Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and demonstrated only for the restricted Boltzmann machine architecture. Here, we estimate generalized Renyi entropies of autoregressive neural quantum states with up to N=256 spins using quantum Monte Carlo methods. A naive "direct sampling" approach performs well for low-order Renyi entropies but fails for larger orders when benchmarked on a 1D Heisenberg model. We therefore propose an improved "conditional sampling" method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling and facilitates calculations of higher-order Renyi entropies in both 1D and 2D Heisenberg models. Access to higher-order Renyi entropies allows for an approximation of the von Neumann entropy as well as extraction of the single copy entanglement. Both methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.
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 …
Rényi-Holevo inequality from $α$-$z$-Rényi relative entropies
Universidad Nacional de La Plata（拉普拉塔国立大学）；Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET)（阿根廷国家科学研究委员会）；Institut de Ciencies Fotoniques (ICFO), The Barcelona Institute of Science and Technology（光科学研究所，巴塞罗那科学与技术研究院）；Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba（科尔多瓦国立大学数学、天文学、物理与计算学院）；Faculty of Physics, Astronomy and Applied Computer Science, Institute of Theoretical Physics, Jagiellonian University（雅盖隆大学理论物理系物理、天文学与应用计算机科学学院）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
The Tsallis and Rényi entropies are important quantities in the information theory, statistics and related fields because the Tsallis entropy is an one parameter generalization of the Shannon entropy and the Rényi entropy includes several useful entropy measures such as the Shannon entropy, Min-entropy and so on, as special choices of its parameter. On the other hand, the discrete-time quantum walk plays important roles in various applications, for example, quantum speed-up algorithm and universal computation. In this paper, we show limiting behaviors of the Tsallis and Rényi entropies for dis…
Recently it was shown that the growth of entanglement in an initially separable state, as measured by the purity of subsystems, can be characterized by a timescale that takes a universal form for any Hamiltonian. We show that the same timescale governs the growth of entanglement for all Rényi entropies. Since the family of Rényi entropies completely characterizes the entanglement of a pure bipartite state, our timescale is a universal feature of bipartite entanglement. The timescale depends only on the interaction Hamiltonian and the initial state.
This paper introduces "swiveled Renyi entropies" as an alternative to the Renyi entropic quantities put forward in [Berta et al., Phys. Rev. A 91, 022333 (2015)]. What distinguishes the swiveled Renyi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Renyi entropies are ordered, which is an important property of the Renyi family of entropies. The swiveled Renyi entropies are however generally dis…
Recently a new quantum generalization of the Renyi divergence and the corresponding conditional Renyi entropies was proposed. Here we report on a surprising relation between conditional Renyi entropies based on this new generalization and conditional Renyi entropies based on the quantum relative Renyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Renyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that…
The Rényi and von Neumann entropies of the thermal state in the generalized uncertainty principle (GUP)-corrected single harmonic oscillator system are explicitly computed within the first order of the GUP parameter $α$. While the von Neumann entropy with $α= 0$ exhibits a monotonically increasing behavior in external temperature, the nonzero GUP parameter makes the decreasing behavior of the von Neumann entropy at the large temperature region. As a result, the von Neumann entropy is maximized at the finite temperature if $α\neq 0$. The Rényi entropy $S_γ$ with nonzero $α$ also exhibits simila…
An Algorithm for Estimating $α$-Stabilizer Rényi Entropies via Purity
Quantum Engineering Centre for Doctoral Training, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering, University of Bristol（量子工程博士培训中心，H.H.威尔斯物理实验室和电气与电子工程系，布里斯托大学）；H.H. Wills Physics Laboratory, University of Bristol（H.H.威尔斯物理实验室，布里斯托大学）
We compute the quantum Renyi relative entropies in an infinite spinless fermionic chain with a defect. Doing a numerical analysis we will show that the resulting quantity depends non trivially on the effective central charge of the theory. Moreover, we will see that an explicit analytic expression can be written for all of them and from that, one can read the quantum fidelity and the relative entropy.
Renyi熵S_q是量子纠缠的有用度量;它们可以从提升到q次幂的约化密度矩阵的迹计算，其中q>=0。对于（d+1）维共形场论，S^{d-1}上的Renyi熵可以从这些理论在（d+1）维de Sitter空间或R x H^d上的热配分函数中提取，其中H^d是d维双曲空间。这些热配分函数又可以分别表示为（d+1）维球面和S ^1 x H^d的分支覆盖上的路径积分。我们计算了d=2时自由无质量标量和费米子的Renyi熵，并展示了如何利用ζ函数正则化，发现S^3的分支覆盖和S^1 × H^2的分支覆盖的计算结果是一致的。大量的自由场的类似计算提供单调插值函数之间的Renyi熵在高斯和平凡的不动点。最后，我们讨论了类似的Renyi熵计算d>2。
We show that the stability theorem of the depolarizing channel holds for the output quantum $p$-Rényi entropy for $p \ge 2$ or $p=1$, which is an extension of the well known case $p=2$. As an application, we present a protocol in which Bob determines whether Alice prepares a pure quantum state close to a product state. In the protocol, Alice transmits to Bob multiple copies of a pure state through a depolarizing channel, and Bob estimates its output quantum $p$-Rényi entropy. By using our stability theorem, we show that Bob can determine whether her preparation is appropriate.
For certain families of finite quantum graphs, we study the question of how eigenfunctions are distributed over the graph. To characterize properties of the distribution, generalized entropies of the Rényi and Tsallis types are considered. The presented approach is similar to entropic uncertainty relations of the Maassen-Uffink type. Using the Riesz theorem, we derive lower bounds on symmetrized generalized entropies of eigenfunctions. A quality of such estimates will depend on boundary conditions used at vertices of the given graph. Rényi and Tsallis entropies of eigenfunctions of star graphs…
We consider Renyi entropies of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Renyi entropy to that of a black hole with hyperbolic horizon; as the Renyi parameter $n$ increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as $n$ is varied, the Renyi entropies will exhibit a phase transition at a critical value of $n$. The location of the phase transition, along with the spectrum of the r…
We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.
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…
We study systematically resource measures of coherence and entanglement based on Rényi relative entropies, which include the logarithmic robustness of coherence, geometric coherence, and conventional relative entropy of coherence together with their entanglement analogues. First, we show that each Rényi relative entropy of coherence is equal to the corresponding Rényi relative entropy of entanglement for any maximally correlated state. By virtue of this observation, we establish a simple operational connection between entanglement measures and coherence measures based on Rényi relative entropi…
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, …
The Rényi entropies $R_{p}[ρ], 0<p<\infty$ of the probability density $ρ_{n,l,m}(\vec{r})$ of a physical system completely characterize the chemical and physical properties of the quantum state described by the three integer quantum numbers $(n,l,m)$. The analytical determination of these quantities is practically impossible up until now, even for the very few systems where their Schrödinger equation is exactly solved. In this work, the Rényi entropies of Rydberg (highly-excited) hydrogenic states are explicitly calculated in terms of the quantum numbers and the parameter $p$. To do that we us…
We introduce and study generalized Rényi entropies defined through the traces of products of ${\rm Tr}_B (|Ψ_i\rangle\langle Ψ_j|)$ where $|Ψ_i\rangle$ are eigenstates of a two-dimensional conformal field theory (CFT). When $|Ψ_i\rangle=|Ψ_j\rangle$ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four-point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to comp…
We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
We discuss the Renyi entanglement entropies of descendant states in critical one-dimensional systems with boundaries, that map to boundary conformal field theories in the scaling limit. We unify the previous conformal-field-theory approaches to describe primary and descendant states in systems with both open and closed boundaries. We provide universal expressions for the first two descendants in the identity family. We apply our technique to critical systems belonging to different universality classes with non-trivial boundary conditions that preserve conformal invariance, and find excellent a…
In this work, we study the Renyi-alpha entropies S_{alpha}(\hat{rho}) = (1 - alpha)^{-1} \ln{Tr(\hat{rho}^{alpha})} of quantum states for N bosons in the phase-space representation. With the help of the Bopp rule, we derive the entropies of Gaussian states in closed form for positive integers alpha = 2,3,4, ... and then, with the help of the analytic continuation, acquire the closed form also for real values of alpha > 0. The quantity S_2(\hat{rho}), primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Renyi-alpha entropies, with positive…
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) …
We show that the new quantum extension of Renyi's α-relative entropies, introduced recently by Muller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Renyi relative entropies depends on the parameter α: for α<1, the …
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 …
Phase-space features of the Wigner flow are examined so to provide a set of continuity equations that describe the flux of quantum information in the phase-space. The reported results suggest that the non-classicality profile of anharmonic (periodic) quantum systems can be consistently obtained in terms of the fluxes of probability, purity and von Neumann entropy. Extensions of the such phase-space quantifiers are also investigated in the context of the so-called Rényi entropy.
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…
Finite-size quantum key distribution rates from Rényi entropies using conic optimization
Luxquanta Technologies S.L.（Luxquanta科技有限公司）；ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学技术研究院）；Universidad de Valladolid（巴利亚多利德大学）
Charged and symmetry-resolved Rényi entropies are entanglement measures quantifying the degree of entanglement within different charge sectors of a theory with a conserved global charge. We use holography to determine the dependence of charged Rényi entropies on small shape deformations away from a spherical or planar entangling surface in general dimensions. This dependence is completely characterized by a single coefficient appearing in the two point function of the displacement operator associated with the Rényi defect. We extract this coefficient using its relation to the one point functio…
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…
We use a Rényi entropy method to prove strong converse theorems for certain information-theoretic tasks which involve local operations and quantum or classical communication between two parties. These include state redistribution, coherent state merging, quantum state splitting, measurement compression with quantum side information, randomness extraction against quantum side information, and data compression with quantum side information. The method we employ in proving these results extends ideas developed by Sharma [arXiv:1404.5940], which he used to give a new proof of the strong converse t…
Rényi entropies are conceptually valuable and experimentally relevant generalisations of the celebrated von Neumann entanglement entropy. After a quantum quench in a clean quantum many-body system they generically display a universal linear growth in time followed by saturation. While a finite subsystem is essentially at local equilibrium when the entanglement saturates, it is genuinely out-of-equilibrium in the growth phase. In particular, the slope of the growth carries vital information on the nature of the system's dynamics, and its characterisation is a key objective of current research. …
