Computing quantum discord is NP-complete

Yichen Huang

doi:10.1088/1367-2630/16/3/033027

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.

Highlights

Quite a few fundamental concepts in quantum mechanics do not have classical analogs: uncertainty relations [7, 12, 48, 49, 75], quantum nonlocality [21, 31, 44, 73], etc
Quantum entanglement [44, 73], defined based on the notion of local operations and classical communication (LOCC), is the most prominent manifestation of quantum correlation. It is a resource in quantum information processing, enabling tasks such as superdense coding [11], quantum teleportation [9] and quantum state merging [41, 42]
The same holds for all descendant protocols of fully quantum Slepian-Wolf (FQSW), where ‘yield’ refers to the amount of entanglement consumed in quantum state merging [62], the amount of distilled entanglement in entanglement distillation [80], the amount of classical information encoded in superdense coding, and the number of teleported qubits in quantum teleportation

Summary

Introduction

Quite a few fundamental concepts in quantum mechanics do not have classical analogs: uncertainty relations [7, 12, 48, 49, 75], quantum nonlocality [21, 31, 44, 73], etc. The classicality problem (detecting whether a given state has zero quantum discord) can be solved in polynomial time [18, 23, 45], but the computational complexity of quantum discord is not known. The hardness proof does not apply to ED, as ED(ρ) can be zero for an entangled state ρ It is an open problem whether computing EC, ER∞, Esq, EsCq, EI , Ib∞ is in NP. It is not clear how large the dimension of ρABC should be so that the right-hand side of (5) is optimal (or sufficiently close to optimal)

NP-completeness of computing quantum discord

NP-completeness of computing constrained Holevo capacity

Applications

Computational complexity of classical states

Conclusion and outlook