Journal of Symbolic Computation
Polynomial time algorithms for modules over finite dimensional algebras
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Mathematics of Operations Research
Random Structures & Algorithms
A deterministic polynomial-time algorithm for approximating mixed discriminant and mixed volume
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
A Linear Lower Bound on the Unbounded Error Probabilistic Communication Complexity
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Matrix integrals and map enumeration: An accessible introduction
Mathematical and Computer Modelling: An International Journal
The Quantum Separability Problem for Gaussian States
Electronic Notes in Theoretical Computer Science (ENTCS)
Combinatorial laplacians and positivity under partial transpose
Mathematical Structures in Computer Science
Quantum Entanglement Analysis Based on Abstract Interpretation
SAS '08 Proceedings of the 15th international symposium on Static Analysis
Equation of motion for entanglement
Quantum Information Processing
Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
SIAM Journal on Optimization
Strong NP-hardness of the quantum separability problem
Quantum Information & Computation
Two slightly-entangled NP-complete problems
Quantum Information & Computation
On independent permutation separability criteria
Quantum Information & Computation
The distillability problem revisited
Quantum Information & Computation
Computational complexity of the quantum separability problem
Quantum Information & Computation
Separability criteria based on the bloch representation of density matrices
Quantum Information & Computation
On global effects caused by locally noneffective unitary operations
Quantum Information & Computation
An analytic approach to the problem of separability of quantum states based upon the theory of cones
Quantum Information Processing
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
Can quantum entanglement detection schemes improve search?
Quantum Information Processing
On the complexity of mixed discriminants and related problems
MFCS'05 Proceedings of the 30th international conference on Mathematical Foundations of Computer Science
Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Maximum Block Improvement and Polynomial Optimization
SIAM Journal on Optimization
Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization
Journal of the ACM (JACM)
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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 Theory, is a natural generalization of matrices with nonnegative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of of completely positive operators, or equivalently, in terms of bipartite density matrices. 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. We introduce a very general class (or promise) of linear subspaces of M(N) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem. The algorithm is a thoroughgoing generalization of algorithms in [23], [26], and its analysis benefits from an operator analog of permanents, so called Quantum Permanents. Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD.