Karol Zyczkowski

Volume of the set of separable states

A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

Negativity of the Wigner function as an indicator of non-classicality

A measure of non-classicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyse this quantity for Fock states, squeezed displaced Fock states and cat-like states defined as coherent superposition of two Gaussian wavepackets.

Induced measures in the space of mixed quantum states

We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N×K composite system, induces a unique measure in the space of N×N mixed states (or in the space of K×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N×N pure states behaves as lnN-1/2.

Random unitary matrices

Methods of constructing random matrices typical of circular unitary and circular orthogonal ensembles are presented. We generate numerically random unitary matrices and show that the statistical properties of their spectra (level-spacing distribution, number variance) and eigenvectors (entropy, participation ratio, eigenvector statistics) confer to the predictions of the random-matrix theory, for both CUE and COE.

Dynamics of quantum entanglement

A model of discrete dynamics of entanglement of a bipartite quantum state is considered. It involves a global unitary dynamics of the system and periodic actions of local bistochastic or decaying channel. For initially pure states the decay of entanglement is accompanied by an increase of von Neumann entropy of the system. We observe and discuss revivals of entanglement due to unitary interaction of subsystems. For some mixed states having different marginal entropies of the subsystems we find an asymmetry in speed of entanglement decay. The entanglement of these states decreases faster, if the depolarizing channel acts on the ‘‘classical’’ subsystem, characterized by smaller marginal entropy.

On the volume of the set of mixed entangled states. 2.

A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural measure is not entangled (is separable). We find analytical lower and upper bounds for this quantity. Numerical calculations give P = 0.632 for N=4 and P=0.384 for N=6, and indicate that P decreases exponentially with N. Analysis of a conditional measure of separability under the condition of fixed purity shows a clear dualism between purity and separability: entanglement is typical for pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable.

Truncations of random unitary matrices

We analyse properties of non-Hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N >M , distributed according to the Haar measure. In this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle. In the limit of large matrices, this ensemble is characterized by the ratio M /N . For the truncated CUE we analytically derive the joint density of eigenvalues and all correlation functions. In the strongly non-unitary case universal Ginibre behaviour is found. For N -M fixed and N to the universal resonance-width distribution with N -M open channels is recovered.

Composed ensembles of random unitary matrices

Composed ensembles of random unitary matrices are defined via products of matrices, each pertaining to a given canonical circular ensemble of Dyson. We investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance. We also discuss the methods of generating random matrices distributed according to invariant Haar measure on the orthogonal and unitary group.

Hilbert–Schmidt volume of the set of mixed quantum states

We compute the volume of the convex (N2 − 1)-dimensional set N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides information about the structure of N. Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step, we analyse volumes of the unitary and orthogonal groups and of the flag manifolds.

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. Using similar techniques, involving the Mellin transform and the Meijer G function, we find exact expressions for the Raney probability distributions, the moments of which are given by a two-parameter generalization of the Fuss-Catalan numbers. These distributions can also be considered as a two-parameter generalization of the Wigner semicircle law.