arXiv subjects

Hans-Juergen Sommers

Publications and source records attributed to Hans-Juergen Sommers.

16 recordsLinked to original sources

arXiv2010

Truncations of Random Orthogonal Matrices

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ 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 non--orthogonality, $M/N=$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.

cond-mat.stat-mech↗

arXiv2010

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. Its radius depends on the number of the Kraus measurement operators, and determines the speed with which an arbitrary initial state converges to the unique invariant state. These spectral properties are characteristic of an ensemble of random quantum maps, which in turn can be described by an ensemble of real random Ginibre matrices. This will be proven in the limit of large dimension.

quant-ph↗

arXiv2009

Exact Fidelity and Full Fidelity Statistics in Regular and Chaotic Surroundings

For a prepared state exact expressions for the time dependent mean fidelity as well as for the mean inverse paricipation ratio are obtained analytically. The distribution function of fidelity in the long time limit and of inverse participation ratio are studied numerically and analytically. Surprising features like fidelity revival and enhanced non--ergodicity are observed. The role of the coupling coefficients and of complexity of background is studied as well.

cond-mat.stat-mech↗

arXiv2009

Random Bures mixed states and the distribution of their purity

Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the Bures measure is proposed. This procedure may serve as an initial step in performing Bayesian approach to quantum state estimation based on the Bures prior. We study the distribution of purity of random mixed states. The moments of the distribution of purity are determined for quantum states generated with respect to the Bures measure. This calculation serves as an exemplary application of the "deform-and-study" approach based on ideas of integrability theory. It is shown that Painlevé equation appeared as a part of the presented theory.

cond-mat.stat-mech↗

arXiv2009

Schur function averages for the real Ginibre ensemble

We derive an explicit simple formula for expectations of all Schur functions in the real Ginibre ensemble. It is a positive integer for all entries of the partition even and zero otherwise. The result can be used to determine the average of any analytic series of elementary symmetric functions by Schur function expansion.

math-ph↗

arXiv2007

Subnormalized states and trace-nonincreasing maps

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M_N induced by partial trace of mixed quantum states distributed uniformly with respect to HS-measure in $M_{N^2}$.

quant-ph↗

arXiv2006

Distribution of G-concurrence of random pure states

Average entanglement of random pure states of an N x N composite system is analyzed. We compute the average value of the determinant D of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distribution P(D). Similar results are obtained for the rescaled N-th root of the determinant, called G-concurrence. We show that in the limit $N\to\infty$ this quantity becomes concentrated at a single point G=1/e. The position of the concentration point changes if one consider an arbitrary N x K bipartite system, in the joint limit $N,K\to\infty$, K/N fixed.

quant-ph↗

arXiv2005

Bures volume of the set of mixed quantum states

We compute the volume of the N^2-1 dimensional set M_N of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N^2-1 dimensional hyper-halfsphere of radius 1/2. For N=2 we obtain the volume of the Uhlmann 3-D hemisphere, embedded in R^4. We find also the area of the boundary of the set M_N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures-Hall normalization constants is derived for an arbitrary N.

quant-ph↗

arXiv2004

Statistical properties of random density matrices

Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according to the Bures measure. The eigenvalues of the random density matrices are analyzed: we derive the eigenvalue distribution for the Bures ensemble which is shown to be broader then the quarter--circle distribution characteristic of the Hilbert--Schmidt ensemble. For measures induced by partial tracing over the environment we compute exactly the two-point eigenvalue correlation function.

quant-ph↗

arXiv2003

Hilbert--Schmidt volume of the set of mixed quantum states

We compute the volume of the convex N^2-1 dimensional set M_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 an information about the complex structure of M_N. Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step we analyze volumes of the unitary and orthogonal groups and of the flag manifolds.

quant-ph↗

arXiv2001

Distribution of proper delay times in quantum chaotic scattering: A crossover from ideal to weak coupling

The probability distribution of the proper delay times during scattering on a chaotic system is derived in the framework of the random matrix approach and the supersymmetry method. The result obtained is valid for an arbitrary number of scattering channels as well as arbitrary coupling to the energy continuum. The case of statistically equivalent channels is studied in detail. In particular, the semiclassical limit of infinite number of weak channels is paid appreciable attention.

cond-mat.dis-nn↗

arXiv2001

Induced measures in the space of mixed quantum states

We analyze 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 x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x 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 $N \times N$ pure states behaves as lnN-1/2.

quant-ph↗

arXiv2001

Reducing nonideal to ideal coupling in random matrix description of chaotic scattering: Application to the time-delay problem

We write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with M statistically equivalent channels at nonideal coupling to that for ideal coupling. Unfolding the phases by their local density leads to universality of their local fluctuations for large M. A relation between the partial time delays and diagonal matrix elements of the Wigner-Smith matrix is revealed for ideal coupling. This helped us in deriving the joint probability distribution of partial time delays and the distribution of the Wigner time delay.

cond-mat.dis-nn↗

arXiv1999

Truncations of random unitary matrices

We analyze 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 derive analytically the joint density of eigenvalues from which easily all correlation functions are obtained. For N-M fixed and N--> infinity the universal resonance-width distribution with N-M open channels is recovered.

chao-dyn↗

arXiv1999

Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices

We consider the spectral form factor of random unitary matrices as well as of Floquet matrices of kicked tops. For a typical matrix the time dependence of the form factor looks erratic; only after a local time average over a suitably large time window does a systematic time dependence become manifest. For matrices drawn from the circular unitary ensemble we prove ergodicity: In the limits of large matrix dimension and large time window the local time average has vanishingly small ensemble fluctuations and may be identified with the ensemble average. By numerically diagonalizing Floquet matrices of kicked tops with a globally chaotic classical limit we find the same ergodicity. As a byproduct we find that the traces of random matrices from the circular ensembles behave very much like independent Gaussian random numbers. Again, Floquet matrices of chaotic tops share that universal behavior. It becomes clear that the form factor of chaotic dynamical systems can be fully faithful to random-matrix theory, not only in its locally time-averaged systematic time dependence but also in its fluctuations.

chao-dyn↗

arXiv1996

Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane

We consider an ensemble of large non-Hermitian random matrices of the form $\hat{H}+i\hat{A}_s$, where $\hat{H}$ and $\hat{A}_s$ are Hermitian statistically independent random $N\times N$ matrices. We demonstrate the existence of a new nontrivial regime of weak non-Hermiticity characterized by the condition that the average of $N\mbox{Tr} \hat{A}_s^2$ is of the same order as that of $\mbox{Tr} \hat{H}^2 $ when $N\to \infty$. We find explicitly the density of complex eigenvalues for this regime in the limit of infinite matrix dimension. The density determines the eigenvalue distribution in the crossover regime between random Hermitian matrices whose real eigenvalues are distributed according to the Wigner semi-circle law and random complex matrices whose eigenvalues are distributed in the complex plane according to the so-called ``elliptic law''.

cond-mat↗