Uncertainty relation for mutual information

We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.

Figure 1

The solid curve is a plot of the quantum mutual information $I(A:B)$ for the asymmetric Werner state (6) with $p=3/4$. The dashed curve is the CQC bound [i.e., $H({\widehat{\sigma }}_X^A:{\widehat{\sigma }}_X^B)$ + $H({\widehat{\sigma }}_Y^A:{\widehat{\sigma }}_Y^B)$] to the quantum mutual information for this state, and the dotted curve is the bound obtained from the classical version of Berta's uncertainty relation [i.e., $S\left(A\right)$ + $log\left(N^A\right)$ − $H\left({\widehat{\sigma }}_X^A\right|{\widehat{\sigma }}_X^B)$ − $H\left({\widehat{\sigma }}_Y^A\right|{\widehat{\sigma }}_Y^B)$]. The shading highlights the difference between the various quantities. For $\eta$ near zero or unity, the CQC relation offers a substantial improvement over Berta's uncertainty relation.

Figure 2

Scatterplots of the quantum mutual information as a function of the sum of the classical mutual informations determined by a pair of unbiased measurement bases. (a)–(c) Plots for our randomly perturbed $2\otimes 2$, $3\otimes 3$, and $4\otimes 4$ boundary states, respectively. The sharp diagonal boundary illustrates that perturbing boundary states do not yield states that violate our CQC relation.