Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the strong non-rigidity of certain tight Euclidean designs
59
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the non-rigidity of Euclidean $t$-designs, namely we study when Euclidean designs (in particular certain tight Euclidean designs) can be deformed keeping the property of being Euclidean $t$-designs. We show that certain tight Euclidean $t$-designs are non-rigid, and in fact satisfy a stronger form of non-rigidity which we call strong non-rigidity. This shows that there are plenty of non-isomorphic tight Euclidean $t$-designs for certain parameters, which seems to have been unnoticed before. We also include the complete classification of tight Euclidean $2$-designs.
rate research
Read More
Relative $t$-designs in the $n$-dimensional hypercube $mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $mathcal{Q}_n$ supported on two shells are rare for large $t$.
Unitary $t$-designs are `good finite subsets of the unitary group $U(d)$ that approximate the whole unitary group $U(d)$ well. Unitary $t$-designs have been applied in randomized benchmarking, tomography, quantum cryptography and many other areas of quantum information science. If a unitary $t$-design itself is a group then it is called a unitary $t$-group. Although it is known that unitary $t$-designs in $U(d)$ exist for any $t$ and $d$, the unitary $t$-groups do not exist for $tgeq 4$ if $dgeq 3$, as it is shown by Guralnick-Tiep (2005) and Bannai-Navarro-Rizo-Tiep (BNRT, 2018). Explicit constructions of exact unitary $t$-designs in $U(d)$ are not easy in general. In particular, explicit constructions of unitary $4$-designs in $U(4)$ have been an open problem in quantum information theory. We prove that some exact unitary $(t+1)$-designs in the unitary group $U(d)$ are constructed from unitary $t$-groups in $U(d)$ that satisfy certain specific conditions. Based on this result, we specifically construct exact unitary $3$-designs in $U(3)$ from the unitary $2$-group $SL(3,2)$ in $U(3),$ and also unitary $4$-designs in $U(4)$ from the unitary $3$-group $Sp(4,3)$ in $U(4)$ numerically. We also discuss some related problems.
Let $q$ be a prime power and $Vcong{mathbb F}_q^n$. A $t$-$(n,k,lambda)_q$ design, or simply a subspace design, is a pair ${mathcal D}=(V,{mathcal B})$, where ${mathcal B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $lambda$ elements of ${mathcal B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group ${rm Aut}({mathcal D})$ acts transitively on ${mathcal B}$. It is shown here that if $tgeq 2$ and ${mathcal D}=(V,{mathcal B})$ is a block-transitive $t$-$(n,k,lambda)_q$ design then ${mathcal D}$ is trivial, that is, ${mathcal B}$ is the set of all $k$-dimensional subspaces of $V$.
We consider $m$-divisible non-crossing partitions of ${1,2,ldots,mn}$ with the property that for some $tleq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapotons $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.
We study tight projective 2-designs in three different settings. In the complex setting, Zauners conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
