Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory

Uncertainty relations capture the essence of the inevitable randomness associated with the outcomes of two incompatible quantum measurements. Recently, Berta et al. [Nature Phys. 6, 659 (2010)] have shown that the lower bound on the uncertainties of the measurement outcomes depends on the correlations between the observed system and an observer who possesses a quantum memory. If the system is maximally entangled with its memory, the outcomes of two incompatible measurements made on the system can be predicted precisely. Here, we obtain an uncertainty relation that tightens the lower bound of Berta et al. by incorporating an additional term that depends on the quantum discord and the classical correlations of the joint state of the observed system and the quantum memory. We discuss several examples of states for which our lower bound is tighter than the bound of Berta et al. On the application side, we discuss the relevance of our inequality for the security of quantum key distribution and show that it can be used to provide bounds on the distillable common randomness and the entanglement of formation of bipartite quantum states.

Figure 1

The right-hand side of our inequality (8) $-2{\log }_{2}c({\sigma }_x,{\sigma }_z)+S\left(A\right|B)+\max \{0,D_A\left({\rho }_{AB}\right)-J_A\left({\rho }_{AB}\right)\}$ (solid blue), and the right-hand side of the Berta et al. inequality (5) $-2{\log }_{2}c({\sigma }_x,{\sigma }_z)+S\left(A\right|B)$ (dashed green), as a function of the noise parameter $p$, when the system and memory are prepared in the two-qubit Werner state (12).

Figure 2

The difference between our bound in Eq. (8) of the main text and that of Berta et al. for Werner states as a function of the dimension $d$ and the parameter $\lambda$.

Figure 3

The difference between our bound in Eq. (8) of the main text and that of Berta et al. for isotropic states as a function of the dimension $d$ and the parameter $\lambda$.