Quantum measurements with prescribed symmetry

We introduce a method to determine whether a given generalised quantum measurement is isolated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measurements (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively.Comment: 10 pages, 2 figure

CP^n, or, entanglement illustrated

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP^3 and give a few new results concerning them.Comment: 28 pages, 9 figure

On the comparison of volumes of quantum states

This paper aims to study the \a-volume of \cK, an arbitrary subset of the set of $N\times N$ density matrices. The \a-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the \a-volume of \cK in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the \a-volume are also established. We employ our results to obtain bounds for the \a-volume of the sets of separable quantum states and of states with positive partial transpose (PPT). Hence, our asymptotic results provide answers for questions listed on page 9 in \cite{K. Zyczkowski1998} for large $N$ in the sense of \a-volume. \vskip 3mm PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.M

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width $\ell$ in the common boundary. We show that such a system has at least one bound state for any $\ell>0$. We find the corresponding eigenvalues and eigenfunctions numerically using the mode--matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase--shift spacing statistics shows that in distinction to closed pseudo--integrable billiards, the present system is essentially non--chaotic. Finally, we illustrate time evolution of wave packets in the present model.Comment: LaTeX text file with 12 ps figure

On the volume of the set of mixed entangled states II

The problem of of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices describing N--dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N > 6, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N=8 and to estimate the volume of the bound entangled states in this case.Comment: 14 pages in Latex, Revtex + epsf; 7 figures in .ps included (one new figure in the revised version, several minor changes

On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert-Schmidt (Euclidean) distance. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system.Comment: 10 pages, 1 figure; ver. 2 - minor changes, new proof of lemma

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 $M_{N^2}$.Comment: LaTeX, 21 pages, 4 Encapsuled PostScript figures, 1 tabl

Geometry of entangled states

Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general K x M problem and characterize the set of effectively different states (which cannot be related by local transformations). Thus we generalize earlier results obtained for the simplest 2 x 2 system, which lead to a stratification of the 6D set of N=4 pure states. We define the concept of absolutely separable states, for which all globally equivalent states are separable.Comment: 16 latex pages, 4 figures in epsf, minor corrections, references updated, to appear in Phys. Rev.

Geometric measure of entanglement and applications to bipartite and multipartite quantum states

The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings (see Shimony 1995 and Barnum and Linden 2001), is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary two-qubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of three-qubit GHZ, W and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.Comment: 13 pages, 11 figures, this is a combination of three previous manuscripts (quant-ph/0212030, quant-ph/0303079, and quant-ph/0303158) made more extensive and coherent. To appear in PR