[Paper Review] Strong NP-Hardness of the Quantum Separability Problem

Sevag Gharibian|arXiv (Cornell University)|Oct 24, 2008

Quantum Computing Algorithms and Architecture40 references6 citations

TL;DR

This paper establishes the strong NP-hardness of the quantum separability problem by showing that determining whether a quantum state is separable remains NP-hard even when the state is at an inverse polynomial distance from the separable set. The result is derived by combining reductions from CLIQUE and robust semidefinite feasibility, extending earlier NP-hardness results that only applied to exponentially close states.

ABSTRACT

Given the density matrix rho of a bipartite quantum state, the quantum separability problem asks whether rho is entangled or separable. In 2003, Gurvits showed that this problem is NP-hard if rho is located within an inverse exponential (with respect to dimension) distance from the border of the set of separable quantum states. In this paper, we extend this NP-hardness to an inverse polynomial distance from the separable set. The result follows from a simple combination of works by Gurvits, Ioannou, and Liu. We apply our result to show (1) an immediate lower bound on the maximum distance between a bound entangled state and the separable set (assuming P != NP), and (2) NP-hardness for the problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking.

Motivation & Objective

• To resolve the complexity of the quantum separability problem (QUSEP) when the input state is bounded away from the separable set by an inverse polynomial distance, rather than exponentially small distance.
• To strengthen prior NP-hardness results for QUSEP by extending them beyond the regime of exponentially close states, where finite-precision computation issues are less severe.
• To establish a lower bound on the maximum Euclidean distance between bound entangled states and the separable set under the assumption that P ≠ NP.
• To prove NP-hardness for determining whether a completely positive trace-preserving map is entanglement-breaking, a fundamental problem in quantum information theory.

Proposed method

• Leverages a many-one reduction from the NP-complete CLIQUE problem to Robust Semidefinite Feasibility (RSDF), as established by Liu.
• Applies a Turing reduction from RSDF to Weak Validity (WVALα), which tests whether a hyperplane approximately separates a convex set.
• Uses a known equivalence between Weak Validity and Weak Membership (WMEMβ) for convex sets, specifically for the set of separable states SM,N.
• Employs a Bloch vector representation of density matrices to relate the trace norm distance between quantum states to the Euclidean distance between their Bloch vectors.
• Combines these reductions to show that WMEMβ(SM,N) is NP-hard for any inverse polynomial β, not just exponentially small β.
• Uses Lemma 8 to relate the Hilbert-Schmidt norm of density matrices to the Euclidean norm of their Bloch vectors, enabling geometric analysis of the separability boundary.

Experimental results

Research questions

• RQ1 Is the quantum separability problem strongly NP-hard when the input state is at an inverse polynomial distance from the separable set?
• RQ2 What is the maximum possible Euclidean distance between a bound entangled state and the set of separable states, assuming P ≠ NP?
• RQ3 Is the problem of determining whether a quantum channel is entanglement-breaking NP-hard?
• RQ4 Can the NP-hardness of the weak membership problem for separable states be extended beyond the exponentially close regime?
• RQ5 What is the role of robust semidefinite feasibility in establishing complexity bounds for quantum state classification?

Key findings

• The quantum separability problem is strongly NP-hard when the input state is at an inverse polynomial distance from the separable set, extending Gurvits' earlier result that only applied to exponentially small distances.
• An immediate lower bound on the maximum Euclidean distance between a bound entangled state and the set of separable states is established under the assumption that P ≠ NP.
• The problem of determining whether a completely positive trace-preserving linear map is entanglement-breaking is shown to be NP-hard.
• The proof relies on a chain of reductions: CLIQUE ≤m RSDF ≤m WVALα(SM,N) ≤T WMEMβ(SM,N), with β being inverse polynomial in the system dimensions.
• The norm of the Bloch vector of the constructed state is shown to be in O(√N), where N is the dimension of the subsystem, supporting the inverse polynomial distance bound.
• The result implies that even with finite-precision input, distinguishing separable from entangled states remains computationally hard when the state is not too close to the boundary of separability.