A quasipolynomial-time algorithm for the quantum separability problem

We present a quasipolynomial-time algorithm for solving the weak membership problem for the convex set of separable, i.e. non-entangled, bipartite density matrices. The algorithm decides whether a density matrix is separable or whether it is eps-away from the set of the separable states in time exp(O(eps^-2 log |A| log |B|)), where |A| and |B| are the local dimensions, and the distance is measured with either the Euclidean norm, or with the so-called LOCC norm. The latter is an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by quantum local operations and classical communication (LOCC) between the parties. We also obtain improved algorithms for optimizing over the set of separable states and for computing the ground-state energy of mean-field Hamiltonians. The techniques we develop are also applied to quantum Merlin-Arthur games, where we show that multiple provers are not more powerful than a single prover when the verifier is restricted to LOCC protocols, or when the verification procedure is formed by a measurement of small Euclidean norm. This answers a question posed by Aaronson et al (Theory of Computing 5, 1, 2009) and provides two new characterizations of the complexity class QMA, a quantum analog of NP. Our algorithm uses semidefinite programming to search for a symmetric extension, as first proposed by Doherty, Parrilo and Spedialieri (Phys. Rev. A, 69, 022308, 2004). The bound on the runtime follows from an improved de Finetti-type bound quantifying the monogamy of quantum entanglement, proved in (arXiv:1010.1750). This result, in turn, follows from a new lower bound on the quantum conditional mutual information and the entanglement measure squashed entanglement.

💡 Research Summary

The paper tackles the weak‑membership version of the quantum separability problem: given a bipartite density matrix ρ on systems A and B, decide whether ρ is separable or at least ε‑far from the set of separable states. Two distance measures are considered: the ordinary Euclidean (2‑norm) distance and the operationally motivated LOCC norm, which quantifies the optimal distinguishing probability when only local operations and classical communication are allowed.

The core algorithmic idea builds on the hierarchy of symmetric extensions introduced by Doherty, Parrilo, and Spedalieri (DPS). A state that admits a k‑symmetric extension is guaranteed to be “k‑extendable”, a sufficient condition for separability. The authors show that it suffices to search for extensions up to a depth k = O(ε⁻² log|A| log|B|). This search is performed via a semidefinite program (SDP) that encodes the existence of a symmetric extension together with the positivity constraints required by quantum mechanics.

A novel technical contribution is an improved de Finetti‑type bound that quantifies the monogamy of entanglement. By establishing a new lower bound on the quantum conditional mutual information I(A;B|C) and relating it to squashed entanglement, the authors prove that if a state fails to have a k‑symmetric extension at the prescribed depth, then its distance from the separable set is at least ε. This bridges the gap between the abstract de Finetti approximation and a concrete, quantitative guarantee needed for algorithmic analysis.

Complexity analysis shows that the SDP has O(|A|·|B|·k) variables and polynomially many constraints, leading to an overall running time of exp(O(k log|A| log|B|)). Substituting the bound on k yields a quasipolynomial runtime exp(O(ε⁻² log|A| log|B|)). This is a dramatic improvement over previously known algorithms whose dependence on ε was exponential, making the method feasible for moderate‑size quantum systems.

Beyond the basic weak‑membership test, the paper demonstrates two important applications. First, it yields an efficient algorithm for optimizing linear functionals over the separable set, which is central to many tasks such as bounding channel capacities or entanglement measures. By embedding the optimization into the same SDP hierarchy, the algorithm returns an ε‑approximate optimum with the same quasipolynomial guarantee. Second, the technique is applied to mean‑field Hamiltonians: the ground‑state energy of a Hamiltonian that is a sum of one‑body terms can be expressed as a minimization over separable states, and the SDP provides a provably close estimate of this energy.

The authors also explore implications for quantum complexity theory. They consider Quantum Merlin‑Arthur (QMA) proof systems where the verifier is restricted either to LOCC measurements or to measurements of small Euclidean norm. In these settings, they prove that multiple provers (the QMA(2) model) are no more powerful than a single prover. Consequently, they obtain two new characterizations of QMA: (i) LOCC‑restricted QMA and (ii) QMA with small‑norm verification operators. This resolves an open question posed by Aaronson et al. (2009) regarding the power of multiple quantum provers under realistic verification constraints.

In summary, the paper delivers a quasipolynomial‑time algorithm for the weak‑membership separability problem, under both Euclidean and LOCC norms, by leveraging symmetric extensions, an improved de Finetti bound, and a fresh analysis of quantum conditional mutual information. It extends the algorithmic toolkit for separability testing, provides practical methods for optimization over separable states and for estimating mean‑field ground energies, and deepens our understanding of the relationship between prover multiplicity and verification restrictions in quantum complexity classes.