ρ-commutative geometry (also known as almost commutative geometry), where the underlying algebras commute up to a numerical factor. Via ρ-Lie-Rinehart pairs, it is shown that the foundational tenets of Carrollian geometry have analogous statements in the almost commutative world. We explicitly build two toy examples: we equip the extended quantum plane and the noncommutative 2-torus with Carrollian structures. This opens up the rigorous study of noncommutative Carrollian geometry via almost commutative geometry.

An error in [1, Theorem 3.3] is pointed out. Due to this, some results have to be corrected. However, the main results, namely the spectrum of the k-Toeplitz operator and the results regarding the exponential decay of eigenvectors, still hold.

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac-Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac-Bianconi coupling, hence, which we call it Dirac-Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higherorder networks beyond the node-based paradigm, while providing a ductile modeling tool for node-and edge-signals.

In this paper, we study the Functional Central Limit Theorem (FCLT) for a strongly damped stochastic Langevin equation with Lipschitz continuous coefficients. The theorem states the convergence of independent random processes toward Brownian motion or more general diffusion processes. This convergence provides a fundamental framework for analyzing limiting behaviors in randomly driven systems. We establish the FCLT for the Langevin equation by investigating the mean square convergence of the asymptotic behavior of its solutions. The major difficulty in our proof is that there exists a component in the solution of the equation which is not an It\^{o} integral, preventing the direct application of the It\^{o} isometry. We overcome this difficulty by adopting the method of integration by parts. Finally, to validate our theoretical framework, we apply the theorem to two distinct physical models, utilizing the classical harmonic oscillator as a basic model and a two-dimensional generalized Langevin system featuring nonlinear, state-dependent damping and multiplicative noise. The numerical simulations confirm the theoretical convergence rates and demonstrate the universality of using the linearized diffusion limit to analyze thermodynamic fluctuations in complex macroscopic scenarios.

We investigate the application of conformable derivatives to model
critical phenomena near continuous phase transition points. By incorporating
a deformation parameter into the differential structure, we derive
unified expressions for thermodynamic observables such as heat capacity,
magnetisation, susceptibility, and coherence length, each exhibiting
a power-law behaviour near the critical temperature. The conformable
derivative framework naturally embeds scale invariance and critical
slowing down into the dynamics without resorting to fully nonlocal
fractional calculus. Modified Ginzburg--Landau equations are constructed
to model superconducting transitions, leading to analytical expressions
for the order parameter and the London penetration depth. Experimental
data from niobium confirm the model's applicability, showing excellent
fits and capturing asymmetric scaling behaviour around the critical
point. This work offers a bridge between classical mean-field theory
and generalised scaling frameworks, with implications for both theoretical
modelling and experimental analysis.

Quantum pseudo-telepathy games, such as the Mermin–Peres magic square and the doily game (DG), theoretically allow players to win with unit probability when using entangled quantum strategies. We quantitatively characterize the quantum advantage in these games and compare it with violations of two Bell inequalities: the Clauser–Horne–Shimony–Holt and the Collins–Gisin inequalities. The analysis is restricted to two families of two-qubit states: modified Werner states and Bell-diagonal states. For each case, we identify and quantify the regions of quantum state space that exhibit either a quantum advantage or a Bell inequality violation, relative to the set of all entangled states. Within these families, the DG encompasses a larger fraction of entangled states than the Mermin–Peres magic square game, although both cover significantly smaller regions of entangled states compared to those where Bell inequalities are violated. Although both approaches are fundamentally linked to quantum contextuality, our analysis of the examined two-qubit state families suggests that Bell inequalities are more effective at revealing entanglement, even if pseudo-telepathy games offer a more intuitive and conceptually appealing perspective.

Classical background fields are a foundational technique in quantum field theory, playing a central role in developments such as the Higgs mechanism. Independently, supersymmetric twisting à la Witten has emerged as a key tool underlying phenomena such as supersymmetric localisation. Although these constructions are traditionally treated as distinct, they arise on an equal footing within the homotopy-algebraic approach to quantum field theory. In this work, we formalise this connection by interpreting both supersymmetric twisting and classical backgrounds as instances of twisting curved quantum -superalgebras. Using the language of homotopy algebras and the Batalin–Vilkovisky formalism, we provide a unified algebraic framework that encompasses topological/holomorphic twists, spontaneous symmetry breaking, computation of anomalies, and supersymmetric localisation à la Festuccia–Seiberg. We examine a variety of applications and examples illustrating this perspective in supersymmetric and general quantum fields theories alike. As a byproduct, we introduce a notion of twisting for quantum -algebras and a homotopy-algebraic reformulation of the one-particle-irreducible effective action.

The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra and a -module V, one may ask about the structure of the sub-ring of -invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.

In quantum control, quantum speed limits provide fundamental lower bounds on the time that is needed to implement certain unitary transformations. Using Lie algebraic methods, we link these speed limits to symmetries of the control Hamiltonians and provide quantitative bounds that can be calculated without solving the controlled system dynamics. In particular we focus on two scenarios: On one hand, we provide bounds on the time that is needed for a control system to implement a given target unitary U and on the other hand we bound the time to implement the most difficult to reach evolution of a target Hamiltonian H_s. We apply our abstract bounds on physically relevant systems like coupled qubits, spin chains, globally controlled Rydberg atoms and NMR molecules and compare our results to the existing literature. We hope that our bounds can aid experimentalists to identify bottlenecks and design faster quantum control systems.

We investigate the application of conformable derivatives to model
critical phenomena near continuous phase transition points. By incorporating
a deformation parameter into the differential structure, we derive
unified expressions for thermodynamic observables such as heat capacity,
magnetisation, susceptibility, and coherence length, each exhibiting
a power-law behaviour near the critical temperature. The conformable
derivative framework naturally embeds scale invariance and critical
slowing down into the dynamics without resorting to fully nonlocal
fractional calculus. Modified Ginzburg--Landau equations are constructed
to model superconducting transitions, leading to analytical expressions
for the order parameter and the London penetration depth. Experimental
data from niobium confirm the model's applicability, showing excellent
fits and capturing asymmetric scaling behaviour around the critical
point. This work offers a bridge between classical mean-field theory
and generalised scaling frameworks, with implications for both theoretical
modelling and experimental analysis.

In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases that are as "far apart" as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct the space of its orthonormal bases as a discrete quotient of the complete flag manifold. We introduce a metric on this space, which corresponds to the "MUBness" distance. This allows us to describe equivalence between sets of mutually unbiased bases in terms of the geometry of this space. The subspace of bases that are unbiased with respect to the standard basis decomposes into orbits under a certain group action, and this decomposition corresponds to the classification of complex Hadamard matrices. More generally, we consider a list of k MUBs, that one wishes to extend. The candidates are points in the subspace comprising all bases which are unbiased with respect to the entire list. This space also decomposes into orbits under a group action, and we prove that points in distinct orbits yield inequivalent MUB lists. Thus, we generalize the relation between complex Hadamard matrices and MUBs. As an application, we identify new symmetries that reduce the parameter space of MUB triples in dimension 4 by a factor of 4.

A special class of para-Hermite Einstein spaces is analyzed. For para-Hermite Einstein spaces the self-dual (SD) Weyl tensor is type-[D] in the Petrov-Penrose classification. The anti-self-dual (ASD) Weyl tensor is algebraically degenerate, equivalently, there exists an ASD congruence of null strings. It is assumed that this ASD congruence is parallely propagated. Thus, the spaces are not only para-Hermite but also Walker. A classification of the spaces according to three criteria is given. Finally, explicit metrics of all admitted Petrov-Penrose types are found.

We study analytically and numerically the time evolution of a nonlinear field described by the nonlinear Schrödinger equation in a chaotic D-shape billiard. In the absence of nonlinearity the system has standard properties of quantum chaos. This model describes a longitudinal light propagation in a multimode D-shape optical fiber and also those in a Kerr nonlinear medium of atomic vapor. We show that, above a certain chaos border of nonlinearity, chaos leads to dynamical thermalization with the Rayleigh–Jeans (RJ) thermal distribution and the formation of the RJ condensate in a vicinity of the ground state accumulating about 80%–90% of total probability. Certain similarities of this phenomenon with the Fröhlich condensate are discussed. Below the chaos border the dynamics is quasi-integrable corresponding to the Kolmogorov–Arnold–Moser integrability. We also describe the time evolution during the process of relaxation to the thermal state and the time dependence of quantum von Neumann and classical Boltzmann entropies during this process. At a strong focusing nonlinearity we show that the wave collapse can take place even at sufficiently high positive energy being very different from the open space case. Finally for the defocusing case we establish the superfluid regime for vortex dynamics at strong nonlinearity. System parameters for optical fiber experimental studies of these effects are also discussed.

Isospectrally patterned lattices (IPL) have recently been shown to exhibit a rich band structure
comprising both regimes of localized as well as extended states. The localized states show
a single center localization behaviour with a characteristic localization length. We derive
a continuum analogue of the IPL which allows us to determine analytically its
eigenvalue spectrum and eigenstates thereby obtaining an expression for the localization length
which involves the ratio of the coupling among the cells of the lattice and the phase gradient across the lattice.
This continuum model breaks chiral symmetry but still shows a pairing of partner states with
positive and negative energies except for the ground state. We perform a corresponding symmetry
analysis which illuminates the continuum models structure as compared to a corresponding chirally
symmetric Hamiltonian.

We analytically find quantitative, non-perturbative bounds to the validity of the rotating-wave approximation (RWA) for the multi-atom generalization of the quantum Rabi model: the Dicke model. Precisely, we bound the norm of the difference between the evolutions of states generated by the Dicke model and its rotating-wave approximated counterpart, that is, the Tavis-Cummings model. The intricate role of the parameters of the model in determining the bounds is discussed and compared with numerical results. Our bounds are intrinsically state-dependent and, in particular, capture a nontrivial dependence on the total angular momentum of the initial state; this behaviour also seems to be confirmed by accompanying numerical results.

Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

A short résumé is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for -symmetric Hamiltonians, where a great number of experiments has been performed, in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos; they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics, they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.

We review the conformal field theory approach to entanglement entropy in 1+1 dimensions. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.

In this topical review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world.

One of the most widely known building blocks of modern physics is Heisenberg’s indeterminacy principle. Among the different statements of this fundamental property of the full quantum mechanical nature of physical reality, the uncertainty relation for energy and time has a special place. Its interpretation and its consequences have inspired continued research efforts for almost a century. In its modern formulation, the uncertainty relation is understood as setting a fundamental bound on how fast any quantum system can evolve. In this topical review we describe important milestones, such as the Mandelstam–Tamm and the Margolus–Levitin bounds on the quantum speed limit, and summarise recent applications in a variety of current research fields—including quantum information theory, quantum computing, and quantum thermodynamics amongst several others. To bring order and to provide an access point into the many different notions and concepts, we have grouped the various approaches into the minimal time approach and the geometric approach, where the former relies on quantum control theory, and the latter arises from measuring the distinguishability of quantum states. Due to the volume of the literature, this topical review can only present a snapshot of the current state-of-the-art and can never be fully comprehensive. Therefore, we highlight but a few works hoping that our selection can serve as a representative starting point for the interested reader.

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.

The Hamiltonian H_eff of an open quantum system consists formally of a first-order interaction term describing the closed (isolated) system with discrete states and a second-order term caused by the interaction of the discrete states via the common continuum of scattering states. Under certain conditions, the last term may be dominant. Due to this term, H_eff is non-Hermitian. Using the Feshbach projection operator formalism, the solution Ψ^E_c of the Schrödinger equation in the whole function space (with discrete as well as scattering states, and the Hermitian Hamilton operator H) can be represented in the interior of the localized part of the system in the set of eigenfunctions ϕ_λ of H_eff. Hence, the characteristics of the eigenvalues and eigenfunctions of the non-Hermitian operator H_eff are contained in observable quantities. Controlling the characteristics by means of external parameters, quantum systems can be manipulated. This holds, in particular, for small quantum systems coupled to a small number of channels. The paper consists of three parts. In the first part, the eigenvalues and eigenfunctions of non-Hermitian operators are considered. Most important are the true and avoided crossings of the eigenvalue trajectories. In approaching them, the phases of the ϕ_λ lose their rigidity and the values of observables may be enhanced. Here the second-order term of H_eff determines decisively the dynamics of the system. The time evolution operator is related to the non-Hermiticity of H_eff. In the second part of the paper, the solution Ψ^E_c and the S matrix are derived by using the Feshbach projection operator formalism. The regime of overlapping resonances is characterized by non-rigid phases of the Ψ^E_c (expressed quantitatively by the phase rigidity ρ). They determine the internal impurity of an open quantum system. Here, level repulsion passes into width bifurcation (resonance trapping): a dynamical phase transition takes place which is caused by the feedback between environment and system. In the third part, the internal impurity of open quantum systems is considered by means of concrete examples. Bound states in the continuum appearing at certain parameter values can be used in order to stabilize open quantum systems. Of special interest are the consequences of the non-rigidity of the phases of ϕ_λ not only for the problem of dephasing, but also for the dynamical phase transitions and questions related to them such as phase lapses and enhancement of observables.

The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called ‘biorthogonal quantum mechanics’, is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.