# Strong NP-hardness of the quantum separability problem

@article{Gharibian2010StrongNO, title={Strong NP-hardness of the quantum separability problem}, author={Sevag Gharibian}, journal={Quantum Inf. Comput.}, year={2010}, volume={10}, pages={343-360} }

Given the density matrix ρ of a bipartite quantum state, the quantum separability prob-lem asks whether ρ is entangled or separable. In 2003, Gurvits showed that this problemis NP-hard if ρ is located within an inverse exponential (with respect to dimension) dis-tance from the border of the set of separable quantum states. In this paper, we extendthis NP-hardness to an inverse polynomial distance from the separable set. The resultfollows from a simple combination of works by Gurvits, Ioannou…

## Figures from this paper

## 151 Citations

Quantum Interactive Proofs and the Complexity of Separability Testing

- PhysicsTheory Comput.
- 2015

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.

Small sets of locally indistinguishable orthogonal maximally entangled states

- MathematicsQuantum Inf. Comput.
- 2014

It is given, for the first time, a proof that such sets of states indeed exist even in the case k < d, and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT).

$k$-extendibility of high-dimensional bipartite quantum states

- Mathematics
- 2015

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…

Two-Message Quantum Interactive Proofs and the Quantum Separability Problem

- Computer Science2013 IEEE Conference on Computational Complexity
- 2013

The quantum separability problem constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK.

Quantum interactive proofs and the complexity of entanglement detection

- Computer Science, PhysicsArXiv
- 2013

This paper identifies a formal connection between physical problems related to entanglement detection and complexity classes in theoretical computer science. In particular, we show that to nearly…

Quantum entanglement, sum of squares, and the log rank conjecture

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2017

The algorithm is based on the sum-of-squares hierarchy and its analysis is inspired by Lovett's proof that the communication complexity of every rank-n Boolean matrix is bounded by Õ(√n).

A quasipolynomial-time algorithm for the quantum separability problem

- Computer ScienceSTOC '11
- 2011

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 State Local Distinguishability via Convex Optimization

- Computer Science
- 2015

A framework based on convex optimization for state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements is built.

Limitations of Sum of Squares for 2-to-4 norm , Separability , and Entangled Games

- Computer Science
- 2016

We introduce a method for using reductions to construct integrality gaps for semidefinite programs (SDPs). These imply new limitations on the sum-of-squares (SoS) hierarchy in two settings where…

An exponential time upper bound for Quantum Merlin-Arthur games with unentangled provers

- MathematicsElectron. Colloquium Comput. Complex.
- 2015

We prove a deterministic exponential time upper bound for Quantum Merlin-Arthur games with k unentangled provers. This is the first non-trivial upper bound of QMA(k) better than NEXP and can be…

## References

SHOWING 1-10 OF 57 REFERENCES

Computational complexity of the quantum separability problem

- Computer ScienceQuantum Inf. Comput.
- 2007

This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic computational complexity.

NP vs QMA_log(2)

- Computer ScienceQuantum Inf. Comput.
- 2010

3-SAT admits a QMAlog(2) protocol with the gap 1/n3+e for every constant e > 0.3, which is shown to be a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap1/24n6.

The complexity of the consistency and N-representability problems for quantum states - eScholarship

- Physics
- 2007

Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this…

Largest separable balls around the maximally mixed bipartite quantum state

- Mathematics
- 2002

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…

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

- MathematicsSTOC '03
- 2003

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.

Quantifying Entanglement

- Physics
- 1997

We have witnessed great advances in quantum information theory in recent years. There are two distinct directions in which progress is currently being made: quantum computation and error correction…

Mixed-state entanglement and quantum error correction.

- Computer SciencePhysical review. A, Atomic, molecular, and optical physics
- 1996

It is proved that an EPP involving one-way classical communication and acting on mixed state M (obtained by sharing halves of Einstein-Podolsky-Rosen pairs through a channel) yields a QECC on \ensuremath{\chi} with rate Q=D, and vice versa, and it is proved Q is not increased by adding one- way classical communication.

Additivity of the classical capacity of entanglement-breaking quantum channels

- Physics
- 2002

We show that for the tensor product of an entanglement-breaking quantum channel with an arbitrary quantum channel, both the minimum entropy of an output of the channel and the…

Characterizing Entanglement

- Physics
- 2001

Quantum entanglement is at the heart of many tasks in quantum information. Apart from simple cases (low dimensions, few particles, pure states), however, the mathematical structure of entanglement is…

Entanglement Breaking Channels

- Mathematics
- 2002

This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in…