Von Neumann Entropy Research Articles

A-unital Operations and Quantum Conditional Entropy

Negative quantum conditional entropy states are key ingredients for information theoretic tasks such as superdense coding, state merging and one-way entanglement distillation. In this work, we ask: how does one detect if a channel is useful in preparing negative conditional entropy states? We answer this question by introducing the class of A-unital channels, which we show are the largest class of conditional entropy non-decreasing channels. We also prove that A-unital channels are precisely the completely free operations for the class of states with non-negative conditional entropy. Furthermore, we study the relationship between A-unital channels and other classes of channels pertinent to the resource theory of entanglement. We then prove similar results for ACVENN: a previously defined, relevant class of states and also relate the maximum and minimum conditional entropy of a state with its von Neumann entropy. The definition of A-unital channels naturally lends itself to a procedure for determining membership of channels in this class. Thus, our work is valuable for the detection of resourceful channels in the context of conditional entropy.

Quantum bit threads and holographic entanglement

Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be interpreted in terms of a set of Planck-thickness bit threads, which can be either classical or quantum. This interpretation uncovers an aspect of the generalized entropy that admits a neat information-theoretic description, namely, the fact that the quantum corrections can be cast in terms of entanglement distillation of the bulk state. We also prove some general properties of our prescription, including nesting and a quantum version of the max multiflow theorem. These properties are used to verify that our proposal respects known inequalities that a von Neumann entropy must satisfy, including subadditivity and strong subadditivity, as well as to investigate the fate of the holographic monogamy. Finally, using the Iyer-Wald formalism we show that for cases with a local modular Hamiltonian there is always a canonical solution to the program that exploits the property of bulk locality. Combining with previous results by Swingle and Van Raamsdonk, we show that the con- sistency of this special solution requires the semi-classical Einstein’s equations to hold for any consistent perturbative bulk quantum state.

Predication and Photon Statistics of a Three-Level System in the Photon Added Negative Binomial Distribution

Statistical and artificial neural network models are applied to forecast the quantum scheme of a three-level atomic system (3LAS) and field, initially following a photon added negative binomial distribution (PANBD). The Mandel parameter is used to detect the photon statistics of a radiation field. Explicit forms of the PANBD are given. The prediction of the Mandel parameter, atomic probability of the 3LAS in the upper state, and von Neumann entropy are obtained using time series and artificial neural network methods. The influence of probability success photons and the number of added photons to the NBD are examined. The total density matrix is used to compute and analyze the time evolution of the initial photonic negative binomial probability distribution that governs the 3LAS–field photon entanglement behavior. It is shown that the statistical quantities are strongly affected by probability success photons and the number of added photons to the NBD. Also, the prediction of quantum entropy is achieved by the time series and neural network.

Strong subadditivity lower bound and quantum channels

We derive the strong subadditivity of the von Neumann entropy with a strict lower bound, dependent on the distribution of quantum correlation in the system. We investigate the structure of states saturating the bounded subadditivity and examine its consequences for the quantum data processing inequality. The quantum data processing achieves a lower bound associated with the locally inaccessible information.

Quantum-Classical Entropy Analysis for Nonlinearly-Coupled Continuous-Variable Bipartite Systems.

The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space—a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties’ degrees of freedom which, in the quantum case, is commonly related to the production of entanglement.

Ground state of asymmetric tops with DMRG: Water in one dimension.

We propose an approach to compute the ground state properties of collections of interacting asymmetric top molecules based on the density matrix renormalization group method. Linear chains of rigid water molecules of varying sizes and density are used to illustrate the method. A primitive computational basis of asymmetric top eigenstates with nuclear spin symmetry is used, and the many-body wave function is represented as a matrix product state. We introduce a singular value decomposition approach in order to represent general interaction potentials as matrix product operators. The method can be used to describe linear chains containing up to 50 water molecules. Properties such as the ground state energy, the von-Neumann entanglement entropy, and orientational correlation functions are computed. The effect of basis set truncation on the convergence of ground state properties is assessed. It is shown that specific intermolecular distance regions can be grouped by their von-Neumann entanglement entropy, which in turn can be associated with electric dipole-dipole alignment and hydrogen bond formation. Additionally, by assuming conservation of local spin states, we present our approach to be capable of calculating chains with different arrangements of the para and ortho spin isomers of water and demonstrate that for the water dimer.

Hydrodynamics of quantum entropies in Ising chains with linear dissipation

We study the dynamics of quantum information and of quantum correlations after a quantum quench, in transverse field Ising chains subject to generic linear dissipation. As we show, in the hydrodynamic limit of long times, large system sizes, and weak dissipation, entropy-related quantities—such as the von Neumann entropy, the Rényi entropies, and the associated mutual information—admit a simple description within the so-called quasiparticle picture. Specifically, we analytically derive a hydrodynamic formula, recently conjectured for generic noninteracting systems, which allows us to demonstrate a universal feature of the dynamics of correlations in such dissipative noninteracting system. For any possible dissipation, the mutual information grows up to a time scale that is proportional to the inverse dissipation rate, and then decreases, always vanishing in the long time limit. In passing, we provide analytic formulas describing the time-dependence of arbitrary functions of the fermionic covariance matrix, in the hydrodynamic limit.

Markov chains with doubly stochastic transition matrices and application to a sequence of non-selective quantum measurements

A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff–von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolution in the future. It is shown that as the discrete time increases these convex polytopes shrink, and the minimum entropy of the probability vectors in them increases. These ideas are applied to a sequence of non-selective measurements (with different projectors in each step) on a quantum system with d-dimensional Hilbert space. The unitary time evolution in the intervals between the measurements, is taken into account. The non-selective measurements destroy stroboscopically the non-diagonal elements in the density matrix. This ‘hermaphrodite’ system is an interesting combination of a classical probabilistic system (immediately after the measurements) and a quantum system (in the intervals between the measurements). Various examples are discussed. In the ergodic example, the system follows asymptotically all discrete paths with the same probability. In the example of rapidly repeated non-selective measurements, we get the well known quantum Zeno effect with ‘frozen discrete paths’ (presented here as a biproduct of our general methodology based on Markov chains with doubly stochastic transition matrices).

Quantum geometric information flows and relativistic generalizations of G. Perelman thermodynamics for nonholonomic Einstein systems with black holes and stationary solitonic hierarchies

We investigate classical and quantum geometric information flow theories (respectively, GIFs and QGIFs) when the geometric flow evolution and field equations for nonholonomic Einstein systems, NES, are derived from Perelman-Lyapunov type entropic type functionals. In this work, the term NES encodes models of gravitational and matter fields interactions and their geometric flow evolution subjected to nonholonomic (equivalently, non-integrable, anholonomic) constraints. There are used canonical geometric variables which allow a general decoupling and integration of systems of nonlinear partial differential equations describing GIFs and QGIFs and (for self-similar geometric flows) Ricci soliton type configurations. Our approach is different from the methods and constructions elaborated for special classes of solutions characterized by area--hypersurface entropy, related holographic and dual gauge--gravity models, and conformal field theories, involving generalizations of the Bekenstein-Hawking entropy and black hole thermodynamics. We formulate the theory of QGIFs which in certain quasi-classical limits encodes GIFs and models with flow evolution of NES. There are analysed the most important properties (inequalities) for NES and defined and computed QGIF versions of the von Neumann, relative and conditional entropy; mutual information, (modified) entanglement and R\'{e}nyi entropy. We construct explicit examples of generic off-diagonal exact and parametric solutions describing stationary solitonic gravitational hierarchies and deformations of black hole configurations. Finally, we show how Perelman's entropy and geometric thermodynamic values, and extensions to GIF and QGIF models can be computed for various new classes of exact solutions which cannot be described following the Bekenstein-Hawking approach.

Exact Time Evolution of Genuine Multipartite Correlations for N-Qubit Systems in a Common Thermal Reservoir

We investigate the dynamical evolution of genuine multipartite correlations for N-qubits in a common reservoir considering a non-dissipative qubits-reservoir model. We derive an exact expression for the time-evolved density matrix by modeling the reservoir as a set of infinite harmonic oscillators with a bilinear form of interaction Hamiltonian. Interestingly, we find that the choice of two-level systems corresponding to an initially correlated multipartite state plays a significant role in potential robustness against environmental decoherence. In particular, the generalized W-class Werner state shows robustness against the decoherence for an equivalent set of qubits, whereas a certain generalized GHZ-class Werner state shows robustness for inequivalent sets of qubits. It is shown that the genuine multipartite concurrence (GMC), a measure of multipartite entanglement of an initially correlated multipartite state, experiences an irreversible decay of correlations in the presence of a thermal reservoir. For the GHZ-class Werner state, the region of mixing parameters for which there exists GMC, shrinks with time and with increase in the temperature of the thermal reservoir. Furthermore, we study the dynamical evolution of the relative entropy of coherence and von-Neumann entropy for the W-class Werner state.

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

We prove that the entanglement entropy of any pure initial state of a bipartite bosonic quantum system grows linearly in time with respect to the dynamics induced by any unstable quadratic Hamiltonian. The growth rate does not depend on the initial state and is equal to the sum of certain Lyapunov exponents of the corresponding classical dynamics. This paper generalizes the findings of [Bianchi et al., JHEP 2018, 25 (2018)], which proves the same result in the special case of Gaussian initial states. Our proof is based on a recent generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum systems [De Palma et al., arXiv:2105.05627]. This technique allows us to extend our result to generic mixed initial states, with the squashed entanglement providing the right generalization of the entanglement entropy. We discuss several applications of our results to physical systems with (weakly) interacting Hamiltonians and periodically driven quantum systems, including certain quantum field theory models.

Benchmark Performance of Digital QKD Platform Using Quantum Permutation Pad

Quantum permutation pad or QPP is a set of quantum permutation gates. QPP has been demonstrated for quantum secure encryption in both classical and quantum computing systems recently, even at a noisy 5-qubit IBMQ systems. In a classical computing system, QPP encryption is implemented as a permutation gate matrix multiplication with information state vectors. In a quantum computing system, QPP is compiled into a quantum encryption circuit in a native quantum computer and encryption is performed through QPP circuit. Leveraging its quantum mechanical characteristics, we report a digital QKD or D-QKD platform using QPP as a quantum mechanical algorithm implemented in classical systems to distribute quantum entropy, generated from physical quantum random number generators or QRNG, and quantum key over the internet. D-QKD interfaces have been developed to support the photonic QKD standard ETSI-014. This makes any systems with ETSI QKD standards compatible with D-QKD. D-QKD offers point-to-point quantum entropy and quantum key distributions as well as point-to-multi-points quantum key synchronizations with speeds 1000x faster than photonic QKD. This paper reports benchmark performance tests and randomness quality tests for pure quantum entropy generated by a QRNG and expanded entropy using the QPP protocol. The work has been funded by the PlanQK <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> project and deployed within the OpenQKD <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> testbed Berlin, operated by Deutsche Telekom.

Información Mutua: Una forma de cuantificar las correlaciones

Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.

Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions

Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl(n) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS± and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.

Observation evidence for entropy switch model of substorm onset

The cause of substorm onset is not yet understood. Chen CX (2016) proposed an entropy switch model, in which substorm onset results from the development of interchange instability. In this study, we sought observational evidence for this model by using Time History of Events and Macroscale Interactions during Substorms (THEMIS) data. We examined two events, one with and the other without a streamer before substorm onset. In contrast to the stable magnetosphere, where the total magnetic field strength is a decreasing function and entropy is an increasing function of the downtail distance, in both events the total magnetic field strength and entropy were reversed before substorm onset. After onset, the total magnetic field strength, entropy, and other plasma quantities fluctuated. In addition, a statistical study was performed. By confining the events with THEMIS satellites located in the downtail region between ~8 and ~12 Earth radii, and 3 hours before and after midnight, we found the occurrence rate of the total magnetic field strength reversal to be 69% and the occurrence rate of entropy reversal to be 77% of the total 205 events.

Entropy in Biochemical Failure

The ab-initio determination of the thermodynamic properties of the hydrolysis of the GTP gamma-phosphate in normal and abnormal cell functions of the RAS protein mutant Thr (Q61) leads to a description of energy cycle deviations in the abnormal mitogen-activated protein kinase cascade. 1 A predictive non-equilibrium probability statement describing the nonlinear changes for these open and finite-lifetime systems follows from reasonable enthalpy and entropy values between the normal and mutated forms based on structures of the GTPase states at the allosteric site. Recent advances in understanding entropy in terms of asymmetric and highly entropic catalysis lead to an investigation of the GTPase entropy, specifically with regard to a failure in catalysis of the phosphate fragment by a water hydrogen positioned by the enzyme. 2 Utilizing a simple atomic metal catalyst surface displacement model, a paradigm that reduces noise from the quantum entanglement plus atomic displacement terms results in the process entropy. The evaluation of entropies within the mixed ionic, covalent and entangled system requires a nonlinear Markovian approach utilizing von Neumann entropies achieved by a systematic accumulation of entangled potentials in a step-wise method. 3 , 4 Determination of the Hamiltonian for the entangled atomic state includes pure and mixed quantum states solved within the Araki–Leib triangle boundary resulting in only hard-entangled states, and the entanglement of Coulombic and Laughlin-like states can be evaluated by slicing the Hilbert spaces and solving the pure states, or mixed states separately, and then summing them. 5 Incorporating the resulting entanglement potentials as well as the Coulombic atomic displacement states into a derivative of the Fokker–Planck equation results in generated and produced entropy. 6

Islands and Page curves in charged dilaton black holes

We study the Page curve for eternal Garfinkle–Horowitz–Strominger dilaton black holes in four dimensional asymptotically flat spacetime by using the island paradigm. The results demonstrate that without the island, the entanglement entropy of Hawking radiation is proportional to time and becomes divergent at late times. While taking account of the existence of the island outside the event horizon, the entanglement entropy stops growing at late times and eventually reaches a saturation value. This value is twice of the Bekenstein–Hawking entropy and consistent with the finiteness of the von Neumann entropy of eternal black holes. Moreover, we discuss the impact of the stringy coefficient n and charge Q on the Page time and the scrambling time respectively. For the non-extremal case, the influence of the coefficient n on them is small compared to the influence of the charge Q. However, for the extremal case, the Page time and the scrambling time become divergent or near vanishing. This implies the island paradigm needs further investigation.

Holographic Kolmogorov-Sinai entropy and the quantum Lyapunov spectrum

In classical chaotic systems the entropy, averaged over initial phase space distributions, follows a universal behavior. While approaching thermal equilibrium it passes through a stage where it grows linearly, while the growth rate, the Kolmogorov-Sinai entropy (rate), is given by the sum over all positive Lyapunov exponents. A natural question is whether a similar relation is valid for quantum systems. We argue that the Maldacena-Shenker-Stanford bound on quantum Lyapunov exponents implies that the upper bound on the growth rate of the entropy, averaged over states in Hilbert space that evolve towards a thermal state with temperature T, should be given by πT times the thermal state’s von Neumann entropy. Strongly coupled, large N theories with black hole duals should saturate the bound. To test this we study a large number of isotropization processes of random, spatially homogeneous, far from equilibrium initial states in large N, mathcal{N} = 4 Super Yang Mills theory at strong coupling and compute the ensemble averaged growth rate of the dual black hole’s apparent horizon area. We find both an analogous behavior as in classical chaotic systems and numerical evidence that the conjectured bound on averaged entropy growth is saturated granted that the Lyapunov exponents are degenerate and given by λi = ±2πT. This fits to the behavior of classical systems with plus/minus symmetric Lyapunov spectra, a symmetry which implies the validity of Liouville’s theorem.

Photon statistics and non-local properties of a two-qubit-field system in the excited negative binomial distribution

In this paper, a quantum scheme for a two-qubit system (2QS) and field initially prepared in the excited negative binomial distribution is presented. The field photon statistics is detected from the evolution of the Mandel parameter, while the evolution of von Neumann entropy detects the nonlocal correlation between the 2QS and radiation field. The concurrence is used to detect the qubit-qubit entanglement during the time evolution. The dynamical properties of single-qubit and two-qubit quantum Fisher information are investigated. We visualize the number of photon excitations on the field in negative binomial states with influence of photon success probability. A connection is provided between the dynamical behaviors of these statistical quantities. We have found that the proposed quantities are strongly influenced by the number of excited photons of the field in negative binomial states and photon success probability.

Classical Computer Assisted Analysis of Small Multiqudit Systems

Quantum computation model is regarded as a model which can overcome barriers in calculations efficiency of problems which appear in modern science. In spite of hardware development, in particular a recent emergence of several different physical installations of the pioneering quantum machines, the contemporary and numerical analysis of problems concerning quantum computing is very important. In the first part of this article, some useful computing techniques for quantum registers processed by a quantum circuits are presented. Applied classical parallel computational techniques are utilised to shorten the whole computational time. New methods of processing state vectors for qudits and density matrices are presented, indicating which operations may be performed in parallel in the context of the implementation of local unitary operations. There is also shown, how to use the reduction operation in parallel implementation of the von Neumann measurement by performing local measurements on a system of qudits. In addition to the purely technical results as described above, the paper includes also a bunch of purely theoretical results which substitute a solid mathematical ground for the computations performed with the help of the computational routines as described in Section III. In particular, a discussion concerning general multi-qudit quantum states through the prism of Entropy and Negativity measures of entanglement included in has been presented. Additionally, the notion of the total entanglement has been introduced. For certain classes of popular multi-qudit states, the introduced deficits of entanglement defined with the use of von Neumann Entropy and Negativity have been discussed. In particular, by the use of Gram matrix technique, the corresponding deficits of entanglement in the analysed states have been computed in an explicite way. Additionally, some new results on AME states for some multiqudit systems are also included in Section V. The last part of the article presents some numerical experiments on multi-qudit entanglement and determination of total entanglement values for convex combinations of GHZ andWstates. Some details concerning technical nature of results are included in the attached Appendix A.