Measurement uncertainty from no-signaling and nonlocality

One of the formulations of Heisenberg uncertainty principle, concerning so-called measurement uncertainty, states that the measurement of one observable modifies the statistics of the other. Here, we derive such a measurement uncertainty principle from two comprehensible assumptions: impossibility of instantaneous messaging at a distance (no-signaling), and violation of Bell inequalities (nonlocality). The uncertainty is established for a pair of observables of one of two spatially separated systems that exhibit nonlocal correlations. To this end, we introduce a gentle form of measurement which acquires partial information about one of the observables. We then bound disturbance of the remaining observables by the amount of information gained from the gentle measurement, minus a correction depending on the degree of nonlocality. The obtained quantitative expression resembles the quantum mechanical formulations, yet it is derived without the quantum formalism and complements the known qualitative effect of disturbance implied by nonlocality and no-signaling.

Figure 1

Lower bound ${\mathcal{D}}_{min}=2\epsilon -\frac{1}{2}(4-{\beta }_{\text{CHSH}})$ [RHS of Eq. (11)] on average total disturbance obtained from nonlocality and no-signaling principle for the case of CHSH inequality for ${\beta }_{\text{CHSH}}=2\sqrt{2}$ corresponding to maximally nonlocal correlations attainable within the framework of quantum mechanics (dashed line) and ${\mathcal{D}}_{min}=\frac{1}{2}\left({\beta }_{\text{CHSH}}-\sqrt{8-4{\left(2\epsilon \right)}^2}\right)$ [RHS of Eq. (15)] obtained from quantum predictions (thick line).

Figure 2

Lower bound ${\mathcal{D}}_{min}=\frac{4}{n}\epsilon -\frac{1}{n}(2n-{\beta }_{\text{chain}})$ [RHS of Eq. (13)] on average total disturbance obtained from nonlocality and no-signaling principle for the case of chain Bell inequality, where we choose ${\beta }_{\text{chain}}=2ncos\left(\frac{\pi }{2n}\right)$ [24] with $n$ denoting the number of observables.

Figure 3

The comparison of lower bound ${\mathcal{D}}_{min}$ on average total disturbance implied by chain Bell inequality (red lines), ${\mathcal{D}}_{min}=\frac{4}{n}\epsilon -\frac{1}{n}(2n-{\beta }_{\text{chain}})$, where ${\beta }_{\text{chain}}=2ncos\left(\frac{\pi }{2n}\right)$, with that implied by generalized chain inequality (black lines), ${\mathcal{D}}_{min}=\frac{4k\epsilon }{n}-2\left[k-{\sum }_{j=1}^{k}cos\left(\frac{(2j-1)\pi }{2n}\right)\right]$ given in Eq. (E19), for $n=100,...,1000$.