On circulant states with positive partial transpose

We construct a large class of quantum "d x d" states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure - that is way we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well known examples of PPT states from the literature and gives rise to a huge family of completely new states.Comment: 15 pages; minor correction

Rotationally invariant multipartite states

We construct a class of multipartite states possessing rotational SO(3) symmetry -- these are states of K spin-j_A particles and K spin-j_B particles. The construction of symmetric states follows our two recent papers devoted to unitary and orthogonal multipartite symmetry. We study basic properties of multipartite SO(3) symmetric states: separability criteria and multi-PPT conditions.Comment: 18 pages; new reference

On partially entanglement breaking channels

Using well known duality between quantum maps and states of composite systems we introduce the notion of Schmidt number of a quantum channel. It enables one to define classes of quantum channels which partially break quantum entanglement. These classes generalize the well known class of entanglement breaking channels.Comment: 9 page

Wigner function for damped systems

Both classical and quantum damped systems give rise to complex spectra and corresponding resonant states. We investigate how resonant states, which do not belong to the Hilbert space, fit the phase space formulation of quantum mechanics. It turns out that one may construct out of a pair of resonant states an analog of the stationary Wigner function.Comment: 18 page

The observables of a dissipative quantum system

A time-dependent product is introduced between the observables of a dissipative quantum system, that accounts for the effects of dissipation on observables and commutators. In the $t \to \infty$ limit this yields a contracted algebra. The general ideas are corroborated by a few explicit examples.Comment: 4 page

Kelvin-Helmholtz instability of relativistic jets - the transition from linear to nonlinear regime

The observed wiggles and knots in astrophysical jets as well as the curvilinear motion of radio emitting features are frequently interpreted as signatures of the Kelvin-Helmholtz (KH) instability (eg. Hardee 1987). We investigate the KH instability of a hydrodynamic jet composed of a relativistic gas, surrounded by a nonrelativistic external medium and moving with a relativistic bulk speed. We show basic nonlinear effects, which become important for a finite amplitude KH modes. Since the KH instability in supersonic jets involves acoustic waves over-reflected on jet boundaries, the basic nonlinear effect relies on the steepening of the acoustic wave fronts, leading to the formation of shocks. It turns our that the shocks appear predominantly in the external nonrelativistic gas, while the internal acoustic waves remain linear for a much longer time. In addition, the external medium "hardens" as soon as the boundary oscillation velocity becomes comparable to the external sound speed. On the other hand, the amplification of internal waves due to the over-reflection is limited by a nonlinearity of the Lorentz $\gamma$ factor. This implies that the sidereal oscillations of the jet boundary, resulting from the K-H instability, are limited to very small amplitudes comparable to a fraction of the jet radius.Comment: TeX, 5 pages, no figures, lecproc.cmm included, To appear in Proceedings of ``Relativistic jets in AGNs'', Krakow, Poland, 27-30 May 1997, M.Ostrowski, M.Sikora, G.Madejski, M. Begelman (eds.

A class of Bell diagonal states and entanglement witnesses

We analyze special class of bipartite states - so called Bell diagonal states. In particular we provide new examples of bound entangled Bell diagonal states and construct the class of entanglement witnesses diagonal in the magic basis.Comment: 17 page

Long-time memory in non-Markovian evolutions

If the dynamics of an open quantum systems is non-Markovian, its {asymptotic} state strongly depends on the initial conditions, even if the dynamics possesses an {invariant} state. This is the very essence of memory effects. In particular, the {asymptotic} state can remember and partially preserve its initial entanglement. Interestingly, even if the non-Markovian evolution relaxes to an equilibrium state, this state needs not be invariant. Therefore, the non-invariance of equilibrium becomes a clear sign of non-Markovianity.Comment: 6 page

Memory in a nonlocally damped oscillator

We analyze the new equation of motion for the damped oscillator. It differs from the standard one by a damping term which is nonlocal in time and hence it gives rise to a system with memory. Both classical and quantum analysis is performed. The characteristic feature of this nonlocal system is that it breaks local composition low for the classical Hamiltonian dynamics and the corresponding quantum propagator.Comment: minor corrections added; title change

Optimal entanglement witnesses for two qutrits

We provide a proof that entanglement witnesses considered recently in [D. Chru\'sci\'nski, F.A. Wudarski, arXiv:1105.4821] are optimal.Comment: 4 page