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Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states

by M\'arton Mesty\'an, Bal\'azs Pozsgay, Ian M. Wanless

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Submission summary

Authors (as registered SciPost users): Balázs Pozsgay
Submission information
Preprint Link: scipost_202308_00029v1  (pdf)
Date submitted: 2023-08-20 17:57
Submitted by: Pozsgay, Balázs
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical


We consider dual unitary operators and their multi-leg generalizations that have appeared at various places in the literature. These objects can be related to multi-party quantum states with special entanglement patterns: the sites are arranged in a spatially symmetric pattern and the states have maximal entanglement for all bipartitions that follow from the reflection symmetries of the given geometry. We consider those cases where the state itself is invariant with respect to the geometrical symmetry group. The simplest examples are those dual unitary operators which are also self dual and reflection invariant, but we also consider the generalizations in the hexagonal, cubic, and octahedral geometries. We provide a number of constructions and concrete examples for these objects for various local dimensions. All of our examples can be used to build quantum cellular automata in 1+1 or 2+1 dimensions, with multiple equivalent choices for the ``direction of time''.

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Reports on this Submission

Anonymous Report 2 on 2023-11-17 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202308_00029v1, delivered 2023-11-17, doi: 10.21468/SciPost.Report.8139


The text discusses the concept of "multi-directional unitarity" and "multi-directional maximally entangled states" in quantum mechanics. These objects are generalizations of dual unitary operators and have special entanglement properties relevant to quantum many-body physics and quantum information theory. The focus is on quantum states with maximal entanglement for all bipartitions that follow from the reflection symmetries of a spatially symmetric arrangement of sites.
The study extends the notion of "dual unitarity" in solvable quantum many-body systems, introducing the concept of multi-directional unitarity in various geometries, including hexagonal, cubic, and octahedral. The text relates these concepts to the broader field of quantum information theory, particularly the study of "absolutely maximally entangled states" (AME), which have maximal bipartite entanglement for all possible bipartitions. AMEs are explored for their applications in quantum error correction codes and tensor network models.
The authors present a unified treatment of multi-directional unitary operators, emphasizing geometric properties and providing various constructions and examples. The discussion includes octahedral geometry, which has not been considered before and highlights spatially symmetric solutions. The text concludes with open problems, suggesting avenues for further research, such as finding complete descriptions of algebraic varieties, considering spatial invariance, exploring factorization in cases with more sites, and studying the quantum circuits arising from these operators.

Just some final comments, we found some small typos that can be corrected during the publication of the article.

All in all, we recommend the publication of the paper.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2023-11-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202308_00029v1, delivered 2023-11-09, doi: 10.21468/SciPost.Report.8082


The manuscript "Multi-directional unitary and maximal entanglement in spatially symmetric quantum states" generalizes the notion of dual unitary matrices to unitaries with similar properties in different geometries, dubbed "multi-directional unitaries".
After recalling the definition of dual unitaries (which are unitaries acting on bipartite systems and keep being unitary under a certain rearrangement of the matrix elements), and their connection to highly entangled states using the operator-state correspondence, the concept of multi-directional unitaries is introduced via a direct generalization. This generalization requires an underlying geometry which defines the rearrangements under which the matrix shall keep being unitary. The considered geometries are squares, hexagons, regular polygons, cubes, octahedra and tetrahedra, as the corresponding multi-directional unitaries have had use cases in the literature.
The main body of the manuscript is then considered with different constructions for thee multi-directional unitaries, starting with trivial ones (the corresponding states of which are tensor products of Bell states), followed by constructions using diagonal unitary matrices, Hadamard matrices, graph states and "classical" constructions, involving unitaries which are represented by permutation matrices.

The manuscript is very well written, in particular, the review of existing literature on related topics, such as AME states and dual unitaries, is extensive. The derivations are correct and the topic is of multi-directional unitaries is certainly of relevance, as they are used, for example, in the construction of quantum cellular automata and appear in the kicked Ising model. As such, the plethora of constructions introduced will be certainly useful in the future, There are, however, some minor points of critique which should be addressed by the authors, which are given below. After an appropriate revision, I recommend publication of the manuscript in SciPost.

Requested changes

1. Using both, Ǔ and U for related matrices tends to be confusing (see also below), especially given the fact that the used operator-state correspondence given in Eq. 2.7 differs from the "standard" Choi correspondence, which is given in Eq. 2.17. Is the introduction of Ǔ really necessary? It seems that the results can also be formulated in terms of U instead of Ǔ, or not? If this is not the case, please justify the use of both notations more than just writing "On the other hand, we also introduce the operator U via...".

2. In Eq. 3.1 and 3.2, I think that PUP should read PǓP instead: P...P is a horizontal along the center, which certainly is part of the square symmetry.

3. After Eq. 5.7, "we see that U is indeed unitary" should read "we see that Ǔ is indeed unitary".

4. In Cpt. 6.1, I do not see why the solution alpha = 1 and beta = 2 requires N to be prime if N > 3. Even in non-prime dimensions, the condition beta^2 not equal to alpha^2 is fulfilled.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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