Entropic uncertainty relations in quantum mechanics

Iwo Białynicki‐Birula, Łukasz Rudnicki

Zhaoan Wang, Bo-Fu Xie, Fei Ming +10 more

Entropic uncertainty relation (EUR) formulates the restriction of the inherent uncertainty of quantum mechanics from an information-theoretic perspective. A tighter lower bound for uncertainty relations can provide information-theoretic security to quantum communication protocols. Recently, a generalized EUR (GEUR) for the measurement of multiple observables in arbitrary many-body systems has been formulated. Here, we experimentally test this GEUR using a four-photon entangled state with a controllable decoherence channel and show that for the tripartite scenario, the GEUR improves the entropic bound from Renes and Boileau's famous results. As an application, we further demonstrate an improvement of the secure key rate in quantum key distribution from the GEUR. Our results extend the test of EURs into multipartite regimes and may find applications in practical quantum cryptography tasks.

Mario Berta, Stephanie Wehner, Mark M. Wilde

The entropic uncertainty relation with quantum side information (EUR-QSI) from (Berta et al 2010 Nat. Phys. 6 659 ) is a unifying principle relating two distinctive features of quantum mechanics: quantum uncertainty due to measurement incompatibility, and entanglement. In these relations, quantum uncertainty takes the form of preparation uncertainty where one of two incompatible measurements is applied. In particular, the 'uncertainty witness' lower bound in the EUR-QSI is not a function of a post-measurement state. An insightful proof of the EUR-QSI from (Coles et al 2012 Phys. Rev. Lett. 108 210405 ) makes use of a fundamental mathematical consequence of the postulates of quantum mechanics known as the non-increase of quantum relative entropy under quantum channels. Here, we exploit this perspective to establish a tightening of the EUR-QSI which adds a new state-dependent term in the lower bound, related to how well one can reverse the action of a quantum measurement. As such, this new term is a direct function of the post-measurement state and can be thought of as quantifying how much disturbance a given measurement causes. Our result thus quantitatively unifies this feature of quantum mechanics with the others mentioned above. We have experimentally tested our theoretical predictions on the IBM quantum experience and find reasonable agreement between our predictions and experimental outcomes.

G. M. Bosyk, M. Portesi, Steeve Zozor

Alejandro Corichi, Angel Garcia, Federico Zadra

The Entropic Uncertainty Relations (EUR) result from inequalities that are intrinsic to the Hilbert space and its dual with no direct connection to the Canonical Commutation Relations. Bialynicky-Mielcisnky obtained them in [1] attending Hilbert spaces with a Lebesgue measure. The analysis of these EUR in the context of singular Hilbert spaces has not been addressed. Singular Hilbert spaces are widely used in scenarios where some discretization of the space (or spacetime) is considered, e.g., loop quantum gravity, loop quantum cosmology and polymer quantum mechanics. In this work, we present an overview of the essential literature background and the road map we plan to follow to obtain the EUR in polymer quantum mechanics.

Ivan Damgård, Serge Fehr, Renato Renner +2 more