اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHow to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of $mathrm{SL}(2,mathbb{C})$
58
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Recently it was reported that the spin-coherent state (SCS) positive-operator-valued measure (POVM) can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. The outcome probability distribution of the SCS POVM for an input quantum state is the generalized $Q$-function, which is defined on the 2-sphere phase space of SCSs. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of applying curved-phase-space correspondences to quantum systems, indicating their experimental utility by explaining how to analyze this measurements performance. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators represent elements of the Lie group $mathrm{SL}(2,mathbb{C})$, a complex version of the usual unitary operators that represent elements of $mathrm{SU}(2)$. Consequently, the associated POVM elements represent points in the 3-hyperboloid $mathrm{SU}(2)backslashmathrm{SL}(2,mathbb{C})$. Three equivalent stochastic techniques, path integral, diffusion (Fokker-Planck) equation, and stochastic differential equations, are applied to show that the POVM quickly limits to the SCS POVM. We apply two basic mathematical tools to the Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan decomposition associated with the symmetric pair $mathrm{SU}(2)subsetmathrm{SL}(2,mathbb{C})$. Informed by these tools, the three stochastic techniques are applied directly to the Kraus operators in a representation independent, and thus geometric, way (independent of any spectral information about the spin components).
قيم البحث
اقرأ أيضاً
In a recent Letter~[PRL textbf{121}, 130404 (2018)], it was announced that the spin-coherent-state POVM can be implemented via a nonadaptive continuous isotropic measurement. In this article, the mathematical concepts used to prove this are explained
in greater depth. Also provided is the more general result of how to implement a generalized-coherent-state POVM for any finite-dimensional unitary representation of a Lie group.
We present the theory of how to achieve phase measurements with the minimum possible variance in ways that are readily implementable with current experimental techniques. Measurements whose statistics have high-frequency fringes, such as those obtain
ed from NOON states, have commensurately high information yield. However this information is also highly ambiguous because it does not distinguish between phases at the same point on different fringes. We provide schemes to eliminate this phase ambiguity in a highly efficient way, providing phase estimates with uncertainty that is within a small constant factor of the Heisenberg limit, the minimum allowed by the laws of quantum mechanics. These techniques apply to NOON state and multi-pass interferometry, as well as phase measurements in quantum computing. We have reported the experimental implementation of some of these schemes with multi-pass interferometry elsewhere. Here we present the theoretical foundation, and also present some new experimental results. There are three key innovations to the theory in this paper. First, we examine the intrinsic phase properties of the sequence of states (in multiple time modes) via the equivalent two-mode state. Second, we identify the key feature of the equivalent state that enables the optimal scaling of the intrinsic phase uncertainty to be obtained. This enables us to identify appropriate combinations of states to use. The remaining difficulty is that the ideal phase measurements to achieve this intrinic phase uncertainty are often not physically realizable. The third innovation is to solve this problem by using realizable measurements that closely approximate the optimal measurements, enabling the optimal scaling to be preserved.
This paper examines the relationship between certain non-commutative analogues of projective 3-space, $mathbb{P}^3$, and the quantized enveloping algebras $U_q(mathfrak{sl}_2)$. The relationship is mediated by certain non-commutative graded algebras
$S$, one for each $q in mathbb{C}^times$, having a degree-two central element $c$ such that $S[c^{-1}]_0 cong U_q(mathfrak{sl}_2)$. The non-commutative analogues of $mathbb{P}^3$ are the spaces $operatorname{Proj}_{nc}(S)$. We show how the points, fat points, lines, and quadrics, in $operatorname{Proj}_{nc}(S)$, and their incidence relations, correspond to finite dimensional irreducible representations of $U_q(mathfrak{sl}_2)$, Verma modules, annihilators of Verma modules, and homomorphisms between them.
We demonstrate the role of measurement back-action of a coherent spin environment on the dynamics of a spin (qubit) coupled to it, by inducing non-classical (Quantum Random Walk like) statistics on its measurement trajectory. We show how the long-lif
e time of the spin-bath allows it to correlate measurements of the qubit over many repetitions. We have used Nitrogen Vacancy centers in diamond as a model system, and the projective single-shot readout of the electron spin at low temperatures to simulate these effects. We show that the proposed theoretical model, explains the experimentally observed statistics and their application for quantum state engineering of spin ensembles towards desired states.
We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thick
ened punctured surface $mathfrak{S} times (0, 1)$ a Laurent polynomial $mathrm{Tr}_lambda^q(K) = mathrm{Tr}_lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm{u}ller space, depending on the choice of an ideal triangulation $lambda$ of the surface $mathfrak{S}$. Along the way, we propose a definition for a $mathrm{SL}_n(mathbb{C})$-version of this invariant.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
