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Quantum Information Processing

Notes on quantum coherence with $$l_1$$-norm and convex-roof $$l_1$$-norm

Jiayao Zhu, Jian Ma, Tinggui Zhang

Published Nov 23, 2021• 1 citation
Publisher
DOI
10.1007/s11128-021-03325-4

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Notes on quantum coherence with l1l_1l1​-norm and convex-roof l1l_1l1​-norm
Bell-type inequalities of l1l_1l1​-norm of coherence
Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .
Superadditivity relations of the l1l_1l1​ l 1 norm of coherence
Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .
Trade-off relations of l1l_1l1​-norm coherence for multipartite systems