اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدCharacterization of informational completeness for covariant phase space observables
167
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for $p=1,2,infty$ of a more general notion of $p$-regularity defined as the norm density of the span of translates of the operator in the Schatten-$p$ class. We show that the relation between zero sets and $p$-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis.
قيم البحث
اقرأ أيضاً
We give a new mathematically rigorous proof for the fact that, when $S$ is a dense subset of $[0,2pi)$, the rotated quadrature operators $Q_theta$, $thetain S$, of a single mode electromagnetic field constitute an informationally complete set of observables.
Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane Pp, for which the phase space is Pp X plane. We examine the consequences of different quantizer operators built from weight functions on th
is phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position and momentum.
It has recently been established that, in a non-demolition measurement of an observable $mathcal{N}$ with a finite point spectrum, the density matrix of the system approaches an eigenstate of $mathcal{N}$, i.e., it purifies over the spectrum of $math
cal{N}$. We extend this result to observables with general spectra. It is shown that the spectral density of the state of the system converges to a delta function exponentially fast, in an appropriate sense. Furthermore, for observables with absolutely continuous spectra, we show that the spectral density approaches a Gaussian distribution over the spectrum of $mathcal{N}$. Our methods highlight the connection between the theory of non-demolition measurements and classical estimation theory.
We introduce a new density for the representation of quantum states on phase space. It is constructed as a weighted difference of two smooth probability densities using the Husimi function and first-order Hermite spectrograms. In contrast to the Wign
er function, it is accessible by sampling strategies for positive densities. In the semiclassical regime, the new density allows to approximate expectation values to second order with respect to the high frequency parameter and is thus more accurate than the uncorrected Husimi function. As an application, we combine the new phase space density with Egorovs theorem for the numerical simulation of time-evolved quantum expectations by an ensemble of classical trajectories. We present supporting numerical experiments in different settings and dimensions.
Quantum mechanics in conical space is studied by the path integral method. It is shown that the curvature effect gives rise to an effective potential in the radial path integral. It is further shown that the radial path integral in conical space can
be reduced to a form identical with that in flat space when the discrete angular momentum of each partial wave is replaced by a specific non-integral angular momentum. The effective potential is found proportional to the squared mean curvature of the conical surface embedded in Euclidean space. The path integral calculation is compatible with the Schrodinger equation modified with the Gaussian and the mean curvature.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
