Numerical shadow and geometry of quantum states

The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.Comment: 19 pages, 5 figure

Numerical shadows: measures and densities on the numerical range

For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined as the probability that the inner product $(Mu,u)$ is equal to $z$, where $u$ stands for a random complex vector from $H_N$, satisfying $||u||=1$. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian $M$ its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional $B$-spline. In the case of a normal $M$ the numerical shadow corresponds to a shadow of a transparent solid simplex in $R^{N-1}$ onto the complex plane. Numerical shadow is found explicitly for Jordan matrices $J_N$, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.Comment: 37 pages, 8 figure

A softer, stronger Lidskii theorem

We provide a new approach to Lidskii's theorem relating the eigenvalues of the difference A-B of two self-adjoint matrices to the eigenvalues of A and B respectively. This approach combines our earlier work on the spectral matching of matrices joined by a normal path with some familiar techniques of functional analysis. It is based, therefore, on general principles and has the additional advantage of extending Lidskii's result to certain pairs of normal matrices. We are also able to treat some related results on spectral variation stemming from the work of Sunder, Halmos and Bouldin

Limited mantle hydration by bending faults at the Middle America Trench

Author Posting. © American Geophysical Union, 2021. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research: Solid Earth 126(1),(2021): e2020JB020982, https://doi.org/10.1029/2020JB020982.Seismic anisotropy measurements show that upper mantle hydration at the Middle America Trench (MAT) is limited to serpentinization and/or water in fault zones, rather than distributed uniformly. Subduction of hydrated oceanic lithosphere recycles water back into the deep mantle, drives arc volcanism, and affects seismicity at subduction zones. Constraining the extent of upper mantle hydration is an important part of understanding many fundamental processes on Earth. Substantially reduced seismic velocities in tomography suggest that outer rise plate‐bending faults provide a pathway for seawater to rehydrate the slab mantle just prior to subduction. Estimates of outer‐rise hydration based on tomograms vary significantly, with some large enough to imply that, globally, subduction has consumed more than two oceans worth of water during the Phanerozoic. We found that, while the mean upper mantle wavespeed is reduced at the MAT outer rise, the amplitude and orientation of inherited anisotropy are preserved at depths >1 km below the Moho. At shallower depths, relict anisotropy is replaced by slowing in the fault‐normal direction. These observations are incompatible with pervasive hydration but consistent with models of wave propagation through serpentinized fault zones that thin to 1 km below Moho. Confining hydration to fault zones reduces water storage estimates for the MAT upper mantle from ∼3.5 wt% to <0.9 wt% H20. Since the intermediate thermal structure in the ∼24 Myr‐old MAT slab favors serpentinization, limited hydration suggests that fault mechanics are the limiting factor, not temperatures. Subducting mantle may be similarly dry globally.National Science Foundation. Grant Numbers: OCE-0625178, OCE-08410632021-06-1

Restricted numerical shadow and geometry of quantum entanglement

The restricted numerical range $W_R(A)$ of an operator $A$ acting on a $D$-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset $R$ of the of set of pure quantum states of dimension $D$. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of $A$ -- a normalized probability distribution on the complex plane supported in $W_R(A)$. Its value at point z \in {\mathbbm C} is equal to the probability that the expectation value $$ is equal to $z$, where $|\psi>$ represents a random quantum state in subset $R$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size $D=N_A N_B$ we analyse the geometry of sets of separable and maximally entangled states of the $N_A \times N_B$ composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on $D=2^3$ dimensional Hilbert space allows us to investigate the structure of the orbits of $GHZ$ and $W$ quantum states of a three--qubit system.Comment: 33 pages, 8 figures, IOP styl

Acute effects of vasoactive drug treatment on brachial artery reactivity

AbstractObjectivesThe goal of this study was to investigate whether concomitant therapy with vasoactive medications alters the results of noninvasive assessment of endothelial function.BackgroundUltrasound assessment of brachial artery flow-mediated dilation is emerging as a useful clinical tool. The current practice of withholding cardiac medications before ultrasound studies has unknown utility and would limit the clinical use of the methodology.MethodsTo determine whether a single dose of a vasoactive drug influences brachial reactivity, we examined flow-mediated dilation and nitroglycerin-mediated dilation in 73 healthy subjects (age 27 ± 6 years). Studies were completed at baseline and 3 h after randomized treatment with a single oral dose of placebo, felodipine (5 mg), metoprolol (50 mg), or enalapril (10 mg). To determine if holding vasoactive therapy for 24 h before study yields different results than continuation of clinically prescribed medications, we examined vascular function in 72 patients (age 57 ± 10 years) with coronary artery disease. Ultrasound studies were performed 24 h after the last dose and again 3 h after patients took their clinically prescribed medications.ResultsIn healthy subjects one dose of all three drugs lowered blood pressure, and metoprolol also lowered heart rate. However, there was no significant effect of treatment on brachial artery dilation. In patients with coronary artery disease on chronic treatment, taking prescribed medications reduced blood pressure and heart rate, but had no significant effect on brachial artery dilation.ConclusionsRecent administration of commonly used nonnitrate vasoactive drugs has no significant effect on brachial reactivity. These findings suggest that current practice of withholding cardiac medications before testing endothelial function may not be necessary, making this methodology more practical for clinical use