Perfect state transfer between real pure states

• URL: http://arxiv.org/abs/2502.08103v2
• Date: Sun, 27 Apr 2025 15:39:32 GMT
• Title: Perfect state transfer between real pure states
• Authors: Chris Godsil, Stephen Kirkland, Hermie Monterde,
• Abstract summary: Pure states correspond to one-dimensional subspaces of $mathbbCn$ represented by unit vectors.<n>We develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Pure states correspond to one-dimensional subspaces of $\mathbb{C}^n$ represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state $\mathbf{x}$ admits perfect state transfer with another real pure state $\mathbf{y}$, (ii) every connected graph admits perfect state transfer between real pure states, and (iii) for any pair of real pure states $\mathbf{x}$ and $\mathbf{y}$ and for any time $\tau$, there exists a real symmetric matrix $M$ such that $\mathbf{x}$ and $\mathbf{y}$ admits perfect state transfer relative to $M$ at time $\tau$. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all $n$-vertex simple unweighted graphs, the least minimum PST time between real pure states relative to the Laplacian is attained by any join graph, while the it is attained by the join of an empty graph and a complete graph of appropriate sizes relative to the adjacency matrix.

Related papers

• Monogamy of Entanglement Bounds and Improved Approximation Algorithms for Qudit Hamiltonians [37.96754147111215]
  We prove new monogamy of entanglement bounds for 2-local qudit Hamiltonian of rank-one projectors without local terms. We certify the ground state energy in terms of the maximum matching of the underlying interaction graph via low-degree sum-of-squares proofs.
  arXiv  Detail & Related papers  (2024-10-21T00:10:51Z)
• Efficient Learning of Balanced Signed Graphs via Iterative Linear Programming [26.334739062500674]
  We propose a fast method to learn a balanced signed graph Laplacian directly from data. Experiments on synthetic and real-world datasets show that our balanced graph learning method outperforms competing methods.
  arXiv  Detail & Related papers  (2024-09-12T06:53:50Z)
• A generalization of quantum pair state transfer [0.0]
  An $s$-pair state in a graph is a quantum state of the form $mathbfe_u+smathbfe_v$. We develop the theory of perfect $s$-pair state transfer in continuous quantum walks.
  arXiv  Detail & Related papers  (2024-04-25T14:45:49Z)
• A Graph is Worth $K$ Words: Euclideanizing Graph using Pure Transformer [47.25114679486907]
  We introduce GraphsGPT, featuring a Graph2Seq encoder that transforms Non-Euclidean graphs into learnable Graph Words. A GraphGPT decoder reconstructs the original graph from Graph Words to ensure information equivalence.
  arXiv  Detail & Related papers  (2024-02-04T12:29:40Z)
• Quantum walks on join graphs [0.0]
  We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian matrix as its associated Hamiltonian. We characterize strong cospectrality, periodicity and perfect state transfer (PST) in a join graph. We demonstrate that the bound $frac2|V(X)|$ is tight for infinite families of graphs.
  arXiv  Detail & Related papers  (2023-12-12T00:33:30Z)
• Three iterations of $(d-1)$-WL test distinguish non isometric clouds of $d$-dimensional points [45.15780579276503]
  We study when the Weisfeiler--Lehman test is complete for clouds of euclidean points represented by complete distance graphs. How many dimensions is enough to distinguish any two non-isometric point clouds in $d$-dimensional Euclidean space? Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $dge 2$, and that only three iterations of the test suffice.
  arXiv  Detail & Related papers  (2023-03-22T18:23:24Z)
• Efficient Signed Graph Sampling via Balancing & Gershgorin Disc Perfect Alignment [51.74913666829224]
  We show that for datasets with strong inherent anti-correlations, a suitable graph contains both positive and negative edge weights. We propose a linear-time signed graph sampling method centered on the concept of balanced signed graphs. Experimental results show that our signed graph sampling method outperformed existing fast sampling schemes noticeably on various datasets.
  arXiv  Detail & Related papers  (2022-08-18T09:19:01Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• $n$-qubit states with maximum entanglement across all bipartitions: A graph state approach [0.0]
  We show that a subset of the 'graph states' satisfy this condition, hence providing a recipe for constructing $k$-uniform states. Finding recipes for construction of $k$-uniform states using graph states is useful since every graph state can be constructed starting from a product state.
  arXiv  Detail & Related papers  (2022-01-14T19:00:09Z)
• Fast Computation of Generalized Eigenvectors for Manifold Graph Embedding [38.902986549367434]
  We leverage existing fast extreme eigenvector computation algorithms for speedy execution. Our embedding is among the fastest in the literature, while producing the best clustering performance for manifold graphs.
  arXiv  Detail & Related papers  (2021-12-15T03:45:39Z)
• Perfect State Transfer in Weighted Cubelike Graphs [0.0]
  A continuous-time quantum random walk describes the motion of a quantum mechanical particle on a graph. We generalize the PST or periodicity of cubelike graphs to that of weighted cubelike graphs.
  arXiv  Detail & Related papers  (2021-09-26T13:44:44Z)
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs [62.997667081978825]
  We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. We show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
  arXiv  Detail & Related papers  (2020-08-11T17:58:13Z)
• Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment [46.145969174332485]
  We propose a fast general metric learning framework that is entirely projection-free. We replace the PD cone constraint in the metric learning problem with possible linear constraints per distances. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes.
  arXiv  Detail & Related papers  (2020-06-15T23:15:12Z)
• Perfect State Transfer on Oriented Graphs [0.0]
  We study the phenomena, unique to oriented graphs, of multiple state transfer. We give a characterization of multiple state transfer, and a new example of a graph where it occurs.
  arXiv  Detail & Related papers  (2020-02-11T20:34:54Z)

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.