Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, Rényi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched Rényi relative entropy for
α∈[0,1/2) and the sandwiched Rényi relative entropy for
α≥1/2. In contrast, the maximal extension had previously been identified only for
α∈[0,2], where it is given by the geometric Rényi relative entropy. In this work, we complete this characterization by proving that for
α>2, the maximal extension is given by the
α-
z Rényi relative entropy with
z=α−1. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.
Abstract We establish a rigorous equivalence between quantum coherence measures defined by the sandwiched Rényi relative entropy and generalized conditional entropies associated with auxiliary quantum systems, providing a direct operational interpretation of sandwiched Rényi relative entropy of coherence as the conditional uncertainty of projective measurement outcomes with quantum memory. Motivated by this connection, we develop a coherence-based approach to quantum uncertainty relations. Within this approach, we introduce a coherence measure induced by the two-parameter α - z -Rényi relative entropy and derive corresponding uncertainty relations for both pure and mixed quantum states. The resulting state-independent bounds recover known results in appropriate limits and yield tighter bounds over a broad parameter regime. Our results provide a systematic connection between quantum coherence, generalized conditional entropies, and uncertainty relations, thereby offering a unified perspective on quantum uncertainty in terms of Rényi-type information measures.
Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.
We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the [Formula: see text]-[Formula: see text]-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo–Rényi inequality. This result leads to a quantum bound for the [Formula: see text]-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.
Quantum coherence, as a direct manifestation of the quantum superposition principle, is a crucial resource in quantum information processing. Block coherence resource theory generalizes the traditional coherence framework by defining coherence via a set of orthogonal projectors. Within this framework, we investigates the construction and comparison of block coherence measures. First, we propose two universal methods for constructing coherence measures and introduce a two-parameter family of measures based on the
α-
z Rényi relative entropy and a family of measures based on the Tsallis relative operator entropy. Second, through theoretical proofs and numerical counterexamples, we compares the ordering relations and numerical magnitudes among different block coherence measures and establishes a series of universal numerical inequalities to constrain their values. Besides, we also use
Cα,1 to show the role of coherence in complex dynamic evolution of the Kominis master equation that includes recombination reactions.
Abstract Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states (2014, Phys. Rev. Lett. 113 , 140401). In BCP framework, a quantum state is called incoherent if it is diagonal in the fixed orthonormal basis, and a coherence measure should satisfy some conditions. For a fixed orthonormal basis, if a quantum state ρ has nonzero imaginary part, then ρ must be coherent. How to quantitatively characterize this fact? In this work, we show that any coherence measure C in BCP framework has the property C ( ρ ) − C (Re ρ )≥0 if C is invariant under state complex conjugation, i.e. C ( ρ ) = C ( ρ *), here ρ *is the conjugate of ρ , Re ρ is the real part of ρ . If C does not satisfy C ( ρ ) = C ( ρ *), we can define a new coherence measure C ′ ( ρ ) = 1 2 [ C ( ρ ) + C ( ρ * ) ] such that C ′ ( ρ ) = C ′ ( ρ * ) . We also establish some similar results for bosonic Gaussian states.