# Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement

@article{DiVincenzo2003UnextendiblePB, title={Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement}, author={David P. DiVincenzo and Tal Mor and Peter W. Shor and John A. Smolin and Barbara M. Terhal}, journal={Communications in Mathematical Physics}, year={2003}, volume={238}, pages={379-410} }

AbstractWe report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We…

## 277 Citations

### From unextendible product bases to genuinely entangled subspaces

- MathematicsPhysical Review A
- 2018

Unextendible product bases (UPBs) are interesting mathematical objects arising in composite Hilbert spaces that have found various applications in quantum information theory, for instance in a…

### Unextendible and uncompletable product bases in every bipartition

- MathematicsNew Journal of Physics
- 2022

Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging from quantum nonlocality to quantum cryptography. A generalized…

### The construction and local distinguishability of multiqubit unextendible product bases

- Mathematics
- 2021

An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It…

### Three-by-three bound entanglement with general unextendible product bases

- Mathematics
- 2011

We discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non-separable positive-partial-transpose states, i.e., states of rank 4…

### Multipartite nonlocality without entanglement in many dimensions

- Computer Science
- 2006

This is the first method to construct a product basis exhibiting nonlocality without entanglement with n parties each holding a system of dimension at least n-1 via a quantum circuit made of controlled discrete Fourier transform gates acting on the computational basis.

### Constructions of Unextendible Special Entangled Bases

- Computer ScienceFrontiers in Physics
- 2022

An efficient method to construct sets of SUEB k is presented, to decompose the whole space into two subspaces such that the rank of each element in one subspace can be easily upper bounded by k while the other one can be generated by two kinds of the special entangled states of “type k”.

### Distance between Bound Entangled States from Unextendible Product Bases and Separable States

- Computer ScienceQuantum Reports
- 2020

It is shown that in most studied cases, witnesses found with the Gilbert algorithm in this work are more optimal than ones obtained by Bandyopadhyay, Ghosh, and Roychowdhury, implying the increased tolerance to experimental imperfections in a realization of the state.

### Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system

- MathematicsScientific reports
- 2016

This paper first constructs a series of orthogonal product bases that are completable but not locally distinguishable in a general m’⊗ n (m’≥ 3 and n’¬3) quantum system, and gives so far the smallest number of locally indistinguishable states of a completable orthogsonal product basis in arbitrary quantum systems.

### Constructing unextendible product bases from multiqubit ones

- PhysicsCommunications in Theoretical Physics
- 2022

The construction of multipartite unextendible product bases (UPBs) is a basic problem in quantum information. We respectively construct two families of 2 × 2 × 4 and 2 × 2 × 2 × 4 UPBs of size…

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