We present a quantum version of the generalized $(h,ϕ)$-entropies, introduced by Salicrú \textit{et al.} for the study of classical probability distributions. We establish their basic properties, and show that already known quantum entropies such as von Neumann, and quantum versions of Rényi, Tsallis, and unified entropies, constitute particular classes of the present general quantum Salicrú form. We exhibit that majorization plays a key role in explaining most of their common features. We give a characterization of the quantum $(h,ϕ)$-entropies under the action of quantum operations, and study their properties for composite systems. We apply these generalized entropies to the problem of detection of quantum entanglement, and introduce a discussion on possible generalized conditional entropies as well.
We study a version of the generalized (h, ϕ)-entropies, introduced by Salicrú et al, for a wide family of probabilistic models that includes quantum and classical statistical theories as particular cases. We extend previous works by exploring how to define (h, ϕ)-entropies in infinite dimensional models.
We consider a two-parameter family of Rényi relative entropies $D_{α,z}(ρ||σ)$ that are quantum generalisations of the classical Rényi divergence $D_α(p||q)$. This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi's axioms for a divergence. We consider the range of the parameters $α,z$ for which the data processing inequality holds. We also investigate a variety of limiting cases for the …
The postulates of thermodynamics were originally formulated for macroscopic systems. They lead to the definition of the entropy, which, for a homogeneous system, is a homogeneous function of order one in the extensive variables and is maximized at equilibrium. We say that the macroscopic systems are extensive and so it is also the entropy. For a mesoscopic system, by definition, the size and the contacts with other systems influence its thermodynamic properties and therefore, if we define an entropy, this cannot be a homogeneous of order one function in the extensive variables. So, mesoscopic …
Quantum conditional entropies from convex trace functionals
Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Mathematical Sciences, University of Copenhagen（哥本哈根大学数学科学系）；Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
Information theory has become an increasingly important research field to better understand quantum mechanics. Noteworthy, it covers both foundational and applied perspectives, also offering a common technical language to study a variety of research areas. Remarkably, one of the key information-theoretic quantities is given by the relative entropy, which quantifies how difficult is to tell apart two probability distributions, or even two quantum states. Such a quantity rests at the core of fields like metrology, quantum thermodynamics, quantum communication and quantum information. Given this …
We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $α$-Rényi entropy, $S_α$, for $α\in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\varepsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\varepsilon$-ball. This minimal state …
The occupancies and entropic entanglement measures for the ground state of two particles in a two-dimensional harmonic anisotropic trap are studied. We implement a method to study the large interaction strength limit for different short- and long-range interaction potentials that allows to obtain the exact entanglement spectrum and several entropies. We show that for long-range interactions, the von Neumann, min-entropy and the family of Rényi entropies remain finite for the anisotropic traps and diverge logarithmically for the isotropic traps. In the short-range interaction case the entanglem…
The so-called $q$-z-\textit{Rényi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant $(0,2)$ symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined di…
We address an information-theoretic approach to noise and disturbance in quantum measurements. Properties of corresponding probability distributions are characterized by means of both the Rényi and Tsallis entropies. Related information-theoretic measures of noise and disturbance are introduced. These definitions are based on the concept of conditional entropy. To motivate introduced measures, some important properties of the conditional Rényi and Tsallis entropies are discussed. There exist several formulations of entropic uncertainty relations for a pair of observables. Trade-off relations f…
The rates of quantum cryptographic protocols are usually expressed in terms of a conditional entropy minimized over a certain set of quantum states. In particular, in the device-independent setting, the minimization is over all the quantum states jointly held by the adversary and the parties that are consistent with the statistics that are seen by the parties. Here, we introduce a method to approximate such entropic quantities. Applied to the setting of device-independent randomness generation and quantum key distribution, we obtain improvements on protocol rates in various settings. In partic…
Quantum Renyi relative entropies provide a one-parameter family of distances between density matrices, which generalizes the relative entropy and the fidelity. We study these measures for renormalization group flows in quantum field theory. We derive explicit expressions in free field theory based on the real time approach. Using monotonicity properties, we obtain new inequalities that need to be satisfied by consistent renormalization group trajectories in field theory. These inequalities play the role of a second law of thermodynamics, in the context of renormalization group flows. Finally, …
This work presents a generalization of the Koashi-Winter relation for $α$-Renyi entropies. This result is based on the Renyi\apos s entropy version of quantum Jensen Shannon divergence. By means of this definition, a classical correlations quantifier $C_α(ρ_{AB}) = \sup_{ξ_{AB}^{M_B}} Q_α(ξ_{AB}^{M_B})$ is proposed, where the optimization is taken over the ensembles $ξ_{AB}^{M_B}$ created by the outputs of the local measurement process. The main result is applied to the capacity of a quantum classical channel over a tripartite pure state $ψ_{ABE}$, that is rated above in function of the probab…
The entropy of a quantum operation, defined as the von Neumann entropy of the corresponding Choi-Jamiołkowski state, characterizes the coupling of the principal system with the environment. For any quantum channel $Φ$ acting on a state of size $N$ one defines the complementary channel $\tilde Φ$, which sends the input state into the state of the environment after the operation. Making use of subadditivity of entropy we show that for any dimension $N$ the sum of both entropies, $S(Φ)+ S(\tilde Φ)$, is bounded from below. This result characterizes the trade-off between the information on the ini…
Quantum coherence is a crucial resource for quantum information processing. By employing the language of coherence orders largely applied in NMR systems, quantum coherence has been currently addressed in terms of multiple quantum coherences (MQCs). Here we investigate the $α$-MQCs, a novel class of multiple quantum coherences which is based on $α$-relative purity, an information-theoretic quantifier analogous to quantum fidelity and closely related to Rényi relative entropy of order $α$. Our framework enables linking $α$-MQCs to Wigner-Yanase-Dyson skew information (WYDSI), an asymmetry monoto…
The requirement that an entropy function be composable is key: it means that the entropy of a compound system can be calculated in terms of the entropy of its independent components. We prove that, under mild regularity assumptions, the only composable generalized entropy in trace form is the Tsallis one-parameter family (which contains Boltzmann-Gibbs as a particular case). This result leads to the use of generalized entropies that are not of trace form, such as Rényi's entropy, in the study of complex systems. In this direction, we also present a characterization theorem for a large class of…
In this paper, we investigate and compare two well-developed definitions of entropy relevant for describing the dynamics of isolated quantum systems: bipartite entanglement entropy and observational entropy. In a model system of interacting particles in a one-dimensional lattice, we numerically solve for the full quantum behavior of the system. We characterize the fluctuations, and find the maximal, minimal, and typical entropy of each type that the system can eventually attain through its evolution. While both entropies are low for some "special" configurations and high for more "generic" one…
Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and Rényi entropies. These quantities capture the desired entanglement properties in non-Her…
In finite dimensions, we provide characterizations of the quantum dynamical semigroups that do not decrease the von Neumann, the Tsallis and the Renyi entropies, as well as a family of functions of density operators strictly related to the Schatten norms. A few remarkable consequences --- in particular, a description of the associated infinitesimal generators --- are derived, and some significant examples are discussed. Extensions of these results to semigroups of trace-preserving positive (i.e., not necessarily completely positive) maps and to a more general class of quantum entropies are als…
We characterize the asymptotic performance of a class of positive operator valued measurements (POVMs) where the only task is to make measurements on independent and identically distributed quantum states on finite-dimensional systems. The POVMs we utilize here can be efficiently described in terms of a reasonably small set of parameters. Their analysis furthers the development of a quantum method of types. They deliver provably optimal performance in asymmetric hypothesis testing and in the transmission of classical messages over quantum channels. We now relate them to the recently developed …
We introduce superdensity operators as a tool for analyzing quantum information in spacetime. Superdensity operators encode spacetime correlation functions in an operator framework, and support a natural generalization of Hilbert space techniques and Dirac's transformation theory as traditionally applied to standard density operators. Superdensity operators can be measured experimentally, but accessing their full content requires novel procedures. We demonstrate these statements on several examples. The superdensity formalism suggests useful definitions of spacetime entropies and spacetime qua…
