Computing quantum discord is NPcomplete
Abstract
We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NPcomplete. Therefore, quantum discord is computationally intractable: the running time of any algorithm for computing quantum discord is believed to grow exponentially with the dimension of the Hilbert space so that computing quantum discord in a quantum system of moderate size is not possible in practice. As byproducts, some entanglement measures (namely entanglement cost, entanglement of formation, relative entropy of entanglement, squashed entanglement, classical squashed entanglement, conditional entanglement of mutual information, and broadcast regularization of mutual information) and constrained Holevo capacity are NPhard/NPcomplete to compute. These complexitytheoretic results are directly applicable in common randomness distillation, quantum state merging, entanglement distillation, superdense coding, and quantum teleportation; they may offer significant insights into quantum information processing. Moreover, we prove the NPcompleteness of two typical problems: linear optimization over classical states and detecting classical states in a convex set, providing evidence that working with classical states is generically computationally intractable.
 Publication:

New Journal of Physics
 Pub Date:
 March 2014
 DOI:
 10.1088/13672630/16/3/033027
 arXiv:
 arXiv:1305.5941
 Bibcode:
 2014NJPh...16c3027H
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 The (published) journal version http://iopscience.iop.org/13672630/16/3/033027/article is more updated than the arXiv versions, and is accompanied with a general scientific summary for nonspecialists in computational complexity