A simple minimax estimator for quantum states

Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. In this article, we review the concept of minimax state estimation, and use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.Comment: 26 pages, 3 figures. v2 contains minor improvements to the text, and an additional appendix on symmetric measurement

G+++ Invariant Formulation of Gravity and M-Theories: Exact BPS Solutions

We present a tentative formulation of theories of gravity with suitable matter content, including in particular pure gravity in D dimensions, the bosonic effective actions of M-theory and of the bosonic string, in terms of actions invariant under very-extended Kac-Moody algebras G+++. We conjecture that they host additional degrees of freedom not contained in the conventional theories. The actions are constructed in a recursive way from a level expansion for all very-extended algebras G+++. They constitute non-linear realisations on cosets, a priori unrelated to space-time, obtained from a modified Chevalley involution. Exact solutions are found for all G+++. They describe the algebraic properties of BPS extremal branes, Kaluza-Klein waves and Kaluza-Klein monopoles. They illustrate the generalisation to all G+++ invariant theories of the well-known duality properties of string theories by expressing duality as Weyl invariance in G+++. Space-time is expected to be generated dynamically. In the level decomposition of E8+++ = E11, one may indeed select an A10 representation of generators Pa which appears to engender space-time translations by inducing infinite towers of fields interpretable as field derivatives in space and time.Comment: Latex 45 pages, 1 figure. Discussion on pages 19 and 20 altered. Appendix B amplified. 4 footnotes added. 2 references added. Acknowledgments updated. Additional minor correction

On the Opening of Branes

We relate, in 10 and 11 dimensional supergravities, configurations of intersecting closed branes with vanishing binding energy to configurations where one of the branes opens and has its boundaries attached to the other. These boundaries are charged with respect to fields living on the closed brane. The latter hosts electric and magnetic charges stemming from dual pairs of open branes terminating on it. We show that charge conservation, gauge invariance and supersymmetry entirely determine these charges and these fields, which can be seen as Goldstone fields of broken supersymmetry. Open brane boundary charges can annihilate, restoring the zero binding energy configuration. This suggests emission of closed branes by branes, a generalization of closed string emission by D-branes. We comment on the relation of the Goldstone fields to matrix models approaches to M-theory.Comment: 13 pages, LaTeX, no figure

Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons

Quantum key distribution is on the verge of real world applications, where perfectly secure information can be distributed among multiple parties. Several quantum cryptographic protocols have been theoretically proposed and independently realized in different experimental conditions. Here, we develop an experimental platform based on high-dimensional orbital angular momentum states of single photons that enables implementation of multiple quantum key distribution protocols with a single experimental apparatus. Our versatile approach allows us to experimentally survey different classes of quantum key distribution techniques, such as the 1984 Bennett \& Brassard (BB84), tomographic protocols including the six-state and the Singapore protocol, and to investigate, for the first time, a recently introduced differential phase shift (Chau15) protocol using twisted photons. This enables us to experimentally compare the performance of these techniques and discuss their benefits and deficiencies in terms of noise tolerance in different dimensions.Comment: 13 pages, 4 figures, 1 tabl

Atoms and Quantum Dots With a Large Number of Electrons: the Ground State Energy

We compute the ground state energy of atoms and quantum dots with a large number N of electrons. Both systems are described by a non-relativistic Hamiltonian of electrons in a d-dimensional space. The electrons interact via the Coulomb potential. In the case of atoms (d=3), the electrons are attracted by the nucleus, via the Coulomb potential. In the case of quantum dots (d=2), the electrons are confined by an external potential, whose shape can be varied. We show that the dominant terms of the ground state energy are those given by a semiclassical Hartree-exchange energy, whose N to infinity limit corresponds to Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates oscillations in the ground state energy as a function of N. These oscillations reflect the dynamics of a classical particle moving in the presence of the Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in general in the case of dots, the motion contains a chaotic component. We compute the correlation effects. They appear at the order N ln N for atoms, in agreement with available data. For dots, they appear at the order N.Comment: 30 pages, 1 figur

Symmetric coupling of four spin-1/2 systems

We address the non-binary coupling of identical angular momenta based upon the representation theory for the symmetric group. A correspondence is pointed out between the complete set of commuting operators and the reference-frame-free subsystems. We provide a detailed analysis of the coupling of three and four spin-1/2 systems and discuss a symmetric coupling of four spin-1/2 systems.Comment: 20 pages, no figure

Source Vacuum Fluctuations of Black Hole Radiance

The emergence of Hawking radiation from vacuum fluctuations is analyzed in conventional field theories and their energy content is defined through the Aharonov weak value concept. These fluctuations travel in flat space-time and carry transplanckian energies sharply localized on cisplanckian distances. We argue that these features cannot accommodate gravitational nonlinearities. We suggest that the very emission of Hawking photons from tamed vacuum fluctuations requires the existence of an exploding set of massive fields. These considerations corroborate some conjectures of Susskind and may prove relevant for the back-reaction problem and for the unitarity issue.Comment: 33 pages, ULB-TH 03/94, 5 figures not included, available on request from F.E. (problem with truncation of long lines

Di-Antiquarks condensation in Color Superconductivity

Some consequences of a classical vector field (chromo-electromagnetic field) coupled to quarks, which undergo to superfluid and/or superconductive states with diquark / diantiquark condensation, are investigated. For this, one scalar field exchange is considered in the lines investigated by Pisarski and Rischke \cite{PISARSKI-RISCHKE} in the mean field approach. Some effects and possible consequences are discussed.Comment: Work presented at the 18th International COnference on Few Body, SP, Brazil, August 200