When we need to estimate a physical quantity or parameter in a quantum system, we prepare an experimental probe, let it interact with the system, and then get a read out from the probe. Such a measurement process almost inevitably involves statistical or systematic errors. Whatever their origins, the effects of statistical errors can be reduced by repeating the measurements and averaging the outcomes. Given a fixed number of measurements (say $N$), the standard way of making repetitive, independent measurements reaches a limit in the estimation precision. It turns out, however, that this limit can be improved considerably by building quantum correlations into the measurements through different strategies. This is essentially what quantum metrology aims to do. Although there are different types of quantum correlations, namely, entanglement and discord, the latter, in contrast to the former, has so far not been suspected to play a role in quantum metrology. In this theoretical paper, however, we propose that quantum discord, in fact, does have a role to play, at least, when the states of the quantum system in question are highly “mixed,” as they are in cases where the system experiences external noises such as thermal fluctuations.

The metrological task that we have set out to perform is to estimate the relative phase of a mixed quantum state, which is important in many quantum phenomena such as quantum interferometry and quantum decoherence. To this end, we describe the mixed state as a qubit, and explore four different measurement strategies, each characterized by a $N$-qubit quantum circuit with a different type of correlation among the $N$ qubits. The first corresponds to the standard strategy; where $N$ is repetitive, independent measurements are made. The second generates only classical correlations among the $N$ qubits; in other words, the qubits share a joint probability distribution. The third and the fourth introduce both classical and nonclassical correlations between the $N$ qubits, where the nonclassical correlation is quantified by both quantum entanglement and quantum discord. For each strategy, the problem of determining the uncertainty in estimating the phase of the qubit then becomes that of computing the quantum Fisher information of the corresponding $N$-qubit circuit.

We find strong evidence that classical correlations by themselves do not lead to enhanced precision in the phase estimation, while the nonclassical correlation, or quantum discord, does: Our most effective strategy achieves a factor-of-$\sqrt{N}$ (quadratic) enhancement in the precision over the standard repetition strategy, even when the entanglement vanishes and the only quantum correlation present is discord. This result is highly significant: It shows that discord is responsible for precision enhancement in certain quantum tasks.