Construction of perfect tensors using biunimodular vectors

Dual unitary gates are highly non-local two-qudit unitary gates that have been studied extensively in quantum many-body physics and quantum information in the recent past. A special subset of dual unitary gates consists of rank-four perfect tensors, which are equivalent to highly entangled multipartite pure states called absolutely maximally entangled (AME) states. In this work, numerical and analytical constructions of dual unitary gates and perfect tensors that are diagonal in a special maximally entangled basis are presented. The main ingredient in our construction is a phase-valued (unimodular) two-dimensional array whose discrete Fourier transform is also unimodular. We obtain perfect tensors for several local Hilbert space dimensions, particularly, in dimension six. A perfect tensor in local dimension six is equivalent to an AME state of four qudits, denoted as AME(4,6), and such a state cannot be constructed from existing constructions of AME states based on error-correcting codes and graph states. The existence of AME(4,6) states featured in well-known open problem lists in quantum information, and was settled positively in Phys. Rev. Lett. 128 080507 (2022). We provide an explicit construction of perfect tensors in local dimension six that can be written in terms of controlled unitary gates in the computational basis, making them amenable for quantum circuit implementations.Comment: 10+9 pages, 3+1 Figures. Comments are welcom

ON ZYGMUND–TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS

Let $P(z)$ be a polynomial of degree $n$, then concerning the estimate for maximum of $|P'(z)|$ on the unit circle, it was proved by S. Bernstein that $\| P'\|_{\infty}\leq n\| P\|_{\infty}$. Later, Zygmund obtained an $L_p$-norm extension of this inequality. The polar derivative $D_{\alpha}[P](z)$ of $P(z)$, with respect to a point $\alpha \in \mathbb{C}$, generalizes the ordinary derivative in the sense that $\lim_{\alpha\to\infty} D_{\alpha}[P](z)/{\alpha} = P'(z).$ Recently, for polynomials of the form $P(z) = a_0 + \sum_{j=\mu}^n a_jz^j,$ $1\leq\mu\leq n$ and having no zero in $|z| 1$, the following Zygmund-type inequality for polar derivative of $P(z)$ was obtained: $$\|D_{\alpha}[P]\|_p\leq n \Big(\dfrac{|\alpha|+k^{\mu}}{\|k^{\mu}+z\|_p}\Big)\|P\|_p, \quad \text{where}\quad |\alpha|\geq1,\quad p>0.$$In this paper, we obtained a refinement of this inequality by involving minimum modulus of $|P(z)|$ on $|z| = k$, which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well

Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler

Ordering and classifying multipartite quantum states by their entanglement content remains an open problem. One class of highly entangled states, useful in quantum information protocols, the absolutely maximally entangled (AME) ones, are specially hard to compare as all their subsystems are maximally random. While, it is well-known that there is no AME state of four qubits, many analytical examples and numerically generated ensembles of four qutrit AME states are known. However, we prove the surprising result that there is truly only {\em one} AME state of four qutrits up to local unitary equivalence. In contrast, for larger local dimensions, the number of local unitary classes of AME states is shown to be infinite. Of special interest is the case of local dimension 6 where it was established recently that a four-party AME state does exist, providing a quantum solution to the classically impossible Euler problem of 36 officers. Based on this, an infinity of quantum solutions are constructed and we prove that these are not equivalent. The methods developed can be usefully generalized to multipartite states of any number of particles.Comment: Rewritten as a regular article and few changes in the title from first version. Close to the published versio

9 × 4 = 6 × 6: Understanding the quantum solution to Euler's problem of 36 officers

The famous combinatorial problem of Euler concerns an arrangement of 36 officers from six different regiments in a 6×6 square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to superpositions of quantum states. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each quantum officer is represented by a superposition of at most 4 classical states. The entire quantum combinatorial design involves 9 Bell bases in nine complementary 4-dimensional subspaces

9 ×\times 4 = 6 ×\times 6: Understanding the quantum solution to the Euler's problem of 36 officers

The famous combinatorial problem of Euler concerns an arrangement of $36$ officers from six different regiments in a $6 \times 6$ square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to quantum states and can be entangled. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each officer is entangled with at most three out of his $35$ colleagues. The entire quantum combinatorial design involves $9$ Bell bases in nine complementary $4$-dimensional subspaces.Comment: 26 page

Thirty-six entangled officers of Euler : quantum solution to a classically impossible problem

The negative solution to the famous problem of $36$ officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME$(4,6)$ of four subsystems with six levels each, equivalently a $2$-unitary matrix of size $36$, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code $(\!(3,6,2)\!)_6$, which saturates the Singleton bound and allows one to encode a $6$-level state into a triplet of such states.Comment: 14 pages, 12 figure

I₂‑DMSO Promoted Deaminative Coupling Reactions of Glycine Esters: Access to 5‑(Methylthio)pyridazin-3(2<i>H</i>)‑ones

An unprecedented, one-step strategy for the synthesis of 5-(methylthio)pyridazin-3(2H)-one derivatives has been developed through iodine triggered deaminative coupling of glycine esters with methyl ketones and hydrazine hydrate in DMSO. These transformations in the absence of hydrazine helped to generate different 3-methylthio-4-oxo-enoates in good yields. Notably, DMSO played multiple roles such as oxidant, methylthiolating reagent, and solvent