Basic properties of the unified entropies are examined. The consideration is mainly restricted to the finite-dimensional quantum case. Bounds in terms of ensembles of quantum states are given. Both the continuity in Fannes' sense and stability in Lesche's sense are shown for wide ranges of parameters. In particular, uniform estimates are obtained for the quantum Rényi entropies. Stability properties in the thermodynamic limit are discussed as well. It is shown that the unified entropies enjoy both the subadditivity and triangle inequality for a certain range of parameters. Non-decreasing of all the unified entropies under projective measurement is proved.
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）
We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning system. This bound is sharper in some situations than previous continuity bounds, which were either based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but did not include a conditioning system. However, we find that the bound does not directly generalize to fully quantum conditional entropies. To investigate possible counterexamples in that setting, …
The entanglement properties of random quantum states or dynamics are important to the study of a broad spectrum of disciplines of physics, ranging from quantum information to high energy and many-body physics. This work investigates the interplay between the degrees of entanglement and randomness in pure states and unitary channels. We reveal strong connections between designs (distributions of states or unitaries that match certain moments of the uniform Haar measure) and generalized entropies (entropic functions that depend on certain powers of the density operator), by showing that Rényi en…
A comparative study of one-dimensional quantum structures which allow analytic expressions for the position and momentum Rényi $R(α)$ and Tsallis $T(α)$ entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms a special status of the Gaussian distribution: at any parameter $α$ it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement $1/2\leqα\leq1$ that is a necessary condition for all other geometries. It is shown that th…
Entropic quantifiers of states lie at the cornerstone of the quantum information theory. While a quantum state can be abstracted as a device that only has outputs, the most general quantum device is a quantum channel that also has inputs. In this work, we extend the entropic quantifiers of states to the ones of channels. In the one-shot and asymptotic scenarios, we propose relative entropies of channels under the task of hypothesis testing. Then, we define the entropy of channels based on relative entropies from the target channel to the completely depolarising channel. We also study propertie…
The quantum Renyi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Renyi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) …
Understanding cause-effect relationships is a crucial part of the scientific process. As Bell's theorem shows, within a given causal structure, classical and quantum physics impose different constraints on the correlations that are realisable, a fundamental feature that has technological applications. However, in general it is difficult to distinguish the set of classical and quantum correlations within a causal structure. Here we investigate a method to do this based on using entropy vectors for Tsallis entropies. We derive constraints on the Tsallis entropies that are implied by (conditional…
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, …
Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them, is said to be steerable from Alice to Bob if the steering assemblage on Bob's side do not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand provide for another characterization of quantum correlations. Contrary to our common intuition conditional entropies for some entangled states can be negative, marking a significant departure…
The growth of Renyi entropies after the injection of energy into a correlated system provides a window upon the dynamics of its entanglement properties. We develop here a simulation scheme by which this growth can be determined in Luttinger liquids systems with arbitrary interactions, even those introducing gaps into the liquid. We apply this scheme to an experimentally relevant quench in the sine-Gordon field theory. While for short times we provide an analytic expression for the growth of the second and third Renyi entropy, to access longer times, we combine our scheme with truncated spectru…
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料中心）；Wilczek Quantum Center, Shanghai Institute for Advanced Studies, University of Science and Technology of China（中国科学技术大学上海高等研究院斐兹克量子中心）；Hefei National Laboratory（合肥国家实验室）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学北京量子器件重点实验室）
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…
We present a scheme for measuring Rényi entropies in generic atomic Hubbard and spin models using single copies of a quantum state and for partitions in arbitrary spatial dimension. Our approach is based on the generation of random unitaries from random quenches, implemented using engineered time-dependent disorder potentials, and standard projective measurements, as realized by quantum gas microscopes. By analyzing the properties of the generated unitaries and the role of statistical errors, with respect to the size of the partition, we show that the protocol can be realized in exisiting AMO …
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…
The von Neumann entropy plays a vital role in quantum information theory. The von Neumann entropy determines, e.g., the capacities of quantum channels. Also, entropies of composite quantum systems are important for future quantum networks, and their characterization is related to the quantum marginal problem. Furthermore, they play a role in quantum thermodynamics. In this thesis the set of quantum entropies of multipartite quantum systems is studied. The problem of its characterization is not new -- however, progress has been sparse, indicating that the problem might be hard and that new meth…
We develop a universal approximation for the Renyi entropies of a pure state at late times in a non-integrable many-body system, which macroscopically resembles an equilibrium density matrix. The resulting expressions are fully determined by properties of the associated equilibrium density matrix, and are hence independent of the details of the initial state, while also being manifestly consistent with unitary time-evolution. For equilibrated pure states in gravity systems, such as those involving black holes, this approximation gives a prescription for calculating entanglement entropies using…
