Non-real Element Research Articles

Kirkwood-Dirac classical pure states

Kirkwood-Dirac (KD) distribution is a representation of quantum states. Recently, KD distribution has been found applications in many areas such as in quantum metrology, quantum chaos and foundations of quantum theory. KD distribution is a quasiprobability distribution, and negative or nonreal elements may signify quantum advantages in certain tasks. A quantum state is called KD classical if its KD distribution is a probability distribution. Since most quantum information processings use pure states as ideal resources, then a key problem is to determine whether a quantum pure state is KD classical. In this paper, we provide some characterizations for the general structure of KD classical pure states. As an application of our results, we prove a conjecture raised by De Bièvre (2021) [56] which finds out all KD classical pure states for discrete Fourier transformation.

On normalized integral table algebras generated by a non-real faithful element of degree 2 with p linear elements

In 1995, Blau completely classified the integral table algebras generated by a faithful element of degree 2, all of whose linear basis elements having degrees a power of 2 are equal to 1. By this classification, one can get that the normalized integral table algebra having exactly one linear element and generated by a faithful non-real element of degree 2 is exactly (Ch(SL(2,5)),Irr(SL(2,5))). In this paper, we investigate normalized integral table algebras having more than one linear elements. We once classified normalized integral table algebra with exactly 2 linear elements (see Li and Chen, 2020 [10]). Here we continue this topic and obtain the classification of normalized table algebra (A,B) generated by a faithful non-real element of degree 2 and |L(B)|=p, where p≥3 is a prime.

Conditions tighter than noncommutation needed for nonclassicality

Kirkwood discovered in 1933, and Dirac discovered in 1945, a representation of quantum states that has undergone a renaissance recently. The Kirkwood–Dirac (KD) distribution has been employed to study nonclassicality across quantum physics, from metrology to chaos to the foundations of quantum theory. The KD distribution is a quasiprobability distribution, a quantum generalization of a probability distribution, which can behave nonclassically by having negative or nonreal elements. Negative KD elements signify quantum information scrambling and potential metrological quantum advantages. Nonreal elements encode measurement disturbance and thermodynamic nonclassicality. KD distributions’ nonclassicality has been believed to follow necessarily from pairwise noncommutation of operators in the distribution’s definition. We show that noncommutation does not suffice. We prove sufficient conditions for the KD distribution to be nonclassical (equivalently, necessary conditions for it to be classical). We also quantify the KD nonclassicality achievable under various conditions. This work resolves long-standing questions about nonclassicality and may be used to engineer quantum advantages.

Schur indices for reality-based algebras with two nonreal basis elements

This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension [Formula: see text] with rational field of definition for which it is a division algebra. The approach has other applications, one of which shows noncommutative association scheme of rank [Formula: see text] must have at least three symmetric relations.

Toward the classification of normalized integral table algebras generated by a faithful non-real element of degree 3 with (I)

The classification of integral table algebras (A, B) generated by a faithful element of degree 2 with was completed by Blau in 1995. After that, it is an interesting topic to classify integral table algebras generated by a faithful non-real element of degree 3 with But it is difficult, so Arad and Chen et al. in 2005 and 2011 studied normalized integral table algebras generated by a faithful non-real element of degree 3 by assuming and they almost classified such table algebras. Here the authors continue this work but not assuming Communicated by Miriam Cohen

Hypertranscendence of the multiple sine function for a complex period

It is known that the multiple sine function for a “rational” period satisfies an algebraic differential equation. However, for a non-“rational” period, the differential algebraicity of the multiple sine function is obscure. In this paper, we prove that, if there exists a non-real element in the set $\{\omega_{j}/\omega_{i}|1\leq i<j\leq r\}$, the multiple sine function $\text{Sin}_{r}(x,(\omega_{1},\cdots,\omega_{r}))$ does not satisfy any algebraic differential equation.

Reality properties of conjugacy classes inG 2

LetG be an algebraic group over a fieldk. We callg eG(k) real ifg is conjugate tog −1 inG(k). In this paper we study reality for groups of typeG 2 over fields of characteristic different from 2. LetG be such a group overk. We discuss reality for both semisimple and unipotent elements. We show that a semisimple element inG(k) is real if and only if it is a product of two involutions inG(k). Every unipotent element inG(k) is a product of two involutions inG(k). We discuss reality forG 2 over special fields and construct examples to show that reality fails for semisimple elements inG 2 over ℚ and ℚp. We show that semisimple elements are real forG 2 overk withcd(k) ≤ 1. We conclude with examples of nonreal elements inG 2 overk finite, with characteristick not 2 or 3, which are not semisimple or unipotent.

On four normalized table algebras generated by a faithful nonreal element of degree 3

On four normalized table algebras generated by a faithful nonreal element of degree 3

Integral Standard Table Algebras with a Faithful Nonreal Element of Degree 3

Integral Standard Table Algebras with a Faithful Nonreal Element of Degree 3

Integral Table Algebras and Bose–Mesner Algebras with a Faithful Nonreal Element of Degree Three

Integral Table Algebras and Bose–Mesner Algebras with a Faithful Nonreal Element of Degree Three

Lobatto-type quadrature formula in rational spaces

This paper constructs the Lobatto-type quadrature formula for the rational space R 2n−1(a1,…,an):= P(x) Π n k=1z.sfnc;x−a k| 2 ,Pϵp 2n−1 with the nonreal elements in { a k } k = 1 n ⊂ Cβ[−1,1] paired by complex conjugation. This Lobatto-type quadrature formula is based on the extreme points of the Chebyshev polynomial of the first kind with the rational space P n ( a 1,…, a n ). Moreover, it is exact for any element in R 2 n − 1 ( a 1,…, a n ).

Inequalities for Rational Functions With Prescribed Poles

AbstractThis paper considers the rational system Pn(a1, a2,……,an) := with nonreal elements in paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in Pn(a1, a2,……an). The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in Pn(a1, a2,……an) are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results

A general transformation with applications to circuit theory

A general functional transformation is introduced and some applications to circuit theory are given. The transformation is most useful in the area of active, nonbilateral circuits. This functional transformation introduces new voltages and currents which are linear combinations of the original voltages and currents plus their derivatives, integrals or more complicated implicit integradifferential relationships. The transformation applied to the currents may be completely unconnected to the transformation applied to the voltages. The transformation may be used for obtaining equivalent circuits of some driving point impedance. Another possibility is obtaining equivalent circuits of some transfer impedance but with different driving point impedances. The transformation handles with equal ease bilateral and non-bilateral circuits as well as circuits containing controlled sources. It may also handle circuits containing non-real elements. Simple numerical examples are given in active, non-bilateral, and passive synthesis, as well as in analysis.

Mahler matrices and the equation QA=AQm

Abstract : Several sets of matrices are defined; each set forms an abelian group. The elements of the matrices are 0, 1, -1 and other roots of unity (or sums of roots of unity). The determinant, characteristic roots, vectors, and elementary divisors are also found. Thus the matrices form a convenient set of test matrices for a routine which purports to solve a matrix. A typical test matrix may involve nonreal elements; if each element of the matrix is replaced by a 2 by 2 matrix in the usual way the roots of the expanded real matrix are the roots of the test matrix and their conjugates. The vectors of the expanded matrix are easily found also. (Author)