bibkey: abdelkhalek2015optimality authors: Kais Abdelkhalek; René Schwonnek; Hans Maassen; Fabian Furrer; Jörg Duhme; Philippe Raynal; Berthold-Georg Englert; Reinhard F. Werner year: 2015 title: “Optimality of entropic uncertainty relations” doi: 10.1142/S0219749915500458 url: https://arxiv.org/abs/1509.00398v1 claim: “Conjecture V.8 (Independence of the optimal states of (α,β)): If ρ is an optimal state for any unitary operator and any α,β>½ satisfying the duality relation (2), then ρ is also an optimal state for all other dual pairs.” strata_touched:

• D5/S3/Quantum/Information/RenyiOptimalStateDependenceRefutation license: citation-only triage: anchor

Optimality of entropic uncertainty relations

K. Abdelkhalek, R. Schwonnek, H. Maassen, F. Furrer, J. Duhme, P. Raynal, B.-G. Englert and R. F. Werner, arXiv:1509.00398v1; International Journal of Quantum Information 13, 1550045 (2015). Subject: quant-ph.

The observables are two orthonormal bases in a finite-dimensional complex Hilbert space. Their overlap matrix is $U_{ij}=⟨x_i|y_j⟩$; $p_X^{\rho }(i)=⟨x_i|\rho |x_i⟩$ and $p_Y^{\rho }(j)=⟨y_j|\rho |y_j⟩$. Section II, p. 4, equation (6) defines the finite-order Rényi entropy by $H_{\alpha }(p)=\log ({\sum }_{j}p(j{)}^{\alpha })/(1-\alpha )$ for $\alpha \ne 1$, and $H_1(p)=-{\sum }_{j}p(j)\log p(j)$ at order one. Its zero-probability terms have their continuous values. The paper states:

The logarithms can be taken in any base (as long as it is always the same base).

With $f(\rho )=(H_{\alpha }(p_X^{\rho }),H_{\beta }(p_Y^{\rho }))$, section II, p. 4, defines the order and optimality:

For any choice we can define the order relation $⊑$ on the state space, so that $\rho ⊑{\rho }^{\prime }$ stands for “$f_1(\rho )\le f_1({\rho }^{\prime })$ and $f_2(\rho )\le f_2({\rho }^{\prime })$”.

We call a state $\rho$ optimal if ${\rho }^{\prime }⊑\rho$ implies $\rho ⊑{\rho }^{\prime }$, and hence $f(\rho )=f({\rho }^{\prime })$.

Conjecture V.8, section V.E, p. 21, reads:

(Independence of the optimal states of $(\alpha ,\beta )$) If $\rho$ is an optimal state for any unitary operator and any $\alpha ,\beta >\frac{1}{2}$ satisfying the duality relation (2), then $\rho$ is also an optimal state for all other dual pairs.

Equation (2) is $1/\alpha +1/\beta =2$. The finite-real-order encoding excludes the extremal pair $\{1/2,\infty \}$, which the conjecture also explicitly excludes. Using natural logarithms multiplies each entropy coordinate by the same positive constant relative to the paper’s base two; the coordinate preorder and optimality are preserved. The Lean carrier includes every positive complex trace-one density matrix, and zero-based indices relabel the paper’s basis outcomes.

Verified locator

• URL: https://arxiv.org/abs/1509.00398v1 — source OptEUR-main.tex, section II (eq:defentropy, deff) and section V.E (conj:independence); PDF pp. 4 and 21, equations (6), (7) and Conjecture V.8.
• DOI: https://doi.org/10.1142/S0219749915500458 — journal citation. ASSUMED-UNVERIFIED: the journal full text and its fidelity to the arXiv conjecture have not been checked.

Scope

The corresponding module refutes the universal conjecture with a qutrit basis projector optimal at $(1,1)$ and not optimal at $(3/5,3)$. The basis projectors reuse OrthogonalRecordEntropy.pointerState on Fin 3. Corollary IV.4 concerns equality states saturating the Maassen–Uffink bound; that is a distinct, narrower statement. No refutation of that corollary, the Fourier-specific conjectures, or a two-dimensional restriction is asserted here. Literature priority beyond the searched scope, including the other citing works, is ASSUMED-UNVERIFIED.