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Construction and Local Equivalence of Dual-Unitary Operators: From Dynamical Maps to Quantum Combinatorial Designs

Suhail Ahmad Rather1,*, S. Aravinda2,†, and Arul Lakshminarayan1

  • 1Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
  • 2Department of Physics, Indian Institute of Technology Tirupati, Tirupati 517619, India

  • *suhailmushtaq@physics.iitm.ac.in
  • †aravinda@iittp.ac.in
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PRX Quantum 3, 040331 – Published 20 December, 2022

DOI: https://doi.org/10.1103/PRXQuantum.3.040331

Abstract

While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization of dual-unitary operators themselves are only partially understood. A nonlinear map on the space of unitary operators has been proposed in Phys. Rev. Lett. 125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries. Here, we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries. A subset of dual-unitary operators having maximum entangling power are 2-unitary operators or perfect tensors and these are equivalent to four-party absolutely maximally entangled states. It is known that they only exist if the local dimension is larger than d=2. We use the nonlinear map, and introduce stochastic variants of it, to construct explicit examples of new dual and 2-unitary operators. A necessary criterion for their local unitary equivalence to distinguish classes is also introduced and used to display various concrete results and a conjecture in d=3. It is known that orthogonal Latin squares provide a “classical combinatorial design” for constructing permutations that are 2-unitary. We extend the underlying design from classical to genuine quantum ones for general dual-unitary operators and give an example of what might be the smallest-sized genuinely quantum design of a 2-unitary in d=4.

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Popular Summary

Entanglement in quantum states gives rise to the famous "spooky action at a distance," wherein particles apparently influence one another instantaneously. This phenomenon of nonlocality has now been tested in many seminal experiments. Apart from states, quantum mechanics contains operators that act to change these states. Nature evolves states via unitary operators that conserve total probability of events. Entanglement in unitary operators leads to entanglement in states. However, the study of such operator entanglement is not as well developed as it is for states. Maximally entangled states in some sense are the most nonlocal ones. While it is easy to construct maximally entangled states, the construction of maximally entangled unitary operators is still surprisingly incomplete and hard. Precisely, such operators have been recently called "dual-unitary" and used by several communities of physicists, for example, in studies of quantum circuits that mimic quantum chaotic many-body systems as well as in models of quantized gravity. This paper describes novel methods of constructing dual-unitary operators in arbitrary dimensions and makes headway in classifying them.

A special class of dual-unitary operators can be used to construct four-party entangled states—called absolutely maximally entangled (AME) states—that are maximally entangled across all possible bipartitions and have many applications. This paper contains constructions of AME states involving quantum analogs of orthogonal Latin squares. Methods to distinguish classes of AME states are introduced, based on which we conjecture that there is only one unique class of AME states of four qutrits (3 level atoms) represented by the permutation P9.

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PRX Quantum 3, 040331– Published 20 December, 2022
  • Received 4 June 2022
  • Revised 16 October 2022
  • Accepted 16 November 2022
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