Abstract

The quantum uncertainty principle famously predicts that there exist measurements that are inherently incompatible, in the sense that their outcomes cannot be predicted simultaneously. In contrast, no such uncertainty exists in the classical domain, where all uncertainty results from ignorance about the exact state of the physical system. Here, we critically examine the concept of preparation uncertainty and ask whether similarly in the quantum regime, some of the uncertainty that we observe can actually also be understood as a lack of information (LOI), albeit a lack of quantum information. We answer this question affirmatively by showing that for the well known measurements employed in BB84 quantum key distribution (Bennett and Brassard 1984 Int. Conf. on Computer System and Signal Processing), the amount of uncertainty can indeed be related to the amount of available information about additional registers determining the choice of the measurement. We proceed to show that also for other measurements the amount of uncertainty is in part connected to a LOI. Finally, we discuss the conceptual implications of our observation to the security of cryptographic protocols that make use of BB84 states.

Highlights

  • The uncertainty principle forms one of the cornerstones of quantum theory

  • We show that even in the quantum case, uncertainty can in part be understood as a lack of information (LOI) that Bob has—namely a lack of quantum information about the register P

  • For the case of d = 2 and BB84 measurements as they are used in quantum key distribution (QKD), this effect is dramatic

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Summary

Introduction

As first observed by Heisenberg [15] and rigorously proven by Kennard [19], it is impossible to perfectly predict the measurement outcomes of both position and momentum observables. This notion was generalised by Robertson to an arbitrary pair of observables [26] showing that uncertainty is an inherent feature of any non-commuting measurements in quantum mechanics. A modern way of capturing the notion of preparation uncertainty is by means of a guessing game [2]. Bob wins the game if he correctly guesses Alice’s measurement outcome X

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