Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry

A PT-symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of PT-symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of PT-symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non-PT-symmetric configurations. This signifies the existence of non-PT-symmetric CPA-lasers.Comment: 11 pages, 3 figures, 1 table, accepted for publication in J. Phys.

Lax-Phillips scattering theory for PT-symmetric \rho-perturbed operators

The S-matrices corresponding to PT-symmetric \rho-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory

A naked singularity stable under scalar field perturbations

We prove the stability of a spacetime with a naked singularity under scalar field perturbations, where the perturbations are regular at the singularity. This spacetime, found by Janis, Newman and Winicour, and independently by Wyman, is sourced by a massless scalar field and also arises as a certain limit of a class of charged dilatonic solutions in string theory. This stability result opens up specific questions for investigation related to the cosmic censorship conjecture and the mechanism by which it is implemented in nature.Comment: 19 pages, version to appear in IJMPD, references adde

Refined algebraic quantisation with the triangular subgroup of SL(2,R)

We investigate refined algebraic quantisation with group averaging in a constrained Hamiltonian system whose gauge group is the connected component of the lower triangular subgroup of SL(2,R). The unreduced phase space is T^*R^{p+q} with p>0 and q>0, and the system has a distinguished classical o(p,q) observable algebra. Group averaging with the geometric average of the right and left invariant measures, invariant under the group inverse, yields a Hilbert space that carries a maximally degenerate principal unitary series representation of O(p,q). The representation is nontrivial iff (p,q) is not (1,1), which is also the condition for the classical reduced phase space to be a symplectic manifold up to a singular subset of measure zero. We present a detailed comparison to an algebraic quantisation that imposes the constraints in the sense H_a Psi = 0 and postulates self-adjointness of the o(p,q) observables. Under certain technical assumptions that parallel those of the group averaging theory, this algebraic quantisation gives no quantum theory when (p,q) = (1,2) or (2,1), or when p>1, q>1 and p+q is odd.Comment: 30 pages. LaTeX with amsfonts, amsmath, amssymb. (v4: Typos corrected. Published version.

Every Hilbert space frame has a Naimark complement

Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory. We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception. So these complements can be used for equiangular frames, RIP property, fusion frames etc. Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.Comment: Changes after Refereein

Entanglement cost of generalised measurements

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_min(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any d-dimensional POVM by E_min(M)/log(d), we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.Comment: 20 pages, plain late

The canonical Naimark extension for the Pauli quantum roulette wheel

We address measurement schemes where certain observables are chosen at random within a set of non-degenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a given probability to be measured, and the statistics of this generalized measurement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable is chosen at random, e.g. according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measurements of Pauli matrices and we explicitly evaluate their canonical Naimark extensions, i.e. their implementation as indirect measurements involving an interaction scheme with a probe system. We thus provide a concrete model to realize the roulette without destroying the signal state, which can be measured again after the measurement, or can be transmitted. Finally, we apply our results to the description of Stern-Gerlach-like experiments on a two-level system.Comment: 8 pages, 2 figures, published on PRA with a different title (the arXiv one was too sexy

Every Hilbert space frame has a Naimark complement

Naimark complements for Hilbert space Parseval frames are one of the most fundamental and useful results in the field of frame theory. We will show that actually all Hilbert space frames have Naimark complements which possess all the usual properties for Naimark complements with one notable exception. So these complements can be used for equiangular frames, RIP property, fusion frames etc. Along the way, we will correct a mistake in a recent fusion frame paper where chordal distances for Naimark complements are computed incorrectly.Comment: Changes after Refereein