Computing quantum discord is NP-complete

TL;DR

This paper proves that computing quantum discord, a measure of quantum correlation, is NP-complete, indicating it is computationally intractable for systems of moderate size, with implications for various quantum information tasks.

Contribution

The paper establishes the NP-completeness of quantum discord computation and related measures, highlighting their inherent computational difficulty.

Findings

Quantum discord computation is NP-complete.

Several entanglement measures are NP-hard or NP-complete to compute.

Problems like classical state detection are also NP-complete.

Abstract

We study the computational complexity of quantum discord (a measure of quantum correlation beyond entanglement), and prove that computing quantum discord is NP-complete. 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 by-products, 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 NP-hard/NP-complete to compute. These complexity-theoretic 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 NP-completeness 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.