Unextendible and strongly uncompletable product bases

• URL: http://arxiv.org/abs/2411.18036v1
• Date: Wed, 27 Nov 2024 04:09:30 GMT
• Title: Unextendible and strongly uncompletable product bases
• Authors: Xiao-Fan Zhen, Hui-Juan Zuo, Fei Shi, Shao-Ming Fei,
• Abstract summary: We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. We construct a UPB with smaller size $d3-3d2+1 in $mathbbCdotimes mathbbCdotimes mathbbCd$, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.
• Score: 4.2270183742578835
• License:
• Abstract: In 2003, DiVincenzo {\it et al}. put forward the question that whether there exists an unextendible product basis (UPB) which is an uncompletable product basis (UCPB) in every bipartition [\href{https://link.springer.com/article/10.1007/s00220-003-0877-6}{DiVincenzo {\it et al}. Commun. Math. Phys. \textbf{238}, 379-410(2003)}]. Recently, Shi {\it et al}. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [\href{https://iopscience.iop.org/article/10.1088/1367-2630/ac9e14}{Shi {\it et al}. New J. Phys. \textbf{24}, 113-025 (2022)}]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size $d^3-3d^2+1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.

Related papers

• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• 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)
• Unextendibility, uncompletability, and many-copy indistinguishable ensembles [77.34726150561087]
  We study unextendibility, uncompletability and analyze their connections to many-copy indistinguishable ensembles. We report a class of multipartite many-copy indistinguishable ensembles for which local indistinguishability property increases with decreasing mixedness.
  arXiv  Detail & Related papers  (2023-03-30T16:16:41Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Construction of multipartite unextendible product bases and geometric measure of entanglement of positive-partial-transpose entangled states [0.0]
  We show that there exist two families UPBs in Hilbert space $mathbbC2otimesmathbbC2otimesmathbbC2otimesmathbbC2otimesmathbbC4$ by merging two different systems of an existing $7$-qubit UPB of size $11$. A new family of $7$-qubit positive-partial-transpose entangled states of rank $27-11$ is constructed.
  arXiv  Detail & Related papers  (2022-12-05T17:42:47Z)
• Average-Case Complexity of Tensor Decomposition for Low-Degree Polynomials [93.59919600451487]
  "Statistical-computational gaps" occur in many statistical inference tasks. We consider a model for random order-3 decomposition where one component is slightly larger in norm than the rest. We show that tensor entries can accurately estimate the largest component when $ll n3/2$ but fail to do so when $rgg n3/2$.
  arXiv  Detail & Related papers  (2022-11-10T00:40:37Z)
• Monogamy of entanglement between cones [68.8204255655161]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones. Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Strong quantum nonlocality for unextendible product bases in heterogeneous systems [15.308818907018546]
  Unextendible product bases (UPBs) can show the phenomenon of quantum nonlocality without entanglement. We construct a series of UPBs with different large sizes in $d_Aotimes d_Botimes d_Cotimes d_D$ for $d_A, d_B, d_C, d_Dq 3$, and we show that these UPBs have strong nonlocality.
  arXiv  Detail & Related papers  (2022-01-01T01:18:37Z)
• Strong nonlocal sets of UPB [4.337598489115445]
  We investigate the construction of 3-qubit UPB with strong nonlocality of different sizes. By means of this structure, a $C4otimes C4otimes C5$ system is obtained based on a $C3otimes C3otimes C4$ system.
  arXiv  Detail & Related papers  (2021-06-16T11:06:47Z)
• On the state space structure of tripartite quantum systems [0.22741525908374005]
  It has been shown that the set of states separable across all the three bipartitions [say $mathcalBint(ABC)$] is a strict subset of the set of states having positive partial transposition (PPT) across the three bipartite cuts [say $mathcalPint(ABC)$] The claim is proved by constructing state belonging to the set $mathPint(ABC)$ but not belonging to $mathcalBint(ABC)$.
  arXiv  Detail & Related papers  (2021-04-14T16:06:58Z)
• Unextendible product bases from tile structures and their local entanglement-assisted distinguishability [16.424004426651326]
  We characterize the condition when a tile structure provides an unextendible product basis (UPB) We show that our UPBs of size $(mn-4lfloorfracm-12rfloor)$ in $mathbbCmotimesmathbbCn$ can be perfectly distinguished by local operations and classical communications.
  arXiv  Detail & Related papers  (2020-03-09T03:01:04Z)

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.