Classical deterministic complexity of Edmonds' Problem and quantum entanglement

@inproceedings{Gurvits2003ClassicalDC,
  title={Classical deterministic complexity of Edmonds' Problem and quantum entanglement},
  author={Leonid I. Gurvits},
  booktitle={STOC '03},
  year={2003}
}
  • L. Gurvits
  • Published in STOC '03 11 March 2003
  • Mathematics
Generalizing a decision problem for bipartite perfect matching, J. Edmonds introduced in [14] the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M(N) contains a nonsingular matrix, where M(N) stands for the linear space of complex NxN matrices. This problem led to many fundamental developments in matroid theory etc.Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum… 
On convex optimization problems in quantum information theory
TLDR
This method allows us to find explicit formulae for the REE and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.
Quantum entanglement, symmetric nonnegative quadratic polynomials and moment problems
Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the
Computing finite-dimensional bipartite quantum separability
TLDR
This work motivates a new interior-point algorithm which, given the expected values of a subset of an orthogonal basis of observables of an otherwise unknown quantum state, searches for an entanglement witness in the span of the subset of Observables, and solves the separability problem.
On the Hardness of the Quantum Separability Problem and the Global Power of Locally Invariant Unitary Operations
Given a bipartite density matrix ρ of a quantum state, the Quantum Separability problem (QUSEP) asks — is ρ entangled, or separable? In this thesis, we first strengthen Gurvits’ 2003 NP-hardness
Quantum entanglement, sum of squares, and the log rank conjecture
TLDR
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).
COMPLEXITY OF THE QUANTUM SEPARABILITY PROBLEM
TLDR
A comprehensive treatment of the deterministic quantum separability problem from a computational perspective and identify relevant open problems is attempted.
Geometry of Entanglement and Quantum Simulators
TLDR
The Jordan-Wigner transformation is used to create a compiler that takes any two-body fermionic Hamiltonian and outputs all qubit gates needed to simulate the time evolution of that Hamiltonian.
The complexity of the consistency and N-representability problems for quantum states - eScholarship
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
Quantum-Inspired Hierarchy for Rank-Constrained Optimization
TLDR
It is proved that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states, and consequently, a complete hierarchy of semideFinite programs are constructed for solving the original problem.
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.
...
...