The **density** **matrices** are positively semi-definite Hermitian **matrices** of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank **density** **matrices** in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten p-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.

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be found using our quantum relative entropy with a suitable uniform prior density matrix.. 27.[r]

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The mathematical theory of quantum mechanics allows us to view quantum states of finite-dimensional systems as Hermitian, positive semi-definite **matrices** with unit trace [1]. It is also known that the combinatorial Laplacian of a graph is a symmetric positive semi-definite matrix. Thus, normalizing this matrix by its trace allows us to view the graph as a quantum state, [2]. A natural next step is to study the information content of the graph by considering the Von Neumann entropy of the graph’s corresponding quantum state, [3]. In that work, it was noted that the Von Neumann entropy may be interpreted as a measure of network regularity. In [4], it was then shown that for scale free networks the Von Neumann entropy of a graph is linearly related to the Shannon entropy of the graph’s ensemble. Correlations between these entropies were observed when the graph’s degree distribution displayed heterogeneity in [5]. It was not until the work of [6] that a well defined operational interpretation of the graph’s Von Neumann entropy was established. Specifically, in that work it was demonstrated that the Von Neumann entropy should be interpreted as the number of entangled bits that are represented by the graph’s quantum state. Further extensions were made in [7], where it was shown that, as the number of vertices grows, almost all connected graphs have a Von Neumann entropy that is greater than the star graph.

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As an outlook, our method can be extended easily to inhomogeneous lattices with man- ageable computational overhead - which facilitates taking into account the trap potential as well as disorder potentials. Even though the initial state used in this work had no cor- relations, it is trivial to incorporate initial correlations (such as a thermal state with ﬁnite J, for example). In order to improve the accuracy, it is also possible to shift the trunca- tion by taking into account three-point correlations (see the Note added below), either fully or only within a ﬁnite range - or to approximate them by suitable functions of the on-site **density** **matrices** ρ μ and two-point correlations ρ ˆ μ corr μ . Finally, our method can be

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α (20) and (7d) becomes (17b) at k=1/2. The reduced **density** **matrices** take the same format as those presented in equations (18a-d). The maximum and least traces corresponding to (18a-d) under critical strength interaction are given in table 5.

Searching in this direction, one immediately encounters the hierarchy of the reduced **density** **matrices** (RDMs) and their cumulant decomposition. (The concept of RDM is of great importance in quantum chemistry, see [17]- [22].) Here it is assumed that the 2-body RDM (2-matrix) is available from perturbation theory or otherwise (hierarchy truncation etc.) and, with the spin structure as a decisive point, it is elucidated how the contraction has to be performed for an extended system in such a way that the 1-body **density** n k ( ) is obtained. In view of the thermodynamic limit, this procedure calls for the cumulant decomposition of the 2-matrix into its Hartree- Fock part γ HF and its nonreducible remainder, the cumulant 2-matrix γ c , which proves to be the source of both n k ( ) and S a,p ( ) q [10]. Diagonalization of γ c yields cumulant geminals and the corresponding weights (i.e. the spectral resolution).

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Additional results of mode spectrum, tomography measurements and reconstructed density matrices for subspace in free space and subspace after transmitting through 2 m of SMF Additional r[r]

The paper is organized in three sections. In section 2,we give the basic definitions and operations on fuzzy **matrices** which will be used in this paper. In section 3, we introduce the hamacher operations on fuzzy **matrices** and focusing on its properties. The De Morgan's law for the hamacher operations are established in section 4.

The purpose of this paper is to point out that the parametrization (2) of a **density** matrix by expectation values suggests a conceptually interesting way to describe the time evolution of a quantum system without invoking its **density** matrix or wave function. Instead, only expec- tation values of Hermitian operators are used which can be measured directly contrary to the wave function. The argument will be given in general terms first, specifying neither the system at hand nor a particular method of state reconstruction. In the main part of this paper the example of a single spin s is worked out explicitly. The discussion at the end puts the results into perspective.

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The paper is organized as follows. In Section 2, un- certainty theory is first introduced in simple word and some basic concepts and properties are given. Then, conventional-AHP ranking method is introduced. Section 3 is the main part of this paper, uncertainty comparison matrix and uncertainty weights are investigated. Section 4 presents two numerical studies to show the applications of the proposed methods. Section 5 discusses the exten- sion of uncertain variable method to interval comparison **matrices** and fuzzy **matrices**, which are transformed into uncertainty comparison **matrices** using linear uncertainty distribution and zigzag uncertainty distribution respec- tively. Section 6 concludes this paper with a brief sum- mary.

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The arrangement of numbers in **matrices** is an extension of our number system and, as we will see, the rules that govern matrix calculations have many similarities with the arithmetic of numbers. **Matrices** are particularly useful in solving complex problems in linear programming.

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For any two sequence spaces X and Y , we denote by (X,Y ) a class of **matrices** A such that Ax ∈ Y for x ∈ X , provided that the series on the right of (1.8) converges for each n . If, in addition, lim Ax = lim x , then we denote such a class by (X,Y ; P ) or (X,Y ) reg .

In this paper, we use CS for medical image reconstruction to improve the speed and efficiency of compression of an image. The matrix that satisfies the condition of RIP or incoherence is chosen as medical image measurement matrix which may scale back the cost of data acquisition and transmission. The selection of measurement matrix are tested on various medical imaging and performance of Hadamard matrix with Discrete wavelet transform techniques improves the overall performance of compressive sensing based medical image reconstruction. In this proposed analysis we have tested various measurement **matrices** along with DCT and DWT for sparsity .It is also analyzed with three kind of measurement persuits like OMP, L1 and Gradient based Peruits .The block diagram of proposed work is shown in Figure 3.The overall system of compressive sensing based medical image reconstruction are measured as four different measuring parameters like Peak Signal to Noise Ratio(PSNR) and Encoding time.

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This paper represent an approach to basic arithmetic between abstract **matrices**, i.e., **matrices** of symbolic dimension with underspecified components. We define a simple basis function that enables the representation of abstract **matrices** composed of arbitrary regions in a single term that supports matrix addition and multiplication by regular arithmetic on terms. This can, in particular, be exploited to obtain general arithmetic closure properties for classes of structured **matrices**. We also describe an approach using alternative basis functions that allow more compact expressions.

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6. Further comments. We observed that netted **matrices** deﬁned using three-term or four-term recurrences with constant coeﬃcients (we call these 3- or 4-netted **matrices**) preserve a four-term recurrence among the entries of their powers. We ask the question: what is the order of the recurrence for higher powers of a 5-netted, and so forth, **matrices**?

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5 Two children spend their pocket money buying books and CDs. One child spends $120 and buys four books and four CDs. The other child buys three CDs and five books and spends $114. Set up a system of simultaneous equations and use **matrices** to find the cost of a single book and a single CD.

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Rotator allocation simply consists of reorganizing the matrices of indexes and, therefore, the matrices of rotations, in such a way that the matrices of rotations have less rotators if p[r]

The fa t that total unimodularity implies balan edness follows, for example, from Camion's theorem [11℄ whi h states that a 0; 1 matrix A is totally unimodular if and only if A does not [r]

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Keywords: Hermitian Matrices, Quaternian Matrices, Singular value, Trace of Matrix, Triple Complex Matrices... Page 121 www.ijiras.com | Email: contact@ijiras.com The proof is complete[r]

thermal annealing at high temperatures. It is observed that when annealed at 800°C, which is well below the melting temperature of Ge (938.3°C), only Ge nucleation occurs. Whereas for both 900°C annealed samples (A- 900 and F-900), Ge nanocrystals usually show nonuni- form distribution of size and **density** within high-k oxide matrix due to the high diffusion rate of Ge atoms, in consistent with the previously reported results [16]. Furthermore, a higher annealing temperature is expected to result in increased critical nucleus size.

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