Uncertainty Relations from Simple Entropic Properties

Coles, Patrick J.; Colbeck, Roger; Yu, Li; Zwolak, Michael

doi:10.1103/physrevlett.108.210405

Cited by 183 publications

(235 citation statements)

References 30 publications

(55 reference statements)

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“…Indeed, our relation has already been employed in [16] for an experimental implementation of bit commitment in the bounded/noisy-storage model. It is an interesting open question whether similarly strong relations can also be obtained with respect to quantum side information [11,20,21]. This would allow security statements for such protocols in terms of the quantum capacity [11] of the storage device, rather than the classical capacity [9] or the entanglement cost [22].…”

Section: Discussionmentioning

confidence: 99%

Min-entropy uncertainty relation for finite-size cryptography

Berta

Wehner

2012

Phys. Rev. A

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Apart from their foundational significance, entropic uncertainty relations play a central role in proving the security of quantum cryptographic protocols. Of particular interest are thereby relations in terms of the smooth min-entropy for BB84 and six-state encodings. Previously, strong uncertainty relations were obtained which are valid in the limit of large block lengths. Here, we prove a new uncertainty relation in terms of the smooth min-entropy that is only marginally less strong, but has the crucial property that it can be applied to rather small block lengths. This paves the way for a practical implementation of many cryptographic protocols. As part of our proof we show tight uncertainty relations for a family of Rényi entropies that may be of independent interest.

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Section: Discussionmentioning

confidence: 99%

Min-entropy uncertainty relation for finite-size cryptography

Berta

Wehner

2012

Phys. Rev. A

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“…In [14], Coles et al extended the 3-space version to a larger class of entropies than the von Neumann entropy. Indeed, it is noteworthy that their proof relies only on some basic axioms of entropy, the most important of which is monotonicity of relative entropy, which is equivalent to SSA [15,16].…”

Section: Introductionmentioning

confidence: 99%

Extended Quantum Conditional Entropy and Quantum Uncertainty Inequalities

Frank

Lieb

2013

Commun. Math. Phys.

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Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. Our proofs utilize the technique of the original derivation of strong subadditivity of the von Neumann entropy.

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“…The latter states that there are certain pairs of observables, such as position and momentum or two orthogonal components of spin angular momentum, which cannot simultaneously be known or jointly measured. Likewise, there are many quantitative statements of this idea, known as uncertainty relations (URs; see, for example, refs [19][20][21][22][23][24][25][26][27][28][29], and modern formulations typically use entropy instead of s.d. as the uncertainty measure, so-called entropic uncertainty relations (EURs) 24 .…”

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confidence: 99%

Equivalence of wave–particle duality to entropic uncertainty

2014

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Interferometers capture a basic mystery of quantum mechanics: a single particle can exhibit wave behaviour, yet that wave behaviour disappears when one tries to determine the particle's path inside the interferometer. This idea has been formulated quantitatively as an inequality, for example, by Englert and Jaeger, Shimony and Vaidman, which upper bounds the sum of the interference visibility and the path distinguishability. Such wave-particle duality relations (WPDRs) are often thought to be conceptually inequivalent to Heisenberg's uncertainty principle, although this has been debated. Here we show that WPDRs correspond precisely to a modern formulation of the uncertainty principle in terms of entropies, namely, the min-and max-entropies. This observation unifies two fundamental concepts in quantum mechanics. Furthermore, it leads to a robust framework for deriving novel WPDRs by applying entropic uncertainty relations to interferometric models. As an illustration, we derive a novel relation that captures the coherence in a quantum beam splitter.

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Uncertainty Relations from Simple Entropic Properties

Cited by 183 publications

References 30 publications

Min-entropy uncertainty relation for finite-size cryptography

Min-entropy uncertainty relation for finite-size cryptography

Extended Quantum Conditional Entropy and Quantum Uncertainty Inequalities

Equivalence of wave–particle duality to entropic uncertainty

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