Entangled States Are Harder to Transfer than Product States.

The distribution of entangled states is a key task of utmost importance for many quantum information processing protocols. A commonly adopted setup for distributing quantum states envisages the creation of the state in one location, which is then sent to (possibly different) distant receivers through some quantum channels. While it is undoubted and, perhaps, intuitively expected that the distribution of entangled quantum states is less efficient than that of product states, a thorough quantification of this inefficiency (namely, of the difference between the quantum-state transfer fidelity for entangled and factorized states) has not been performed.

Geometric structure of thermal cones.

The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future, and incomparable regions. In this work, we analyze the structure of the resulting thermal cones, i.

Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices.

A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρ^{AB} such that both of its reduced density matrices ρ^{A} and ρ^{B} of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρ^{A} and ρ^{B}, which is bounded from below by the rescaled Bures distance and from above by the root infidelity.

Transition from order to chaos in reduced quantum dynamics.

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 is chaotic, while varying r we find the universal period-doubling behavior characteristic to one-dimensional maps: period-2 dynamics starts at r_{1}≈0.

Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem.

The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six.

Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs.

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2.

Universal Spectra of Random Lindblad Operators.

To understand the typical dynamics of an open quantum system in continuous time, we introduce an ensemble of random Lindblad operators, which generate completely positive Markovian evolution in the space of the density matrices. The spectral properties of these operators, including the shape of the eigenvalue distribution in the complex plane, are evaluated by using methods of free probabilities and explained with non-Hermitian random matrix models. We also demonstrate the universality of the spectral features.

Entanglement of Three-Qubit Random Pure States.

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) .

Spectral density of generalized Wishart matrices and free multiplicative convolution.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3.

Jarzynski equality for quantum stochastic maps.

Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e.

Extremal spacings between eigenphases of random unitary matrices and their tensor products.

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 composite quantum system consisting of k subsystems.

Better late than never: information retrieval from black holes.

We show that, in order to preserve the equivalence principle until late times in unitarily evaporating black holes, the thermodynamic entropy of a black hole must be primarily entropy of entanglement across the event horizon. For such black holes, we show that the information entering a black hole becomes encoded in correlations within a tripartite quantum state, the quantum analogue of a one-time pad, and is only decoded into the outgoing radiation very late in the evaporation. This behavior generically describes the unitary evaporation of highly entangled black holes and requires no specially designed evolution.

Collective uncertainty entanglement test.

For a given pure state of a composite quantum system we analyze the product of its projections onto a set of locally orthogonal separable pure states. We derive a bound for this product analogous to the entropic uncertainty relations. For bipartite systems the bound is saturated for maximally entangled states and it allows us to construct a family of entanglement measures, we shall call collectibility.

Product of Ginibre matrices: Fuss-Catalan and Raney distributions.

Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distributions P(s)(x), such that their moments are equal to the Fuss-Catalan numbers of order s. We find a representation of the Fuss-Catalan distributions P(s)(x) in terms of a combination of s hypergeometric functions of the type (s)F(s-1). The explicit formula derived here is exact for an arbitrary positive integer s, and for s=1 it reduces to the Marchenko-Pastur distribution.

Truncations of random orthogonal matrices.

Statistical properties of nonsymmetric real random matrices of size M, obtained as truncations of random orthogonal N×N matrices, are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong nonorthogonality, M/N=const, the behavior typical to real Ginibre ensemble is found.

Universal bounds for the Holevo quantity, coherent information, and the Jensen-Shannon divergence.

The mutual information between the sender of a classical message encoded in quantum carriers and a receiver is fundamentally limited by the Holevo quantity. Using strong subadditivity of entropy, we prove that the Holevo quantity is not larger than an exchange entropy. This implies an upper bound for coherent information.

Universality of spectra for interacting quantum chaotic systems.

We analyze a model quantum dynamical system subjected to periodic interaction with an environment, which can describe quantum measurements. Under the condition of strong classical chaos and strong decoherence due to large coupling with the measurement device, the spectra of the evolution operator exhibit an universal behavior. A generic spectrum consists of a single eigenvalue equal to unity, which corresponds to the invariant state of the system, while all other eigenvalues are contained in a disk in the complex plane.

Experimental simulation of quantum graphs by microwave networks.

We present the results of experimental and theoretical study of irregular, tetrahedral microwave networks consisting of coaxial cables (annular waveguides) connected by T joints. The spectra of the networks were measured in the frequency range 0.0001-16 GHz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity Delta(3) (L).

Quantum iterated function systems.

An iterated function system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum IFS (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting, a QIFS consists of completely positive maps acting in the space of density operators.

Quantization of classical maps with tunable Ruelle-Pollicott resonances.

We investigate the correspondence between the decay of correlation in classical systems, governed by Ruelle-Pollicott resonances, and the properties of the corresponding quantum systems. For this purpose we construct classical dynamics with controllable resonances together with their quantum counterparts. As an application of such tunable resonances we reveal the role of Ruelle-Pollicott resonances for the localization properties of quantum energy eigenstates.

Entropy computing via integration over fractal measures.

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems, one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems.

Irreversible quantum baker map.

We propose a generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be quantized by means of a unitary Floquet operator. A corresponding quantum system is constructed as a completely positive map acting in the space of density matrices.