Diagonal Operators Research Articles

Optimal superposition enhancement of a mixed state by stochastic diagonal superposition-free operations

The resource theory of superposition is a generalization of coherence theory. A rigorous framework for quantifying superposition of linear independent states has been given by previous studies. In this paper, the optimal superposition enhancement of a mixed state is studied. Based on the l1 measure of superposition the optimal superposition enhancement value of a mixed state can be obtained under the stochastic diagonal superposition-free operations(or incomplete diagonal superposition-free operations). In addition, the optimal probability corresponding to the optimal superposition enhancement value of a mixed state is also discussed. The results show that although the superposition-free operations can decrease the superposition of a state, the superposition enhancement of a state is possible, a single mixed state can be probabilistically transformed into another state with higher superposition than before by superposition-free operations. These studies will give us a better understanding of the superposition manipulation and the resource theory of superposition.

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

Anderson's theorem and A-spectral radius bounds for semi-Hilbertian space operators

Let H be a complex Hilbert space and let A be a positive bounded linear operator on H. Let T be an A-bounded operator on H. For rank(A)=n<∞, we show that if WA(T)⊆D‾(={λ∈C:|λ|≤1}) and WA(T) intersects ∂D(={λ∈C:|λ|=1}) at more than n points, then WA(T)=D‾. In particular, when A is the identity operator on Cn, then this leads to Anderson's theorem in the complex Hilbert space Cn. We introduce the notion of A-compact operators to study analogous result when the space H is infinite dimensional. Further, we develop an upper bound for the A-spectral radius of n×n operator matrices with entries are commuting A-bounded operators, where A=diag(A,A,…,A) is an n×n diagonal operator matrix. Several inequalities involving A-spectral radius of A-bounded operators are also given.

Some Integrals Involving Meijer’s and Hypergeometric Functions Using the Coherent State Technique

In this paper, we have examined a particular case of coherent states, defined so that their structure constants depend only on the products of the energy eigenvalues of the examined systems. In this manner, we have built all three kinds of coherent states (Barut–Girardello, Klauder–Perelomov, and Gazeau–Klauder). From the equation of the unitary operator decomposition, we have highlighted and used a so-called fundamental integral, to obtain some new integrals involving Meijer’s and hypergeometric functions. All calculations are made using the properties of the diagonal operator ordering technique. This is implicitly proved that the coherent state technique can be useful not only in different branches of physics (quantum mechanics, quantum optics, quantum information theory, and so on) but also in the deduction of new integrals involving generalized Meijer’s and hypergeometric functions. This approach can be considered as suitable “feedback” from physics to mathematics.

On η-biharmonic hypersurfaces in pseudo-Riemannian space forms

Abstract In this paper, η-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type $\begin{array}{} \displaystyle M^3_2 \end{array}$ in $\begin{array}{} \displaystyle \mathbb{E}^4_2 \end{array}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.

On Faithful Matrix Representations of q-Deformed Models in Quantum Optics

Consider the q -deformed Lie algebra, t q : K ^ 1 , K ^ 2 q = 1 − q K ^ 1 K ^ 2 , K ^ 3 , K ^ 1 q = s K ^ 3 , K ^ 1 , K ^ 4 q = s K ^ 4 , K ^ 3 , K ^ 2 q = t K ^ 3 , K ^ 2 , K ^ 4 q = t K ^ 4 , and K ^ 4 , K ^ 3 q = r K ^ 1 , where r , s , t ∈ ℝ − 0 , subject to the physical properties: K ^ 1 and K ^ 2 are real diagonal operators, and K ^ 3 = K ^ 4 † , ( † is for Hermitian conjugation). The q -deformed Lie algebra, t q is introduced as a generalized model of the Tavis–Cummings model (Tavis and Cummings 1968, Bashir and Sebawe Abdalla 1995), namely, K ^ 1 , K ^ 2 = 0 , K ^ 1 , K ^ 3 = − 2 K ^ 3 , K ^ 1 , K ^ 4 = 2 K ^ 4 , K ^ 2 , K ^ 3 = K ^ 3 , K ^ 2 , K ^ 4 = K ^ 4 , and K ^ 4 , K ^ 3 = K ^ 1 , which is subject to the physical properties K ^ 1 and K ^ 2 are real diagonal operators, and K ^ 3 = K ^ 4 † . Faithful matrix representations of the least degree of t q are discussed, and conditions are given to guarantee the existence of the faithful representations.

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N^{m+1}_t(c) of dimension m+1 and index t (mge 2 and 1 le tle m). We shall study hypersurfaces {M}^{m}_{t'} which are polyharmonic of order r (briefly, r-harmonic), where rge 3 and either t'=t or t'=t-1. Let A denote the shape operator of {M}^{m}_{t'}. Under the assumptions that {M}^{m}_{t'} is CMC and {{rm{Tr}}},{A}^{2} is a constant, we shall obtain the general condition which determines that {M}^{m}_{t'} is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r ge 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

On η-biharmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms

In this paper, η-biharmonic hypersurfaces Mrn with constant scalar curvature in a pseudo-Riemannian space form are studied. Under the assumption that Mrn has diagonalizable shape operator with at most six distinct principal curvatures, we prove that Mrn has constant mean curvature. As an application, we obtain a partial classification result of these hypersurfaces and show that (nc)-biharmonic hypersurfaces must be minimal.

Spectral analysis for finite rank perturbation of diagonal operator in non-archimedean Banach space of countable type

In this paper we are concerned with the spectral analysis for the classes of finte rank perturbations of diagonal operators in the form A = D+F where D is a diagonal operator and F = u’1 ⊗v’1+...+u’m ⊗vm is an operator of finite rank in the non-archimedean Banach space of countable type. We compute the spectrum of A using the theory of Fredholm operators in non archimedean setting and the concept of essential spectrum for linear operators.

Quantum state preparation and nonunitary evolution with diagonal operators

Realizing non-unitary transformations on unitary-gate based quantum devices is critically important for simulating a variety of physical problems including open quantum systems and subnormalized quantum states. We present a dilation based algorithm to simulate non-unitary operations using probabilistic quantum computing with only one ancilla qubit. We utilize the singular-value decomposition (SVD) to decompose any general quantum operator into a product of two unitary operators and a diagonal non-unitary operator, which we show can be implemented by a diagonal unitary operator in a 1-qubit dilated space. While dilation techniques increase the number of qubits in the calculation, and thus the gate complexity, our algorithm limits the operations required in the dilated space to a diagonal unitary operator, which has known circuit decompositions. We use this algorithm to prepare random sub-normalized two-level states on a quantum device with high fidelity. Furthermore, we present the accurate non-unitary dynamics of two-level open quantum systems in a dephasing channel and an amplitude damping channel computed on a quantum device. The algorithm presented will be most useful for implementing general non-unitary operations when the SVD can be readily computed, which is the case with most operators in the noisy intermediate-scale quantum computing era.

Reducing subspaces for rank-one perturbations of normal operators

We study the existence of reducing subspaces for rank-one perturbations of diagonal operators and, in general, of normal operators of uniform multiplicity one. As we will show, the spectral picture will play a significant role in order to prove the existence of reducing subspaces for rank-one perturbations of diagonal operators whenever they are not normal. In this regard, the most extreme case is provided when the spectrum of the rank-one perturbation of a diagonal operator $T=D + u\otimes v$ (uniquely determined by such expression) is contained in a line, since in such a case $T$ has a reducing subspace if and only if $T$ is normal. Nevertheless, we will show that it is possible to exhibit non-normal operators $T=D + u\otimes v$ with spectrum contained in a circle either having or lacking non-trivial reducing subspaces. Moreover, as far as the spectrum of $T$ is contained in any compact subset of the complex plane, we provide a characterization of the reducing subspaces $M$ of $T$ such that the restriction $T\mid _M$ is normal. In particular, such characterization allows us to exhibit rank-one perturbations of completely normal diagonal operators (in the sense of Wermer) lacking reducing subspaces. Furthermore, it determines completely the decomposition of the underlying Hilbert space in an orthogonal sum of reducing subspaces in the context of a classical theorem due to Behncke on essentially normal operators.

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation.

Best n-Term Approximation of Diagonal Operators and Application to Function Spaces with Mixed Smoothness

In this paper we give exact values of the best n-term approximation widths of diagonal operators between ℓp(ℕ)and ℓq(ℕ)with 0 < p,q ≤ ∞. The result will be applied to obtain the asymptotic constants of best n-term approximation widths of embeddings of function spaces with mixed smoothness by trigonometric system.

Rationality and C2-cofiniteness of certain diagonal coset vertex operator algebras

In this paper, it is shown that the diagonal coset vertex operator algebra C(Lg(k+2,0),Lg(k,0)⊗Lg(2,0)) is rational and C2-cofinite in case g=so(2n), n≥3 and k is an admissible number for gˆ. It is also shown that the diagonal coset vertex operator algebra C(Lsl2(k+4,0),Lsl2(k,0)⊗Lsl2(4,0)) is rational and C2-cofinite in case k is an admissible number for sl2ˆ. Furthermore, irreducible modules of C(Lsl2(k+4,0),Lsl2(k,0)⊗Lsl2(4,0)) are classified in case k is a positive odd integer.

On a class of non-Hermitian Hamiltonians with tridiagonal matrix representation

We show that some non-Hermitian Hamiltonian operators with tridiagonal matrix representation may be quasi Hermitian or similar to Hermitian operators. In the class of Hamiltonian operators discussed here the transformation is given by a Hermitian, positive-definite, diagonal operator. We show that there is an important difference between open boundary conditions and periodic ones. We illustrate the theoretical results by means of two simple, widely used, models.

New unitary equivalences for some operator matrices with applications

In this paper, first we consider cross-diagonal, in particular off-diagonal, operator matrices; those operator matrices such that all entries except those on the main and off diagonals are zero. We show that this operator matrix is unitarily equivalent to a block diagonal operator matrix whose diagonal blocks are all two-by-two, except at most one of them which is one-by-one. Then, using this unitary equivalence, we show that any left circulant operator matrix is unitarily equivalent to a direct sum of a one-by-one operator and an off-diagonal operator matrix. As an application, we give equalities for the numerical radius of some important operator matrices. In particular, for the following left circulant operator matrix, we show that where is the nth root of unity. Meanwhile, some inequalities for general operator matrices are obtained.

On diagonal operators between the sequence (LF)-spaces l_p({mathcal {V}})

Diagonal (multiplication) operators acting between a particular class of countable inductive spectra of Fréchet sequence spaces, called sequence (LF)-spaces, are investigated. We prove results concerning boundedness, compactness, power boundedness, and mean ergodicity. Furthermore, we determine when a diagonal operator is Montel and reflexive. We analyze the spectra in terms of the system of weights defining the spaces.

Quantum simulation in the semi-classical regime

Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter&amp;#x210F;, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of&amp;#x210F;and the precision&amp;#x03B5;are obtained. It is found that the number of required qubits,m, scales only logarithmically with respect to&amp;#x210F;. When the solution has bounded derivatives up to order&amp;#x2113;, the symmetric Trotting method has gate complexityO((&amp;#x03B5;&amp;#x210F;)&amp;#x2212;12polylog(&amp;#x03B5;&amp;#x2212;32&amp;#x2113;&amp;#x210F;&amp;#x2212;1&amp;#x2212;12&amp;#x2113;)),provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented withpoly(m)operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of&amp;#x210F;. The gate complexity in this case is reduced toO(&amp;#x03B5;&amp;#x2212;12polylog(&amp;#x03B5;&amp;#x2212;32&amp;#x2113;&amp;#x210F;&amp;#x2212;1)),with&amp;#x2113;again indicating the smoothness of the solution.

Diagonal unitary and orthogonal symmetries in quantum theory: II. Evolution operators

We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.

Finite rank perturbations of normal operators: Spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.