Abstract
As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.
Highlights
The uncertainty principle is one of the most important principle in quantum physics
We find that if we take a complete set of mutually unbiased bases (MUBs) measurements into account, we can obtain an uncertainty equality that the sum of measurement uncertainties over all MUBs on a subsystem in the presence of quantum memory is equal to a fixed quantity determined by the initial state
It has been shown that after a complete set of MUB measurements on one partite, the total uncertainty on the other partite is exactly given by the purities of the initial system and the memory
Summary
The uncertainty principle is one of the most important principle in quantum physics. It implies the impossibility of simultaneously determining the definite values of incompatible observables. The uncertainty principle was first formulated via the standard deviation of a pair of complementary observables, known as the Heisenberg’s uncertainty principle[1] ΔxΔp ≥ ħ/2 for the coordinate x and the momentum p in an infinite dimensional Hilbert space. Later the Robertson−Schrödinger uncertainty inequality[2,3] presented an uncertainty relation for two arbitrary observables in a finite dimensional Hilbert space. Instead of the standard deviation of observables, the uncertainty principle can be elegantly formulated in terms of entropies related to measurement bases
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