Marek Kuś

Polish Academy of Sciences | PAN · Center for Theoretical Physics

If η is a contact form on a manifold M such that the orbits of the Reeb vector field R form a simple foliation F on M, then the presymplectic 2-form dη on M induces a symplectic structure ω on the quotient manifold N=M/F. We call (M,η) a contactification of the symplectic manifold (N,ω). First, we present an explicit geometric construction of conta...

The one-dimensional system of particles with a $1/x^2$ repulsive potential is known as the Calogero-Moser system. Its classical version can be generalised by substituting the coupling constants with additional degrees of freedom, which span the $\mathfrak{so}(N)$ or $\mathfrak{su}(N)$ algebra with respect to Poisson brackets. We present the quantum...

We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric...

Many-body one-dimensional systems with 1/x2 interactions are known as Calogero-Moser systems. The so called ordinary system with a common coupling constant for all interacting pairs can be generalized into models with couplings which evolve as additional degrees of freedom. This can be done via unitary reduction of a linear matrix model or, alterna...

We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric...

We study a damped kicked top dynamics of a large number of qubits (N→∞) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant r∈[0,1], which plays the role of the single control parameter. In the parameter range for which the classical dynamics...

We study a damped kicked top dynamics of a large number of qubits ($N \rightarrow \infty$) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant $r\in [0,1]$, which plays the role of the single control parameter. In the parameter range for which...

We use spin coherent states to compare classical and quantum evolution of a simple paradigmatic, discrete-time quantum dynamical system exhibiting chaotic behavior in the classical limit. The spin coherent states are employed to define a phase-space quasidistribution for quantum states (P-representation). It can be, in principle, used for a direct...

We discuss various examples of classical Calogero-Moser models with internal degrees of freedom. These models besides of having some attractive properties, like the complete integrability, are of interest eg., in studying spectral properties of quantum chaotic systems. The role of internal degrees of freedom is important in at least two aspects. Fi...

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in suffi...

We use the general setting for contrast (potential) functions in statistical and information geometry provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. We study the case when the two-form is degenerate and show how in suffi...

We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. If the two-form is non-degenerate, it de...

In this article we undertake a diagnosis of the state of ontology as it is currently practiced in Poland. We point to the strengths of Polish ontology and the aspects that should be improved in order for Polish ontology to flourish further.We cover different styles of thinking in Polish ontology, as well as different methodologies. We address threa...

In this article we answer the question of why categories are becoming more and more popular in physics, mathematics and philosophy. The article presents a review of the role of categories in the philosophy of mathematics, in the foundations of mathematics, in metaphysics and in quantum mechanics. Our claim is that category theory is a formal ontolo...

The fidelity susceptibility measures the sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here, we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum syste...

We consider a system of two particles, each with large angular momentum j, in the singlet state. The probabilities of finding projections of the angular momenta on selected axes are determined. The generalized Bell inequalities involve these probabilities and we study them using statistical methods. We show that most of Bell's inequalities cannot b...

We demonstrate that the proper general setting for contrast (potential) functions in statistical and information geometry is the one provided by Lie groupoids and Lie algebroids. The contrast functions are defined on Lie groupoids and give rise to two-forms and three-forms on the corresponding Lie algebroid. If the two-form is non-degenerate, it de...

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philos...

I show how classical and quantum physics approach the problem of randomness and probability. Contrary to popular opinions, neither we can prove that classical mechanics is a deterministic theory, nor that quantum mechanics is a nondeterministic one. In other words it is not possible to show that randomness in classical mechanics has a purely episte...

The fidelity susceptibility measures sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. W...

We consider the classical limit of a system of two particles, each with arbitrary angular momentum j, in a state with zero total angular momentum. The state is maximally entangled and therefore exhibits non-classical features. To compare the quantum system with its classical counterpart the probabilities of finding projections of the angular moment...

An exact analytical treatment of the dynamical problem for time-dependent 2×2 pseudo-Hermitian su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter ν that allows for distinguishing two different dynam...

An exact analytical treatment of the dynamical problem for time-dependent 2x2 pseudo-hermitian su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent new scenarios are identified within both classical and quantum contexts. Such a class is spanned by a positive parameter $\nu$ that allows to distinguish two differen...

We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we obtain also results concerning...

We show that the propositional system of a many-box model is always a set-representable effect algebra. In particular cases of 2-box and 1-box models, it is an orthomodular poset and an orthomodular lattice, respectively. We discuss the relation of the obtained results with the so-called Local Orthogonality principle. We argue that non-classical pr...

In the previous chapter, we classified Hamiltonians H and Floquet operators F by their groups of canonical transformations.

A classical Hamiltonian system is called time-reversal invariant if from any given solution x(t), p(t) of Hamilton’s equations an independent solution x′(t′), p′(t′), is obtained with t′ = −t and some operation relating x and p′ to the original coordinates x and momenta p.

Regular classical trajectories of dissipative systems eventually end up on limit cycles or settle on fixed points. Chaotic trajectories, on the other hand, approach so-called strange attractors whose geometry is determined by Cantor sets and their fractal dimension. In analogy with the Hamiltonian case, the two classical possibilities of simple and...

This chapter will focus mainly on the kicked rotator that displays global classical chaos in its cylindrical phase space for sufficiently strong kicking. The chaotic behavior takes the form of “rapid” quasi-random jumps of the phase variable around the cylinder and “slow” diffusion of the conjugate angular momentum p along the cylinder.

The precursor of quantum mechanics due to Bohr and Sommerfeld was already seen by Einstein as applicable only to classically integrable dynamics.

Here we want to consider classical autonomous systems with f degrees of freedom and 2f pairs of canonical variables pi, qi.

As we have seen in Chap. 5, representing determinants as Gaussian integrals over anticommuting alias Grassmann variables makes for great simplifications in computing averages over the underlying matrices.

The first attempt to understand why spectral fluctuations of quantum chaotic systems are faithfully mirrored by random matrices was proposed by Pechukas.

This chapter will present classical Hamiltonian mechanics to the extent needed for the semiclassical endeavors to follow.

First hints at the usefulness of the sigma model for studying spectral statistics of individual dynamics appeared in 2005. The semiclassical construction of the non-oscillatory part of the two-point correlator of the quantum level density for classically chaotic systems

A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces .

This by now classic text provides an excellent introduction to and survey of the still-expanding field of quantum chaos. For this long-awaited fourth edition, the original text has been thoroughly modernized. The topics include a brief introduction to classical Hamiltonian chaos, a detailed exploration of the quantum aspects of nonlinear dynamics,...

This progress report covers recent developments in the area of quantum randomness, which is an extraordinarily interdisciplinary area that belongs not only to physics, but also to philosophy, mathematics, computer science, and technology. For this reason the article contains three parts that will be essentially devoted to different aspects of quant...

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed...

In the quantum logic framework we show that the no-signaling box model is a particular type of tensor product with single box logics. Such notion of a tensor product is too strong to apply in the category of logics of quantum mechanical systems. In the light of the obtained results, the statement that no-signaling box models are generalizations of...

We show that the propositional system of a many-box model is always a set representable effect algebra. In particular case of 2-box and 1-box models it is an orthomodular poset and orthomodular lattice respectively. We discuss relation of obtained results with the so-called Local Orthogonality principle. We argue that the non-classical properties o...

We provide mathematicaly rigorous justification of using term "probability" in connection to the so called non-signalling theories,known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-sig...

In the quantum logic framework we show that the no-signaling box model is a particular type of tensor product of the logics of single boxes. Such notion of tensor product is too strong to apply in the category of logics of quantum mechanical systems. Consequently, we show that the no-signaling box models cannot be considered as generalizations of q...

The premise of this note is the following observation: the formalism of Bohrification is a natural place for the interpretation of general non-signalling theories. We demonstrate it through an analysis of so-called box-worlds, a popular framework for the discussion of systems exhibiting super-quantum correlations. In particular, we show that non-si...

We construct a new set of generalized coherent states, the electron-hole coherent states, for a (quasi-)spin particle on the infinite line. The definition is inspired by applications to the Bogoliubov-de Gennes equations where the quasi-spin refers to electron- and hole-like components of electronic excitations in a superconductor. Electron-hole co...

Using a quantum logic approach we analyze the structure of the so-called non-signaling theories respecting relativistic causality, but allowing correlations violating bounds imposed by quantum mechanics such as CHSH inequality. We discuss the relations among such theories, quantum mechanics, and classical physics. Our main result is the constructio...

We study mechanisms that allow one to synchronize the quantum phase of two qubits relative to a fixed basis. Starting from one qubit in a fixed reference state and the other in an unknown state, we find that contrary to the impossibility of perfect quantum cloning, the quantum-phase can be synchronized perfectly through a joined unitary operation....

We show how to reduce the nonlinear Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, to a hierarchy of matrix Riccati equations using the cominuscule induction. The construction works for all reductive Lie algebras with no simple factors of type G2, F4 or E8. A corresponding hierarchy of...

Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure Fermionic Gaussian states, the corres...

For several types of correlations: mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions and non-Gaussian correlations in fermionic systems we estimate the fraction of non-correlated states among the density matrices with the same spectra. We prove that for the pu...

We analyze form the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geom...

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to th...

We show that the non-linear autonomus Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. The result generalizes our previous one concerning the complex Lie algebra of the special l...

The recent paper by Hul et al. (Phys. Rev. Lett. 109, 040402 (2012), see Ref. [7]) addresses an important mathematical problem whether scattering properties of wave systems are uniquely connected to their shapes? The analysis of the isoscattering microwave networks presented in this paper indicates a negative answer to this question. In this paper...

Extremal spacings between eigenphases of random unitary matrices of size N pertaining to circular ensembles are investigated. Explicit probability distributions for the minimal spacing for various ensembles are derived for N=4. We study ensembles of tensor product of k random unitary matrices of size n which describe independent evolution of a comp...

We give a general solution to the question when the convex hulls of orbits of quantum states on a finite-dimensional Hilbert space under unitary actions of a compact group have a non-empty interior in the surrounding space of all density states. The same approach can be applied to study convex combinations of quantum channels. The importance of bot...

A scheme to generate long-range spin-spin interactions between three-level ions in a chain is presented, providing a feasible experimental route to the rich physics of well-known SU(3) models. In particular, we demonstrate different signatures of quantum chaos which can be controlled and observed in experiments with trapped ions.

The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in pa...

We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons and fermions. In each case properties of an entanglement measures are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entangl...

DOI:https://doi.org/10.1103/PhysRevA.87.029904

Nonlocal properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix U equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N2) we find its asymptotic behavior...

We introduce detector-level entanglement, a unified entanglement concept for identical particles that takes into account the possible deletion of many-particle which-way information through the detection process. The concept implies a measure for the effective indistinguishability of the particles, which is controlled by the measurement setup and w...

Using techniques from symplectic geometry, we prove that a pure state of three qubits is up to local unitaries uniquely determined by its one-particle reduced density matrices exactly when their ordered spectra belong to the boundary of the, so called, Kirwan polytope. Otherwise, the states with given reduced density matrices are parameterized, up...

We present a general algorithm for finding all classes of pure multiparticle states equivalent under stochastic local operations and classical communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also...

The famous question of Mark Kac "Can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator can be legitimately extended to scattering systems. In the modified version one asks whether the geometry of a vibrating system can be determined by scatteri...

We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map on coadjoint orbits of local transformations (unitary transformations which do not change entanglement), thus the geometry of coadjoint orbits provides a partial c...

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.

We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical-half-numerical method that allows to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT ent...

Majorana representation of quantum states by a constellation of n 'stars' (points on the sphere) can be used to describe any pure state of a simple system of dimension n+1 or a permutation symmetric pure state of a composite system consisting of n qubits. We analyze the variance of the distribution of the stars, which can serve as a measure of the...

We elaborate the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. The entanglement is characterized in terms of generalized Segre maps, supplementing thus an algebraic approach to the problem by a more geometric point of view.

We give a criterion of classicality for mixed states in terms of expectation values of a quantum observable. Using group representation theory we identify all cases when the criterion can be computed exactly in terms of the spectrum of a single operator.

We present a description of locally equivalent states in terms of symplectic geometry. Using the moment map between local orbits in the space of states and coadjoint orbits of the local unitary group we reduce the problem of local unitary equivalence to an easy part consisting of identifying the proper coadjoint orbit and a harder problem of the ge...

The exact formulae are given for the total intensity and spectrum of resonance fluorescence for a driving laser field of gaussian lineshape.

We analyze the concept of entanglement for multipartite system with bosonic and fermionic constituents and its generalization to systems with arbitrary parastatistics. We use the representation theory of symmetry groups to formulate a unified approach to this problem in terms of simple tensors with appropriate symmetry. For an arbitrary parastatist...

We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions.

We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using Kostant-Sternberg theorem, we show that separable states form a unique Kaehler orbit, whereas orbi...

We give a universal recipe for constructing nonlinear entanglement witnesses able to detect non-classical correlations in arbitrary systems of distinguishable and/or identical particles for an arbitrary number of constituents. The constructed witnesses are expressed in terms of expectation values of observables. As such they are, at least in princi...

We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.

We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, “most classical” can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons,...

We study extremality in various sets of states that have positive partial transposes. One of the tools we use for this purpose is the recently formulated criterion allowing to judge if a given state is extremal in the set of PPT states. First we investigate qubit–ququart states and show that the only candidates for extremal PPT entangled states (PP...

We consider transformation maps on the space of states which are symmetries in the sense of Wigner. Due to the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body, to be able to contribute to the classification of positi...

We investigate the parameter dynamics of eigenvalues of Hamiltonians (‘level dynamics’) defined on symmetric spaces relevant to condensed matter and particle physics. In particular we: (1) identify the appropriate reduced manifold on which the motion takes place, (2) identify the correct Poisson structure ensuring the Hamiltonian character of the r...

DOI:https://doi.org/10.1103/PhysRevE.78.019904

The mechanism for entanglement of two flux qubits each interacting with a single mode electromagnetic field is discussed. By performing a Bell state measurement (BSM) on photons we find the two qubits in an entangled state depending on the system parameters. We discuss the results for two initial states and take into consideration the influence of...

Relations between states and maps, which are known for quantum systems in finitedimensional Hilbert spaces, are formulated rigorously in geometrical terms with no use of coordinate (matrix) interpretation. In a tensor product realization they are represented simply by a permutation of factors. This leads to natural generalizations for infinite-dime...

We give a simple method of analysing the "motion" of (quasi)-energy levels of nonintegrable quantum systems when a nonintegrability parameter is changed. In both cases of autonomous systems and periodically kicked ones it offers a particularly simple explanation of the Hamiltonian character and complete integrability of the resulting equations of "...

We reduce the question whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement a...

We developed an approach to a quantitative characterization of entanglementproperties of, possibly mixed, bi- and multipartite quantum states of arbitraryfinite dimension. Particular emphasis was given to: 1) the derivation ofreliable estimates which allow for an efficient evaluation of entanglementmeasures, 2) construction of measures of entanglem...

We address several problems concerning the geometry of the space of Hermitian operators on a finite-dimensional Hilbert space, in particular the geometry of the space of density states and canonical group actions on it. For quantum composite systems we discuss and give examples of entanglement measures.