The Infimum Norm of Completely Positive Maps

Let A  be a unital C* -algebra, let L: A→B(H)  be a linear map, and let ∅: A→B(H)  be a completely positive linear map. We prove the property in the following:  is completely positive}=inf {||T*T+TT*||1/2:  L= V*TπV  which is a minimal commutant representation with isometry} . Moreover, if L=L* , then  is completely positive  . In the paper we also extend the result  is completely positive}=inf{||T||: L=V*TπV}  [3 , Corollary 3.12].

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

Generalizations of Douady’s magic formula

We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components H of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on H to the set of angles of external rays landing on the real axis.

How Many Inflections are There in the Lyapunov Spectrum?

Iommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Study of Linear Transformation and its Application

Linear functions are commonly referred to as “linear map”, “linear operator” or “linear transformation”. In some cases, the term “homomorphism” will be used, which refers to functions from one kind of set that respect any structures on the sets; The two structures on vector spaces, scalar multiplication, and addition, are respected by linear maps from vector spaces. A linear function is commonly written with a capital L to indicate its linearity, but because we’re only looking at very specific functions, we may alternatively express it with merely f. Matrices are a helpful tool for linear transformation calculations. It’s important to understand how to find the matrix of a linear transformation as well as the properties of matrices.

Completely order bounded maps on non-commutative $${\varvec{L_p}}$$-spaces

We define norms on $$L_p({\mathcal {M}}) \otimes M_n$$ L p ( M ) ⊗ M n where $${\mathcal {M}}$$ M is a von Neumann algebra and $$M_n$$ M n is the space of complex $$n \times n$$ n × n matrices. We show that a linear map $$T: L_p({\mathcal {M}}) \rightarrow L_q({\mathcal {N}})$$ T : L p ( M ) → L q ( N ) is decomposable if $${\mathcal {N}}$$ N is an injective von Neumann algebra, the maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n have a common upper bound with respect to our defined norms, and $$p = \infty $$ p = ∞ or $$q = 1$$ q = 1 . For $$2p< q < \infty $$ 2 p < q < ∞ we give an example of a map $$T$$ T with uniformly bounded maps $$T \otimes Id_{M_n}$$ T ⊗ I d M n which is not decomposable.

Using Direct Acyclic Graphs to Enhance Skeleton-Based Action Recognition with a Linear-Map Convolution Neural Network

Research on the human activity recognition could be utilized for the monitoring of elderly people living alone to reduce the cost of home care. Video sensors can be easily deployed in the different zones of houses to achieve monitoring. The goal of this study is to employ a linear-map convolutional neural network (CNN) to perform action recognition with RGB videos. To reduce the amount of the training data, the posture information is represented by skeleton data extracted from the 300 frames of one film. The two-stream method was applied to increase the accuracy of recognition by using the spatial and motion features of skeleton sequences. The relations of adjacent skeletal joints were employed to build the direct acyclic graph (DAG) matrices, source matrix, and target matrix. Two features were transferred by DAG matrices and expanded as color texture images. The linear-map CNN had a two-dimensional linear map at the beginning of each layer to adjust the number of channels. A two-dimensional CNN was used to recognize the actions. We applied the RGB videos from the action recognition datasets of the NTU RGB+D database, which was established by the Rapid-Rich Object Search Lab, to execute model training and performance evaluation. The experimental results show that the obtained precision, recall, specificity, F1-score, and accuracy were 86.9%, 86.1%, 99.9%, 86.3%, and 99.5%, respectively, in the cross-subject source, and 94.8%, 94.7%, 99.9%, 94.7%, and 99.9%, respectively, in the cross-view source. An important contribution of this work is that by using the skeleton sequences to produce the spatial and motion features and the DAG matrix to enhance the relation of adjacent skeletal joints, the computation speed was faster than the traditional schemes that utilize single frame image convolution. Therefore, this work exhibits the practical potential of real-life action recognition.