Complete measurements of quantum observables

We define a complete measurement of a quantum observable (POVM) as a measurement of the maximally refined version of the POVM. Complete measurements give information from the multiplicities of the measurement outcomes and can be viewed as state preparation procedures. We show that any POVM can be measured completely by using sequential measurements or maximally refinable instruments. Moreover, the ancillary space of a complete measurement can be chosen to be minimal.Comment: Based on talk given in CEQIP 2012 conferenc

The modern tools of quantum mechanics (A tutorial on quantum states, measurements, and operations)

This tutorial is devoted to review the modern tools of quantum mechanics, which are suitable to describe states, measurements, and operations of realistic, not isolated, systems in interaction with their environment, and with any kind of measuring and processing devices. We underline the central role of the Born rule and and illustrate how the notion of density operator naturally emerges, together the concept of purification of a mixed state. In reexamining the postulates of standard quantum measurement theory, we investigate how they may formally generalized, going beyond the description in terms of selfadjoint operators and projective measurements, and how this leads to the introduction of generalized measurements, probability operator-valued measures (POVM) and detection operators. We then state and prove the Naimark theorem, which elucidates the connections between generalized and standard measurements and illustrates how a generalized measurement may be physically implemented. The "impossibility" of a joint measurement of two non commuting observables is revisited and its canonical implementations as a generalized measurement is described in some details. Finally, we address the basic properties, usually captured by the request of unitarity, that a map transforming quantum states into quantum states should satisfy to be physically admissible, and introduce the notion of complete positivity (CP). We then state and prove the Stinespring/Kraus-Choi-Sudarshan dilation theorem and elucidate the connections between the CP-maps description of quantum operations, together with their operator-sum representation, and the customary unitary description of quantum evolution. We also address transposition as an example of positive map which is not completely positive, and provide some examples of generalized measurements and quantum operations.Comment: Tutorial. 26 pages, 1 figure. Published in a special issue of EPJ - ST devoted to the memory of Federico Casagrand

Constraints for quantum logic arising from conservation laws and field fluctuations

We explore the connections between the constraints on the precision of quantum logical operations that arise from a conservation law, and those arising from quantum field fluctuations. We show that the conservation-law based constraints apply in a number of situations of experimental interest, such as Raman excitations, and atoms in free space interacting with the multimode vacuum. We also show that for these systems, and for states with a sufficiently large photon number, the conservation-law based constraint represents an ultimate limit closely related to the fluctuations in the quantum field phase.Comment: To appear in J. Opt. B: Quantum Semiclass. Opt., special issue on quantum contro

Instabilities in Zakharov Equations for Laser Propagation in a Plasma

F.Linares, G.Ponce, J-C.Saut have proved that a non-fully dispersive Zakharov system arising in the study of Laser-plasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields non-vanishing at infinity, the system develops strong instabilities of Hadamard's type, implying that the Cauchy problem is strongly ill-posed

Scattering and small data completeness for the critical nonlinear Schroediger equation

We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity

Measurement schemes for the spin quadratures on an ensemble of atoms

We consider how to measure collective spin states of an atomic ensemble based on the recent multi-pass approaches for quantum interface between light and atoms. We find that a scheme with two passages of a light pulse through the atomic ensemble is efficient to implement the homodyne tomography of the spin state. Thereby, we propose to utilize optical pulses as a phase-shifter that rotates the quadrature of the spins. This method substantially simplifies the geometry of experimental schemes.Comment: 4pages 2 figure

Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space dimensions with mass resonance, exhibits long range scattering phenomena. Modified wave operators and solutions are constructed for these equations. We also show that the modified wave operators can be chosen such that they linearize the non-linear representation of the Poincar\'e group defined by the NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy