Reducing computational complexity of quantum correlations

Abstract

We address the issue of reducing the resource required to compute information-theoretic quantum correlation measures like quantum discord and quantum work deficit in two qubits and higher dimensional systems. We show that determination of the quantum correlation measure is possible even if we utilize a restricted set of local measurements. We find that the determination allows us to obtain a closed form of quantum discord and quantum work deficit for several classes of states, with a low error. We show that the computational error caused by the constraint over the complete set of local measurements reduces fast with an increase in the size of the restricted set, implying usefulness of constrained optimization, especially with the increase of dimensions. We perform quantitative analysis to investigate how the error scales with the system size, taking into account a set of plausible constructions of the constrained set. Carrying out a comparative study, we show that the resource required to optimize quantum work deficit is usually higher than that required for quantum discord. We also demonstrate that minimization of quantum discord and quantum work deficit is easier in the case of two-qubit mixed states of fixed ranks and with positive partial transpose in comparison to the corresponding states having non-positive partial transpose. Applying the methodology to quantum spin models, we show that the constrained optimization can be used with advantage in analyzing such systems in quantum information-theoretic language. For bound entangled states, we show that the error is significantly low when the measurements correspond to the spin observables along the three Cartesian coordinates, and thereby we obtain expressions of quantum discord and quantum work deficit for these bound entangled states.