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Marek Kuś

Publications and source records attributed to Marek Kuś.

At least 37 records · Page 2Linked to original sources
arXiv2014

Classical simulation of fermionic linear optics augmented with noisy ancillas

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 corresponding computation scheme is classically simulable. We present an analytic characterisation of the set of convex-Gaussian states in the first non-trivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by De Melo et al. and to study in detail the geometrical properties of the set of convex-Gaussian states.

quant-ph↗
arXiv2013

Wei-Norman equations for classical groups

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 linear group. We show that it cannot be extended to all simple Lie algebras, in particular to the exceptional $G_2$ algebra.

math-ph↗
arXiv2013

A universal framework for entanglement detection

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 entanglement measure reduces to the well-known many particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a non trivial lower bound of thus obtained generalized concurrence.

math-ph↗
arXiv2013

Critical sets of the total variance of state detect all SLOCC entanglement classes

We present a general algorithm for finding all classes of pure multiparticle states equivalent under Stochastic Local Operations and Classsical 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 discuss the Morse indices of critical points which have the interpretation of the number of independent non-local perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.

math-ph↗
arXiv2013

Convexity of momentum map, Morse index, and quantum entanglement

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 geometric invariant theory (GIT) and momentum map geometry we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many `asymptotically' equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well defined measure of entanglement) of the state Var[v] attains maximum. We provide an algorithm for finding critical sets of Var[v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using the recent developments in the momentum map geometry known as Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

math-ph↗
arXiv2013

Quantum chaos in SU_3 models with trapped ions

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.

cond-mat.quant-gas↗
arXiv2013

Convex bodies of states and maps

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 both problems stems from the fact that, usually, only sets with non-vanishing volumes in the embedding spaces of all states or channels are of practical importance. For the group of local transformations on a bipartite system we characterize maximally entangled states by properties of a convex hull of orbits through them. We also compare two partial characteristics of convex bodies in terms of largest balls and maximum volume ellipsoids contained in them and show that, in general, they do not coincide. Separable states, mixed-unitary channels and k-entangled states are also considered as examples of our techniques.

math-ph↗
arXiv2013

Wei-Norman equations for a unitary evolution

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 particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.

math.CA↗
arXiv2012

When is a pure state of three qubits determined by its single-particle reduced density matrices?

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 to local unitary equivalence, by two real variables. Given inevitable experimental imprecisions, this means that already for three qubits a pure quantum state can never be reconstructed from single-particle tomography. We moreover show that knowledge of the reduced density matrices is always sufficient if one is given the additional promise that the quantum state is not convertible to the Greenberger--Horne--Zeilinger (GHZ) state by stochastic local operations and classical communication (SLOCC), and discuss generalizations of our results to an arbitary number of qubits.

quant-ph↗
arXiv2012

Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?

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 scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the, so called, transplantation relation.

quant-ph↗
arXiv2012

Bipartite entanglement, spherical actions and geometry of local unitary orbits

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 classification of different entanglement classes. To achieve the full classification a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions and distinguishable particles.

math-ph↗
arXiv2012

Entanglement of Identical Particles and the Detection Process

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 which quantifies the extent to which the (anti-)symmetrization of the wave-function impacts on physical observables. Initially indistinguishable particles can gain or loose entanglement on their transition to distinguishability, and their quantum statistical behavior depends on their initial entanglement. Our results show that entanglement cannot be attributed to a state of identical particles alone, but that the detection process has to be incorporated in the analysis.

quant-ph↗
arXiv2012

On detection of quasiclassical states

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.

math-ph↗
arXiv2011

Barycentric measure of quantum entanglement

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 degree of non-coherence for simple systems, or an entanglement measure for composed systems. Dynamics of the Majorana points induced by a unitary dynamics of the pure state is investigated.

quant-ph↗
arXiv2011

Geometry of the local equivalence of states

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 geometry of fibers of the moment map. We give a detailed analysis of the properties of orbits of "equally entangled states". In particular we show connections between certain symplectic properties of orbits such as their isotropy and coisotropy with effective criteria of local unitary equivalence.

math-ph↗
arXiv2010

Symplectic geometry of entanglement

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 orbits of entanglement states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement and we show how to calculate it for various multiparticle systems providing also simple criteria of separability. The presented method is general and can be applied also under different additional symmetry conditions stemming, eg. from the indistinguishability of particles.

math-ph↗
arXiv2009

Searching for extremal PPT entangled states

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 (PPTES) have ranks of the state and its partial transposition (5,5) or (5,6) (equivalently (6,5)). Then, examples of extremal states of (5,5) type and the so--called edge states of type (5,6) are provided. We also make an attempt to explore the set of PPT states with ranks (5,6). Finally, we discuss what are the possible configurations of ranks of density matrices and their respective partial transposition in general three-qubit and four-qubit symmetric states for which there may exist extremal entangled PPT states. For instance in the first case we show that the only possibilities are (4,4,4) and (4,4,5).

quant-ph↗