Dvoretzky's theorem and the complexity of entanglement detection

@article{Aubrun2017DvoretzkysTA,
  title={Dvoretzky's theorem and the complexity of entanglement detection},
  author={Guillaume Aubrun and Stanisław J. Szarek},
  journal={arXiv: Quantum Physics},
  year={2017},
  volume={5202017},
  pages={1-20}
}
Dvoretzky's theorem and the complexity of entanglement detection, Discrete Analysis 2017:1, 20 pp. Let $H$ be a Hilbert space. A _state_ on $H$ is a linear operator $\rho:H\to H$ such that tr$(\rho)=1$ and tr$(\rho P)\geq 0$ for every orthogonal projection $P$. A linear operator satisfying just the second condition is called _positive_. If $H=H_1\otimes H_2$ then an important piece of information about $\rho$ is the extent to which it can be split up into a part that acts on $H_1$ and a part… 
8 Citations

Tables from this paper

Lower Bounds for Testing Complete Positivity and Quantum Separability
TLDR
It is shown that learning an unknown completely positive distribution on the grid $[d] \times [d]$ requires $\Omega(d/\epsilon^2)$ samples, and that the true complexity of quantum entanglement may in fact be higher.
Limitations of Semidefinite Programs for Separable States and Entangled Games
TLDR
A new method for using reductions to construct integrality gaps for SDPs, meaning instances where the SDP value is far from the true optimum, is introduced, based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory.
The Set of Separable States has no Finite Semidefinite Representation Except in Dimension $$3\times 2$$
Given integers n $\geq$ m, let Sep(n,m) be the set of separable states on the Hilbert space $\mathbb{C}^n \otimes \mathbb{C}^m$. It is well-known that for (n,m)=(3,2) the set of separable states has
USING RANDOM MATRICES IN QUANTUM INFORMATION THEORY
TLDR
The method of moments is presented, one of the most successful methods used to study the spectra of large random matrices, and integration over Gaussian spaces is discussed.
On polyhedral approximations of the positive semidefinite cone
  • Hamza Fawzi
  • Mathematics, Computer Science
    Math. Oper. Res.
  • 2021
TLDR
Lower bounds on the size of linear programs that approximate the positive semidefinite cone are proved and hypercontractivity of the noise operator on the hypercube is demonstrated, demonstrating that there is no generic way of approximating semidfinite programs with compact linear programs.
APPLICATIONS OF RANDOM MATRICES IN QUANTUM INFORMATION THEORY
These are notes for five lectures given at the second School of pthe program “Operator Algebras, Groups and Applications to Quantum Information” held in May 2019 at the ICMAT in Mardid. The goal of
On approximations of the PSD cone by a polynomial number of smaller-sized PSD cones
TLDR
Lower bounds on N are proved to achieve a good approximation of D by considering two constructions of an approximating set and it is shown that any set S that approximates D within a constant approximation ratio must have superpolynomial S+-extension complexity.

References

SHOWING 1-10 OF 48 REFERENCES
$k$-extendibility of high-dimensional bipartite quantum states
The idea of detecting the entanglement of a given bipartite state by searching for symmetric extensions of this state was first proposed by Doherty, Parrilo and Spedialeri. The complete family of
Limitations of Semidefinite Programs for Separable States and Entangled Games
TLDR
A new method for using reductions to construct integrality gaps for SDPs, meaning instances where the SDP value is far from the true optimum, is introduced, based on new limitations on the sum-of-squares (SoS) hierarchy in approximating two particularly important sets in quantum information theory.
A quasipolynomial-time algorithm for the quantum separability problem
TLDR
A quasipolynomial-time algorithm for solving the weak membership problem for the convex set of separable, i.e. non-entangled, bipartite density matrices and an improved de Finetti-type bound quantifying the monogamy of quantum entanglement are presented.
Quantum Interactive Proofs and the Complexity of Separability Testing
TLDR
Strong hardness results are obtained by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.
Classical deterministic complexity of Edmonds' Problem and quantum entanglement
TLDR
It turns out that one of the most important cases when Edmonds' problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable) bipartite density matrices.
Largest separable balls around the maximally mixed bipartite quantum state
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral ${l}_{p}$ norms for $1l~pl~\ensuremath{\infty},$ of separable (unentangled) matrices around
Complete family of separability criteria
We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state rho to a larger number of parties satisfying certain symmetry properties. It
Tensor products of convex sets and the volume of separable states on N qudits (10 pages)
This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we
Computational complexity of the quantum separability problem
  • L. Ioannou
  • Computer Science
    Quantum Inf. Comput.
  • 2007
TLDR
This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic computational complexity.
Strong NP-hardness of the quantum separability problem
TLDR
An immediate lower bound on the maximum distance between abound entangled state and the separable set is shown and NP-hardness for the problem of determining whether a completely positive trace-preserving linearmap is entanglement-breaking is extended to an inverse polynomial distance.
...
1
2
3
4
5
...