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In quantum computing, Trotter estimates are critical for enabling efficient simulation of quantum systems and quantum dynamics, help implement complex quantum algorithms, and provide a systematic way to control approximate errors. In this paper, we extend the analysis of Trotter-Suzuki approximations, including third and higher orders, to Jordan-Banach algebras. We solve an open problem in our earlier paper on the existence of second-order Trotter formula error estimation in Jordan-Banach algebras. To illustrate our work, we apply our formula to simulate Trotter-factorized spins, and show improvements in the approximations. Our approach demonstrates the adaptability of Trotter product formulas and estimates to non-associative settings, which offers new insights into the applications of Jordan algebra theory to operator dynamics., Comment: some updates on section 5

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with Jordan product formulas are investigated in the setting of JB-algebras. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. We also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements., Comment: 12 pages

We initiate the study of relative operator entropies and Tsallis relative operator entropies in the setting of JB-algebras. We establish their basic properties and extend the operator inequalities on relative operator entropies and Tsallis relative operator entropies to this setting. In addition, we improve the lower and upper bounds of the relative operator $(\alpha, \beta)$-entropy in the setting of JB-algebras that were established in Hilbert space operators setting by Nikoufar [18, 20]. Though we employ the same notation as in the classical setting of Hilbert space operators, the inequalities in the setting of JB-algebras have different connotations and their proofs requires techniques in JB-algebras., Comment: 16 pages. arXiv admin note: text overlap with arXiv:2012.13127

In this paper, the notion of operator means in the setting of JB-algebras is introduced and their properties are studied. Many identities and inequalities are established, most of them have origins from operators on Hilbert space but they have different forms and connotations, and their proofs require techniques in JB-algebras., Comment: 14

In this paper, we investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on relative operator entropies still hold in these broader settings. In addition, we improve the lower and upper bounds of the relative operator $(\alpha, \beta)$-entropy established by Nikoufar which refined the bounds for the relative operator entropy obtained by Fujii and Kamei., Comment: 12

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (factorization, residual finiteness, property (T)) under extensions of discrete quantum groups., Comment: 17 pages + references; edits following referee comments; to appear in JNCG

We show that the discrete duals of the universal unitary quantum groups and orthogonal quantum groups have Kirchberg's factorization property when n is different from 3., Comment: 13 pages. Corrected typos and a few obscurities in the first version, mostly in the introductory section

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the author's first paper on simple quantum groups are shown to be simple using results of Raum and Weber., Comment: 25 pages

The notion of normal quantum subgroup introduced in algebraic context by Parshall and Wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author. As applications, a quantum analog of the third fundamental isomorphism theorem for groups is obtained, which is used along with the equivalence theorem to obtain results on structure of quantum groups with property F and quantum groups with property FD. Other results on normal quantum subgroups for tensor products, free products and crossed products are also proved., Comment: 28 pages. Improved version incorporating referee's suggestions and corrections. To appear in Journal of Algebra

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups $B_u(Q)$ for $Q \in GL(n, {\mathbb C})$ satisfying $Q \bar{Q} = \pm I_n$, $n \geq 2$; (b) The quantum automorphism groups $A_{aut}(B, \tau)$ of finite dimensional $C^*$-algebras $B$ endowed with the canonical trace $\tau$ %endowed with a tracial functional $tr$ when $\dim(B) \geq 4$, including the quantum permutation groups $A_{aut}(X_n)$ on $n$ points ($n \geq 4$); (c) The standard deformations $K_q$ of simple compact Lie groups $K$ and their twists $K_q^u$, as well as Rieffel's deformation $K_J$., Comment: AMS-LATEX file, 49 pages

We introduce a discrete deformation of Rieffel type for finite (quantum) groups. Using this, we give a non-trivial example of a finite quantum group of order 18. We also give a deformation of finite groups of Lie type by using their maximal abelian subgroups., Comment: AMSLATEX, 21 pages. Accepted for publication in Commun Math. Phys. Corrected an error occured at the end of the proof of Theorem 3.2, as well as some typos, as found by the referee

We classify the compact quantum groups $A_u(Q)$ (resp. $B_u(Q)$) up to isomorphism when $Q>0$ (resp. when $Q \bar{Q} \in {\mathbb R} I_n$). We show that the general $A_u(Q)$'s and $B_u(Q)$'s for arbitrary $Q$ have explicit decompositions into free products of these special $A_u(Q)$'s and $B_u(Q)$'s., Comment: 13 pages. To appear in Journal of Operator Theory. Details left out in the original version are filled in

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of a different nature from ergodic actions of compact Lie groups. In particular, we construct: (1). ergodic actions of the compact quantum groups $A_u(Q)$ on the Powers factors; (2). ergodic actions of the compact quantum groups $A_u(n)$ on the hyperfinite II_1 factor; (3). ergodic actions of the compact quantum groups $A_u(Q)$ on the Cuntz algebras; (4). ergodic actions of general compact quantum groups on their homogeneous spaces and an example of a non-homogeneous classical space that nevertheless admits an ergodic action of a compact quantum group., Comment: AMSLATEX, 22 pages, Accepted for publication in Commun. Math. Phys. A few minor changes are made according to the referee's suggestion

By introducing a result that guarantees a given bialgebra to be a Hopf algebra under a natural condition, we show that the quantum automorphism group of the algebra k[x] of polynomials over a field k (of any characteristic) is the universal quantum a x + b group, generalizing the fact that the automorphism group of k[x] is the a x + b group. The q-deformation of the a x + b group is then seen as one among a certain family ${\mathcal A}_{q, n}$ of quantum subgroups of this universal quantum group., Comment: AMSLATEX, 9 papges

We determine the quantum automorphism groups of finite spaces and find they are all compact quantum groups in the sense of Woronowicz. This solves a problem of Connes for finite spaces., Comment: AMSLATEX, 21 pages. To appear in Commun Math Phys