اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدLog-Convex set of Lindblad semigroups acting on $N$-level system
139
0
0.0
(
0
)
تأليف
Fereshte Shahbeigi
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We analyze the set ${cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ${cal A}_N^Q$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that for mixed Weyl channels the hyper-decoherence commutes with the dynamics, so that decohering a quantum accessible channel we obtain a bistochastic matrix form the set ${cal A}_N^C$ of classical maps accessible by a semigroup. Focusing on $3$-level systems we investigate the geometry of the sets of quantum accessible maps, its classical counterpart and the support of their spectra. We demonstrate that the set ${cal A}_3^Q$ is not included in the set ${cal U}^Q_3$ of quantum unistochastic channels, although an analogous relation holds for $N=2$. The set of transition matrices obtained by hyper-decoherence of unistochastic channels of order $Nge 3$ is shown to be larger than the set of unistochastic matrices of this order, and yields a motivation to introduce the larger sets of $k$-unistochastic matrices.
قيم البحث
اقرأ أيضاً
It is commonly accepted that a deviation of the Wigner quasiprobability distribution of a quantum state from a proper statistical distribution signifies its nonclassicality. Following this ideology, we introduce the global indicator $mathcal{Q}_N$ fo
r quantification of classicality-quantumness correspondence in the form of the functional on the orbit space $mathcal{O}[mathfrak{P}_N]$ of the $SU(N)$ group adjoint action on the state space $mathfrak{P}_N$ of an $N$-dimensional quantum system. The indicator $mathcal{Q}_{N}$ is defined as a relative volume of a subspace $mathcal{O}[mathfrak{P}^{(+)}_N] subset mathcal{O}[mathfrak{P}_N],,$ where the Wigner quasiprobability distribution is positive. An algebraic structure of $mathcal{O}[mathfrak{P}^{(+)}_N]$ is revealed and exemplified by a single qubit $(N=2)$ and single qutrit $(N=3)$. For the Hilbert-Schmidt ensemble of qutrits the dependence of the global indicator on the moduli parameter of the Wigner quasiprobability distribution has been found.
We introduce the concepts of Poisson brackets for classical noise, and of canonically conjugate Wiener processes (symplectic noise). Phase space diffusions driven by these processes are considered and the general form of a stochastic process preservi
ng the full (system and noise) Poisson structure is obtained. We show that, once the classical stochastic model is required to preserve the joint system and noise Poisson bracket, it has much in common with quantum markovian models.
From a geometric point of view, Paulis exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of $1$-fermion and $N$-fermion density matrices, therefore it coincides with the convex hull of the pure $N$-represent
able $1$-fermion density matrices. Consequently, the description of ground state physics through $1$-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the $1$-body $N$-representability problem to ensemble states with fixed spectrum $mathbf{w}$, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the $1$-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of $mathbf{w}$. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes.
In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Renyi entropies, as well as a family of functions of density operators strictly related to the Schatten n
orms. A few remarkable consequences --- in particular, a description of the associated infinitesimal generators --- are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are also considered.
In this work, we investigate the possibility of compressing a quantum system to one of smaller dimension in a way that preserves the measurement statistics of a given set of observables. In this process, we allow for an arbitrary amount of classical
side information. We find that the latter can be bounded, which implies that the minimal compression dimension is stable in the sense that it cannot be decreased by allowing for small errors. Various bounds on the minimal compression dimension are proven and an SDP-based algorithm for its computation is provided. The results are based on two independent approaches: an operator algebraic method using a fixed point result by Arveson and an algebro-geometric method that relies on irreducible polynomials and Bezouts theorem. The latter approach allows lifting the results from the single copy level to the case of multiple copies and from completely positive to merely positive maps.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
