Quantum Permutation Groups Research Articles

Correction to: Tracing the orbitals of the quantum permutation group

Correction to: Tracing the orbitals of the quantum permutation group

Entanglement-invertible channels

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].

A Concrete Model for the Quantum Permutation Group on 4 Points

In 2019, Jung–Weber gave an example of a concrete magic unitary M, which defines a C * -algebraic model of the quantum permutation group S 4 + . We show with the help of a computer that there exist no polynomials up to degree 50 separating the entries of M from the generators of C ( S 4 + ) . This indicates that the magic unitary M might already define a faithful model of S 4 + .

From Quantum Automorphism of (Directed) Graphs to the Associated Multiplier Hopf Algebras

This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that K [Mq(n)] is a quadratic algebra, and can be equipped with a multiplier Hopf ∗-algebra structure in the sense of quantum permutation groups developed byWang and an observation by Rollier–Vaes. In our next paper, we will propose the study of multiplier Hopf graph algebras. The current paper can be viewed as a precursor to this upcoming work, serving as a crucial intermediary bridging the gap between the abstract concept of multiplier Hopf algebras and the well-developed field of graph theory, thereby establishing connections between them! This survey review paper is dedicated to the 78th birthday anniversary of Professor Alfons Van Daele.

Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$^\ast $-algebra $B$, which includes the quantum permutation group $S_N^+$, where $N = \dim B$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $\Gamma $ lead to isomorphic $\ast $-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $1$-boundedness between the various algebras associated with these quantum groups.

Infinite quantum permutations

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups.We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.

Tannaka–Krein Reconstruction and Ergodic Actions of Easy Quantum Groups

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

Quantum Channels with Quantum Group Symmetry

In this paper we will demonstrate that any compact quantum group can be used as symmetries for quantum channels, which leads us to the concept of covariant channels. We then unearth the structure of the convex set of covariant channels by identifying all extreme points under the assumption of multiplicity-free condition for the associated fusion rule, which provides a wide generalization of the results of Mozrzymas et al. (J Math Phys 58(5):052204, 2017). The presence of quantum group symmetry in contrast to the group symmetry will be highlighted in the examples of quantum permutation groups and \(SU_q(2)\). In the latter example, we will see the necessity of the Heisenberg picture coming from the non-Kac type condition. This paper ends with the covariance with respect to projective representations, which leads us back to Weyl covariant channels and its fermionic analogue.

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.

Sinkhorn Algorithm for Quantum Permutation Groups

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.

Bisynchronous Games and Factorizable Maps

We introduce a new class of non-local games and corresponding densities, which we call bisynchronous. Bisynchronous games are a subclass of synchronous games and exhibit many interesting symmetries when the algebra of the game is considered. We develop a close connection between these non-local games and the theory of quantum groups which recently surfaced in studies of graph isomorphism games. When the number of inputs is equal to the number of outputs, we prove that a bisynchronous density arises from a trace on the quantum permutation group. Each bisynchronous density gives rise to a completely positive map, and we prove that these maps are factorizable maps.

Nonlocal games and quantum permutation groups

We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x∈V(X) and y∈V(Y) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature.Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory.

Models of quantum permutations

For N≥4 we present a series of ⁎-homomorphisms φn:C(SN+)→Bn where SN+ is the quantum permutation group. They are not necessarily representations of the quantum group SN+ but they yield good and somewhat “small” operator algebraic models of quantum permutation matrices. In the inverse limit however they produce a quantum group which turns out to be isomorphic to SN+; the latter fact is thanks to an argument by Alexandru Chirvasitu and Paweł Józiak building on topological generation.

Topological generation and matrix models for quantum reflection groups

We prove that the von Neumann algebras of quantum permutation groups and quantum reflection groups have the Connes embedding property. We do this by establishing several new topological generation results for the quantum permutation groups SN+ and the quantum reflection groups HNs+. We use these results to show that these quantum groups admit sufficiently many “matrix models”. In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain “flat” matrix models for SN+ are inner faithful.

QUANTUM INCREASING SEQUENCES GENERATE QUANTUM PERMUTATION GROUPS

AbstractWe answer a question of Skalski and Sołan (2016) about inner faithfulness of the Curran’s map of extending a quantum increasing sequence to a quantum permutation. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran’s map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon (2018).

Higher transitive quantum groups: theory and models

We investigate the notion of $k$-transitivity for the quantum permutation groups $G\subset S_N^+$, with a brief review of the known $k=1,2$ results, and with a study of what happens at $k\geq 3$. We then discuss matrix modelling questions for the algebras

Quantum reflections, random walks and cut-off

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of reflections. We also study random walks on quantum reflection groups and more generally on free wreath products of finite groups by quantum permutation groups.

Modeling questions for quantum permutations

Given a quantum permutation group [Formula: see text], with orbits having the same size [Formula: see text], we construct a universal matrix model [Formula: see text], having the property that the images of the standard coordinates [Formula: see text] are projections of rank [Formula: see text]. Our conjecture is that this model is inner faithful under suitable algebraic assumptions, and is in addition stationary under suitable analytic assumptions. We prove this conjecture for the classical groups, and for several key families of group duals.

Remarks on Hopf Images and Quantum Permutation Groups

AbstractMotivated by a question of A. Skalski and P. M. Sołtan (2016) about inner faithfulness of S. Curran’s map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on points. This enables us not only to answer the aforementioned question in the positive for the case where n = 4, k = z, but also to classify the automorphisms of , describe all the embeddings O−1(2) ⊂ and show that all the copies of O−1(2) ⊂ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.

Isolated Partial Hadamard Matrices and Related Topics

We study the partial Hadamard matrices [Formula: see text] which are isolated, under the assumption that the entries [Formula: see text] are roots of unity, or more generally, under the assumption that the combinatorics of H comes from vanishing sums of roots of unity. We first review the various conjectures on the subject, and then we present several new results, regarding notably the isolation of the master Hadamard matrices, [Formula: see text], and the structure of the isolated matrices arising via the McNulty-Weigert construction. We discuss then the notion of isolation, in some related contexts, of the magic unitary matrices, and of the quantum permutation groups.