Constructions of $k$-uniform states in heterogeneous systems

• URL: http://arxiv.org/abs/2305.12769v1
• Date: Mon, 22 May 2023 06:58:16 GMT
• Title: Constructions of $k$-uniform states in heterogeneous systems
• Authors: Keqin Feng, Lingfei Jin, Chaoping Xing and Chen Yuan
• Abstract summary: We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power.
• Score: 65.63939256159891
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A pure quantum state of $n$ parties associated with the Hilbert space $\CC^{d_1}\otimes \CC^{d_2}\otimes\cdots\otimes \CC^{d_n}$ is called $k$-uniform if all the reductions to $k$-parties are maximally mixed. The $n$ partite system is called homogenous if the local dimension $d_1=d_2=\cdots=d_n$, while it is called heterogeneous if the local dimension are not all equal. $k$-uniform sates play an important role in quantum information theory. There are many progress in characterizing and constructing $k$-uniform states in homogeneous systems. However, the study of entanglement for heterogeneous systems is much more challenging than that for the homogeneous case. There are very few results known for the $k$-uniform states in heterogeneous systems for $k>3$. We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. The first construction is derived from the error correcting codes by establishing a connection between irredundant mixed orthogonal arrays and error correcting codes. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power. The second construction is derived from a matrix $H$ meeting the condition that $H_{A\times \bar{A}}+H^T_{\bar{A}\times A}$ has full rank for any row index set $A$ of size $k$. These matrix construction can provide more flexible choices for the local dimensions, i.e., the local dimensions can be any integer (not necessarily prime power) subject to some constraints. Our constructions imply that for any positive integer $k$, one can construct $k$-uniform states of a heterogeneous system in many different Hilbert spaces.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Pseudorandom and Pseudoentangled States from Subset States [49.74460522523316]
  A subset state with respect to $S$, a subset of the computational basis, is [ frac1sqrt|S|sum_iin S |irangle. We show that for any fixed subset size $|S|=s$ such that $s = 2n/omega(mathrmpoly(n))$ and $s=omega(mathrmpoly(n))$, a random subset state is information-theoretically indistinguishable from a Haar random state even provided
  arXiv  Detail & Related papers  (2023-12-23T15:52:46Z)
• Dimension-free Remez Inequalities and norm designs [48.5897526636987]
  A class of domains $X$ and test sets $Y$ -- termed emphnorm -- enjoy dimension-free Remez-type estimates. We show that the supremum of $f$ does not increase by more than $mathcalO(log K)2d$ when $f$ is extended to the polytorus.
  arXiv  Detail & Related papers  (2023-10-11T22:46:09Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Multipartite entanglement and quantum error identification in $D$-dimensional cluster states [0.0]
  We show how to create $m$-uniform states using local gates or interactions. We show how to achieve larger $m$ values using quasi-$D$ dimensional cluster states. This opens the possibility to use cluster states to benchmark errors on quantum computers.
  arXiv  Detail & Related papers  (2023-03-27T18:00:02Z)
• Quantum $k$-uniform states from quantum orthogonal arrays [0.0]
  We give infinite classes of 2-uniform states of $N$ systems with dimension of prime power $dgeq 2$ for arbitrary $Ngeq 5$. We also give 3-uniform states of $N$-qubit systems for arbitrary $Ngeq 6$ and $Nneq 7,8,9,11$.
  arXiv  Detail & Related papers  (2023-03-27T08:43:35Z)
• Scattering data and bound states of a squeezed double-layer structure [77.34726150561087]
  A structure composed of two parallel homogeneous layers is studied in the limit as their widths $l_j$ and $l_j$, and the distance between them $r$ shrinks to zero simultaneously. The existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.
  arXiv  Detail & Related papers  (2020-11-23T14:40:27Z)
• $k$-Uniform states and quantum information masking [15.308818907018546]
  A pure state of $N$ parties with local dimension $d$ is called a $k$-uniform state if all the reductions to $k$ parties are maximally mixed. We show that when $dgeq 4k-2$ is a prime power, there exists a $k$-uniform state for any $Ngeq 2k$ (resp. $2kleq Nleq d+1$)
  arXiv  Detail & Related papers  (2020-09-26T01:27:45Z)
• Constructions of $k$-uniform states from mixed orthogonal arrays [18.378398718548016]
  We study $k$-uniform states in heterogeneous systems whose local dimensions are mixed. We present two constructions of $2$-uniform states in heterogeneous systems. We show that some $k$-uniform bases can not be distinguished by local operations and classical communications.
  arXiv  Detail & Related papers  (2020-06-07T08:35:22Z)
• Absolutely maximally entangled states in tripartite heterogeneous systems [10.073311016204238]
  We introduce the concept of irreducible AME states as the basic elements to construct AME states with high local dimensions. We identify the AME states in which kinds of heterogeneous systems are irreducible. In addition, we propose a method to construct $k$-uniform states of more parties using two existing AME states.
  arXiv  Detail & Related papers  (2020-01-23T21:35:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.