Characterizing Nonclassical Correlations via Local Quantum Uncertainty

Quantum mechanics predicts that measurements of incompatible observables carry a minimum uncertainty which is independent of technical deficiencies of the measurement apparatus or incomplete knowledge of the state of the system. Nothing yet seems to prevent a single physical quantity, such as one spin component, from being measured with arbitrary precision. Here, we show that an intrinsic quantum uncertainty on a single observable is ineludible in a number of physical situations. When revealed on local observables of a bipartite system, such uncertainty defines an entire class of bona fide measures of nonclassical correlations. For the case of $2\times d$ systems, we find that a unique measure is defined, which we evaluate in closed form. We then discuss the role that these correlations, which are of the “discord” type, can play in the context of quantum metrology. We show in particular that the amount of discord present in a bipartite mixed probe state guarantees a minimum precision, as quantified by the quantum Fisher information, in the optimal phase estimation protocol.

Figure 1

Quantum correlations trigger local quantum uncertainty. Let us consider a bipartite state $\rho$. An observer on subsystem $A$ is equipped with a quantum meter, a measurement device whose error bar shows the quantum uncertainty only. (Note that, in order to access such a quantity, the measurement of other observables that are defined on the full bipartite system may be required, in a procedure similar to state tomography) (a) If $\rho$ is uncorrelated or contains only classical correlations [darker (brown) inner shade], i.e., $\rho$ is of the form $\rho =\sum _i{p}_i|i⟩⟨i{|}_A\otimes {\sigma }_{iB}$ (with $\{|i⟩\}$ an orthonormal basis for $A$) [8, 9, 10], the observer can measure at least one observable on $A$ without any intrinsic quantum uncertainty. (b) If $\rho$ contains a nonzero amount of quantum correlations [lighter (yellow) outer shade], as quantified by entanglement for pure states [5] and quantum discord in general [10], any local measurement on $A$ is affected by quantum uncertainty. The minimum quantum uncertainty associated to a single measurement on subsystem $A$ can be used to quantify discord in the state $\rho$, as perceived by the observer on $A$. In this Letter, we adopt the Wigner-Yanase skew information [16] to measure the quantum uncertainty on local observables.

Figure 2

The plot shows different contributions to the error bar of spin measurements on subsystem $A$ in a Werner state [5] $\rho =p|{\varphi }^{+}⟩⟨{\varphi }^{+}|+(1-p)\mathbb{I}/4,p\in [0,1]$, of two qubits $A$ and $B$. The solid red line is the variance ${\mathrm{Var}}_{\rho }({\sigma }_z^A)$ of the ${\sigma }_z^A$ operator, which amounts to the total statistical uncertainty. The dashed blue curve represents the local quantum uncertainty ${\mathcal{U}}_A(\rho )$, which in this case is $\mathcal{I}(\rho ,{\sigma }_z^A)$ (any local spin direction achieves the minimum for this class of states). The dotted green curve depicts the (normalized) linear entropy $S_L(\rho )=(4/3)(1-\mathrm{Tr}\{{\rho }^2\})$ of the global state $\rho$, which measures its mixedness. Notice that the Werner state is separable for $p\le 1/3$ but it always contains discord for $p>0$.

Figure 3

Quantum correlations-assisted parameter estimation. A probe state $\rho$ of a bipartite system $AB$ is prepared, and a local unitary transformation depending on an unobservable parameter $\phi$ acts on subsystem $A$, transforming the global state into ${\rho }_{\phi }$. By means of a suitable measurement at the output, one can construct an (unbiased) estimator $\tilde{\phi }$ for $\phi$. The quality of the estimation strategy is benchmarked by the variance of the estimator. For a given probe state $\rho$, the optimal measurement at the output returns an estimator ${\tilde{\phi }}_{\mathrm{best}}$ for $\phi$ with the minimum allowed variance given by the inverse of the QFI $\mathcal{F}({\rho }_{\phi })$, according to the quantum Cramér-Rao bound [33]. In the prototypical case of optical phase estimation, the present scheme corresponds to a Mach-Zender interferometer. Restricting to pure inputs, research in quantum metrology [31] has shown that in this case, entangled probes allow us to beat the shot noise limit $\mathcal{F}\propto n$ ($n$ being the input mean photon number) and reach ideally the Heisenberg scaling $\mathcal{F}\propto n^2$. However, recent investigations have revealed how in the presence of realistic imperfections, the achieved precision quickly degrades to the shot noise level [35, 36, 37]. For mixed bipartite probes, we show that the QFI is bounded from below by the amount of quantum correlations in the probe state $\rho$ as quantified by the LQU.