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… 

Unextendible and uncompletable product bases in every bipartition

Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging bound entangled states, quantum nonlocality without

From unextendible product bases to genuinely entangled subspaces

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

The construction and local distinguishability of multiqubit unextendible product bases

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

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

TLDR
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

TLDR
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

TLDR
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

TLDR
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

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 eight by
...

References

SHOWING 1-10 OF 30 REFERENCES

Unextendible product bases and bound entanglement

An unextendible product basis( UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and

Entanglement purification via separable superoperators

TLDR
This work uses the fact that every EPP is a separable superoperator to give a new upper bound on the rate of EPPs on Bell- diagonal states, and thus on the capacity of Bell-diagonal channels.

Quantum nonlocality without entanglement

We exhibit an orthogonal set of product states of two three-state particles that nevertheless cannot be reliably distinguished by a pair of separated observers ignorant of which of the states has

Product Bases in Quantum Information Theory

We review the role of product bases in quantum information theory. We prove two conjectures which were made in DiVincenzo et al., quant-ph/9908070, namely the existence of two sets of bipartite

Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?

It is shown that if a mixed state can be distilled to the singlet form it must violate partial transposition criterion [A. Peres, Phys. Rev. Lett. 76, 1413 (1996)]. It implies that there are two

Purification of noisy entanglement and faithful teleportation via noisy channels.

TLDR
Upper and lower bounds on the yield of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$ are given, showing $D(M)>0$ if $\bra{Psi-}M\ket-}>\half$.

RIGOROUS TREATMENT OF DISTILLABLE ENTANGLEMENT

TLDR
A new definition of distillable entanglement is given which removes the constraint that the distilation protocol produce an output of constant dimension, but could conceivably overestimate the true value.

General teleportation channel, singlet fraction and quasi-distillation

We prove a theorem on direct relation between the optimal fidelity $f_{max}$ of teleportation and the maximal singlet fraction $F_{max}$ attainable by means of trace-preserving LQCC action (local