Abstract
We consider the entanglement marginal problem, which consists of deciding whether a number of reduced density matrices are compatible with an overall separable quantum state. To tackle this problem, we propose hierarchies of semidefinite programming relaxations of the set of quantum state marginals admitting a fully separable extension. We connect the completeness of each hierarchy to the resolution of an analog classical marginal problem and thus identify relevant experimental situations where the hierarchies are complete. For finitely many parties on a star configuration or a chain, we find that we can achieve an arbitrarily good approximation to the set of nearest-neighbour marginals of separable states with a time (space) complexity polynomial (linear) on the system size. Our results even extend to infinite systems, such as translation-invariant systems in 1D, as well as higher spatial dimensions with extra symmetries.
Highlights
Recent quantum experiments with cold atoms and superconducting qubits have demonstrated different degrees of control over systems of size ranging from about 50 [1] to several hundred qubits [2]
The application of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of relaxations [3] to characterize the set of separable states, or some entanglement detection methods based on incomplete information [4] would require solving an optimization problem where one of the variables corresponds to the full density matrix of the system
In Proposition 4, the set of state ensembles to which this hierarchy converges to. This set corresponds to the set of marginals of an overall separable state iff the classical marginal problem is trivial in the considered marginal scenario
Summary
Recent quantum experiments with cold atoms and superconducting qubits have demonstrated different degrees of control over systems of size ranging from about 50 [1] to several hundred qubits [2]. The application of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of relaxations [3] to characterize the set of separable states, or some entanglement detection methods based on incomplete information [4] would require solving an optimization problem where one of the variables corresponds to the full density matrix of the system. We find that, in many cases of physical interest, the corresponding hierarchies have a time complexity polynomial on the system size, and a memory complexity linear on the system size This allows us to certify entanglement in systems of hundreds of sites. In Proposition 4, the set of state ensembles to which this hierarchy converges to As we argue, this set corresponds to the set of marginals of an overall separable state iff the classical marginal problem is trivial in the considered marginal scenario.
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