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Fractal structures appearing in solutions of certain quantum problems are investigated. We prove the previously announced results concerning the existence and properties of fractal states for the Schrödinger equation in the infinite one-dimensional well. In particular, we show that for this problem, there exist solutions in the form of fractal quantum carpets: the probability density P(x,t) forms a fractal surface with dimension Dxy, while its cross-sections Pt(x) and Px(t) typically form fractal graphs with dimensions Dx and Dt, respectively, where Dxy=2+Dx/2 and Dt=1+Dx/2 (almost everywhere). [ABSTRACT FROM AUTHOR]

A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

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. To this end, in this work, we consider n -independent amplitude-damping channels, acting in parallel, i.e., each, locally, on one part of an n -qubit state. We derive exact analytical results for the fidelity decrease, with respect to the case of product states, in the presence of entanglement in the initial state, for up to four qubits. Interestingly, we find that genuine multipartite entanglement has a more detrimental effect on the fidelity than two-qubit entanglement. Our results hint at the fact that, for larger n -qubit states, the difference in the average fidelity between product and entangled states increases with increasing single-qubit fidelity, thus making the latter a less trustworthy figure of merit.

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.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a d-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behavior based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorization-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit. Finally, we also generalize the construction of thermal cones to account for probabilistic transformations.

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. In the single-qubit case, we provide a semianalytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analog of the Wasserstein distance of the order of 2 on the set of density matrices. We introduce an associated measure of proximity of quantum states, called swap fidelity, and discuss its properties and applications in quantum machine learning.

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}≈0.3181, while the next bifurcation occurs at r_{2}≈0.5387. In parallel with period-4 oscillations observed for r≤r_{3}≈0.5672, we identify a secondary bifurcation diagram around r≈0.544, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until r≤r_{∞}∼0.578, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order. Finally, for r=1 the model reduces to the standard undamped chaotic kicked top.

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. This special state deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))_{6}, which saturates the Singleton bound and allows one to encode a six-level state into a triplet of such states.

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. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.

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. The notion of an ensemble of random generators of Markovian quantum evolution constitutes a step towards categorization of dissipative quantum chaos.

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 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.

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. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

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., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are applied for exemplary systems subjected to quantum channels acting on a finite-dimensional Hilbert space.

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. In the asymptotic case, as the total dimension N=n(k) becomes large, the nearest neighbor distribution P(s) becomes Poissonian, but statistics of extreme spacings P(s(min)) and P(s(max)) reveal certain deviations from the Poissonian behavior.

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. Our work suggests the existence of a matter-field sum rule for any fundamental theory.

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. In the case M=N-L with fixed L, a universal distribution of resonance widths is recovered.