Fundamental Limits on the Speed of Evolution of Quantum States

This paper reports on some new inequalities of Margolus-Levitin-Mandelstam-Tamm-type involving the speed of quantum evolution between two orthogonal pure states. The clear determinant of the qualitative behavior of this time scale is the statistics of the energy spectrum. An often-overlooked correspondence between the real-time behavior of a quantum system and the statistical mechanics of a transformed (imaginary-time) thermodynamic system appears promising as a source of qualitative insights into the quantum dynamics.Comment: 6 pages, 1 eps figur

Optimal transfer of an unknown state via a bipartite operation

A fundamental task in quantum information science is to transfer an unknown state from particle $A$ to particle $B$ (often in remote space locations) by using a bipartite quantum operation $\mathcal{E}^{AB}$. We suggest the power of $\mathcal{E}^{AB}$ for quantum state transfer (QST) to be the maximal average probability of QST over the initial states of particle $B$ and the identifications of the state vectors between $A$ and $B$. We find the QST power of a bipartite quantum operations satisfies four desired properties between two $d$-dimensional Hilbert spaces. When $A$ and $B$ are qubits, the analytical expressions of the QST power is given. In particular, we obtain the exact results of the QST power for a general two-qubit unitary transformation.Comment: 6 pages, 1 figur

Density of states of a two-dimensional electron gas in a non-quantizing magnetic field

We study local density of electron states of a two-dimentional conductor with a smooth disorder potential in a non-quantizing magnetic field, which does not cause the standart de Haas-van Alphen oscillations. It is found, that despite the influence of such ``classical'' magnetic field on the average electron density of states (DOS) is negligibly small, it does produce a significant effect on the DOS correlations. The corresponding correlation function exhibits oscillations with the characteristic period of cyclotron quantum $\hbar\omega_c$.Comment: 7 pages, including 3 figure

Effects of two dimensional plasmons on the tunneling density of states

We show that gapless plasmons lead to a universal $(\delta\nu(\epsilon)/\nu\propto |\epsilon|/E_F)$ correction to the tunneling density of states of a clean two dimensional Coulomb interacting electron gas. We also discuss a counterpart of this effect in the "composite fermion metal" which forms in the presence of a quantizing perpendicular magnetic field corresponding to the half-filled Landau level. We argue that the latter phenomenon might be relevant for deviations from a simple scaling observed by A.Chang et al in the tunneling $I-V$ characteristics of Quantum Hall liquids.Comment: 12 pages, Latex, NORDITA repor

Intertwining Operators And Quantum Homogeneous Spaces

In the present paper the algebras of functions on quantum homogeneous spaces are studied. The author introduces the algebras of kernels of intertwining integral operators and constructs quantum analogues of the Poisson and Radon transforms for some quantum homogeneous spaces. Some applications and the relation to $q$-special functions are discussed.Comment: 20 pages. The general subject is quantum groups. The paper is written in LaTe

On the tensor convolution and the quantum separability problem

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product is separable or entangled. We show that the tensor convolution defined for certain mappings on an almost arbitrary locally compact abelian group, give rise to formulation of an equivalent problem to the separability one.Comment: 13 pages, two sections adde

The Right to a Glass Box: Rethinking the Use of Artificial Intelligence in Criminal Justice

Artificial intelligence (“AI”) increasingly is used to make important decisions that affect individuals and society. As governments and corporations use AI more pervasively, one of the most troubling trends is that developers so often design it to be a “black box.” Designers create AI models too complex for people to understand or they conceal how AI functions. Policymakers and the public increasingly sound alarms about black box AI. A particularly pressing area of concern has been criminal cases, in which a person’s life, liberty, and public safety can be at stake. In the United States and globally, despite concerns that technology may deepen pre-existing racial disparities and overreliance on incarceration, black box AI has proliferated in areas such as: DNA mixture interpretation; facial recognition; recidivism risk assessments; and predictive policing. Despite constitutional criminal procedure protections, judges have often embraced claims that AI should remain undisclosed in court. Both champions and critics of AI, however, mistakenly assume that we inevitably face a trade-off: black box AI may be incomprehensible, but it performs more accurately. But that is not so. In this Article, we question the basis for this assumption, which has so powerfully affected judges, policymakers, and academics. We describe a mature body of computer science research showing how “glass box” AI—designed to be fully interpretable by people—can be more accurate than the black box alternatives. Indeed, black box AI performs predictably worse in settings like the criminal system. After all, criminal justice data is notoriously error prone, and it may reflect preexisting racial and socioeconomic disparities. Unless AI is interpretable, decisionmakers like lawyers and judges who must use it will not be able to detect those underlying errors, much less understand what the AI recommendation means. Debunking the black box performance myth has implications for constitutional criminal procedure rights and legislative policy. Judges and lawmakers have been reluctant to impair the perceived effectiveness of black box AI by requiring disclosures to the defense. Absent some compelling—or even credible—government interest in keeping AI black box, and given the substantial constitutional rights and public safety interests at stake, we argue that the burden rests on the government to justify any departure from the norm that all lawyers, judges, and jurors can fully understand AI. If AI is to be used at all in settings like the criminal system—and we do not suggest that it necessarily should—the presumption should be in favor of glass box AI, absent strong evidence to the contrary. We conclude by calling for national and local regulation to safeguard, in all criminal cases, the right to glass box AI

Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu

SCD Patterns Have Singular Diffraction

Among the many families of nonperiodic tilings known so far, SCD tilings are still a bit mysterious. Here, we determine the diffraction spectra of point sets derived from SCD tilings and show that they have no absolutely continuous part, that they have a uniformly discrete pure point part on the z-axis, and that they are otherwise supported on a set of concentric cylinder surfaces around this axis. For SCD tilings with additional properties, more detailed results are given.Comment: 11 pages, 2 figures; Accepted for Journal of Mathematical Physic

Detection of a Moving Rigid Solid in a Perfect Fluid

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid-solid system fills the whole two dimensional space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in [3], that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, like ellipses for instance, for which the problem of detection admits a unique solution. Using complex analysis, we prove that the well-posedness of the inverse problem is equivalent to the solvability of an infinite set of nonlinear equations. This result allows us to show that when the solid enjoys some symmetry properties, it can be partially detected. Besides, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. Finally, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.Comment: 19 pages, 14 figure