Optimality of entropic uncertainty relations

Abstract

The entropic uncertainty relation proven by Maassen and Uffink for arbitrary pairs of two observables is known to be non-optimal. Here, we call an uncertainty relation optimal, if the lower bound can be attained for any value of either of the corresponding uncertainties. In this work we establish optimal uncertainty relations by characterising the optimal lower bound in scenarios similar to the Maassen-Uffink type. We disprove a conjecture by Englert et al. and generalise various previous results. However, we are still far from a complete understanding and, based on numerical investigation and analytical results in small dimension, we present a number of conjectures. arXiv:1509.00398v1 [quant-ph] 1 Sep 2015

Entanglement Detection by Violations of Noisy Uncertainty Relations: A Proof of Principle

“…However, for this experiment, it turns out that the dominant part of the noise regime can be well explained (see Fig. 4) by the action of the channel (13), with a spin-flip noise parameter α ≈ 0.2, estimated from experimental data.…”

Entanglement detection with scrambled data

“…In Ref. [15], Theorem V.2 states that for two concave functionals f 1 , f 2 on the state space, for any state ρ, there is a pure state |ψ such that f 1 (|ψ ψ|) f 1 (ρ) and f 2 (|ψ ψ|) f 2 (ρ). Furthermore, it is shown in Theorem V.3 that the state |ψ can additionally be chosen real if the inputs of the functionals are linked by a real unitary matrix.…”

“…According to Theorem V.3 in Ref. [15], if two entropies S 1 and S 2 are considered where the measurement bases are related by a real unitary transformation, then for any state ρ, there is always a pure and real state |ψ with S 1 (|ψ ψ|) S 1 (ρ) and S 2 (|ψ ψ|) S 2 (ρ). As in our case σ x = Hσ z H † where H is the Hadamard matrix, it is sufficient to consider pure real states to obtain minimal S (2) zz for given S (2) xx .…”

“…6 in Ref. [15]. Any separable state ρ can be written as the mixture of two states on the topological boundary of the space of separable states.…”

Entropic uncertainty relations for successive measurements of canonically conjugate observables

“…Entropic uncertainty relations are currently the subject of active research (see the reviews [11][12][13] and references therein). Questions of their optimality are addressed in [14]. Other approaches are based on the sum of variances [15,16] and on majorization relations [17][18][19][20][21].…”