Local quantum uncertainty and quantum discord

This morning’s session was closed with a nice talk by Gerardo Adesso, titled “Characterizing nonclassical correlations via local quantum uncertainty”. Gerardo started with a clear introduction to classical and quantum correlations and quantum discord, and discussed even controversial issues such as the (possible) role of quantum discord in the DQC1 model of quantum computation

Then, Gerardo recalled a (very!) old measure called the Wigner-Yanase (W-Y) skew information (1963), and used it as a measure of the “quantum part” of the variance of a given measurement. In a simple illustration of this, a mixed state may have part of the variance in a given measurement’s results due to the (classical) mixture, and part due to intrinsic quantum indeterminacy of the measurement outcomes, and it is this last part which the Wigner-Yanase skew information measures. For pure states, the W-Y skew information coincides with the variance; for mixed states, it is typically smaller. In general, it is zero if and only if state and observable commute. I think there may have been other reasons for its adoption as a measure of quantum (intrinsic) uncertainty, but I can’t recall, perhaps I should look up the original 1963 Wigner-Yanase paper…

We can then ask the question: which quantum states unavoidably exhibit quantum uncertainty? For global observables, the answer is: none! as we can define an observable in terms of the eigenstates of the density matrix, with no variance at all.

For local observables, on the other hand, Gerardo proved that a quantum state has quantum uncertainty (as measured by the W-Y skew info) if and only if they have non-zero discord. We can define the Local Quantum Uncertainty as the minimum W-Y skew info over all local observables. This quantity is computable for 2 x d systems, which is a definite advantage over discord itself.

Gerardo wrapped up his talk with an application of his Local Quantum Uncertainty to metrology. I didn’t follow the details, but apparently it serves as an upper bound for the variance of phase estimation with mixed probes and optimal quantum measurements.


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