Von Neumann Entropy Research Articles

Random projection using random quantum circuits

The random sampling task performed by Google's Sycamore processor gave us a glimpse of the “quantum supremacy era.” This has definitely shed some light on the power of random quantum circuits in this abstract task of sampling outputs from the (pseudo)random circuits. In this paper, we explore a practical near-term use of local random quantum circuits in dimensional reduction of large low-rank data sets. We make use of the well-studied dimensionality reduction technique called the random projection method. This method has been extensively used in various applications such as image processing, logistic regression, entropy computation of low-rank matrices, etc. We prove that the matrix representations of local random quantum circuits with sufficiently shorter depths [∼O(n)] serve as good candidates for random projection. We demonstrate numerically that their projection abilities are not far off from the computationally expensive classical principal components analysis on MNIST and CIFAR-100 image datasets. We also benchmark the performance of quantum random projection against the commonly used classical random projection in the tasks of dimensionality reduction of image data sets and computing von Neumann entropies of large low-rank density matrices. And finally, using variational quantum singular value decomposition, we demonstrate a near-term implementation of extracting the singular vectors with dominant singular values after quantum random projecting a large low-rank matrix to lower dimensions. All such numerical experiments unequivocally demonstrate the ability of local random circuits to randomize a large Hilbert space at sufficiently shorter depths with robust retention of properties of large data sets in reduced dimensions. Published by the American Physical Society 2024

Quantum Entropy and the Mechanism of Gravity

Quantum entropy measures the number of possible microstates of position and momentum due to quantum uncertainty. Black hole entropy is an example of quantum entropy. Classical entropy, like the Gibbs entropy of a gas, does not depend on quantum uncertainty. This paper is the first to define the quantum entropy of ordinary matter, termed “atomic entropy”. The atomic entropy at the surface of a spherical object depends on its mass and is inversely proportional to its surface area. The entropy scale factor (ESF) is a theory that all changes in the scale of spacetime are due to differences in entropy. Defining atomic entropy allows the ESF to make predictions for systems including ordinary matter. In the ESF, gravitational objects are attracted to one another because gravitational entropy, a form of quantum entropy, increases as the distance between these objects decreases. The increase in gravitational entropy increases the total entropy of the system, in accordance with the second law of thermodynamics. This paper is also the first to use the gradients in the scale of space and time predicted by the ESF to derive Newtonian gravity. Explaining the mechanism of gravity in this way provides a link between quantum physics and gravity.

Quantum Entropy and Reinforcement Learning for Distributed Denial of Service Attack Detection in Smart Grid

Quantum Entropy and Reinforcement Learning for Distributed Denial of Service Attack Detection in Smart Grid

Quantum to classical parton evolution in the QGP

We study the time evolution of the density matrix of a high energy quark in the presence of a dense QCD background that is modeled as a stochastic Gaussian color field. At late times, we find that only the color singlet component of the quark’s reduced density matrix survives the in-medium evolution and that the density matrix becomes asymptotically diagonal in both transverse position and momentum spaces. In addition, we observe an accelerated entropy growth due to the larger phase space being explored by the quark and that the quantum and classical quark entropies converge at late times. We further observe that the quark state loses all memory of the initial condition. Combined with the fact that the reduced density matrix satisfies Boltzmann-diffusion transport, we conclude that the quark reduced density matrix can be interpreted as a classical phase space distribution.

Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives.

This article is devoted to developing an approach for manipulating the von Neumann entropy S(ρ(t)) of an open two-qubit system with coherent control and incoherent control inducing time-dependent decoherence rates. The following goals are considered: (a) minimizing or maximizing the final entropy S(ρ(T)); (b) steering S(ρ(T)) to a given target value; (c) steering S(ρ(T)) to a target value and satisfying the pointwise state constraint S(ρ(t))≤S¯ for a given S¯; (d) keeping S(ρ(t)) constant at a given time interval. Under the Markovian dynamics determined by a Gorini-Kossakowski-Sudarshan-Lindblad type master equation, which contains coherent and incoherent controls, one- and two-step gradient projection methods and genetic algorithm have been adapted, taking into account the specifics of the objective functionals. The corresponding numerical results are provided and discussed.

Tip of the Quantum Entropy Cone.

Relations among von Neumann entropies of different parts of an N-partite quantum system have direct impact on our understanding of diverse situations ranging from spin systems to quantum coding theory and black holes. Best formulated in terms of the set Σ_{N}^{*} of possible vectors comprising the entropies of the whole and its parts, the famous strong subaddivity inequality constrains its closure Σ[over ¯]_{N}^{*}, which is a convex cone. Further homogeneous constrained inequalities are also known. In this Letter we provide (nonhomogeneous) inequalities that constrain Σ_{N}^{*} near the apex (the vector of zero entropies) of Σ[over ¯]_{N}^{*}, in particular showing that Σ_{N}^{*} is not a cone for N≥3. Our inequalities apply to vectors with certain entropy constraints saturated and, in particular, they show that while it is always possible to upscale an entropy vector to arbitrary integer multiples it is not always possible to downscale it to arbitrarily small size, thus answering a question posed by Winter.

Attaining the maximum Bose–Einstein condensation in a finite-size hard-core boson model

We propose a model for hard-core bosons in a lattice which allows to achieve the optimal occupation number predicted by Tennie et al (2017 Phys. Rev. B 96 064502) for a finite number of sites. The model is based on an extension of the Hamiltonian of the so-called Hubbard star, whose quantum properties are studied by means of quantum information descriptors such as the von Neumann entropy and the mutual information. These metrics are analyzed as a function of the one- and two-particle reduced density matrices, allowing to explore the relationship between condensation and entanglement by means of a control parameter that, under a given limit, connects our findings with previous results. All developments comprised in this article have been derived by analytical methods.

Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique

Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique

Geometric versus Entropic Gaussian Correlations in an Open Quantum System of Two Bosonic Modes

Different types of geometric and entropic quantum correlation quantifiers are studied for a system composed of two resonant bosonic modes embedded in a thermal bath. The description of the evolution of the correlation measures is formulated in the framework of the theory of open systems, based on completely positive quantum dynamical semigroups, by using both a geometric and entropic quantification of total nonclassical correlations of Gaussian states. We consider the special case when the initial squeezed thermal state of the system preserves its form in time. We show that time evolution of the measures strongly depends on the parameters characterising the initial state of the system (squeezing parameter and average thermal photon numbers of the two modes) and of the thermal environment (temperature of the thermal bath and dissipation rate). In the limit of large times all the considered measures asymptotically tend to zero value, corresponding to an asymptotic bimodal uncorrelated product state. We make a comparison between the behaviour of the evolution in time of the Gaussian geometric quantum correlations and Gaussian entropic quantum correlations.

Phase transition in von Neumann entropy from replica symmetry breaking

We study the entanglement transition in monitored Brownian SYK chains in the large-N limit. Without measurement the steady state n-th Rényi entropy is obtained by summing over a class of solutions, and is found to saturate to the Page value in the n → 1 limit. In the presence of measurements, the analytical continuation n → 1 is performed using the cyclic symmetric solution. The result shows that as the monitoring rate increases, a continuous von Neumann entanglement entropy transition from volume-law to area-law occurs at the point of replica symmetry unbreaking.

Stochastic Adder Circuits with Improved Entropy Output.

Random pulse computing (RPC), the third paradigm along with digital and quantum computing, draws inspiration from biology, particularly the functioning of neurons. Here, we study information processing in random pulse computing circuits intended for the summation of numbers. Based on the information-theoretic merits of entropy budget and relative Kolmogorov-Sinai entropy, we investigate the prior art and propose new circuits: three deterministic adders with significantly improved output entropy and one exact nondeterministic adder that requires much less additional entropy than the previous art. All circuits are realized and tested experimentally, using quantum entropy sources and reconfigurable logic devices. Not only the proposed circuits yield a precise mathematical result and have output entropy near maximum, which satisfies the need for building a programmable random pulse computer, but also they provide affordable hardware options for generating additional entropy.

Some New Results Involving Past Tsallis Entropy of Order Statistics.

This work focuses on exploring the properties of past Tsallis entropy as it applies to order statistics. The relationship between the past Tsallis entropy of an ordered variable in the context of any continuous probability law and the past Tsallis entropy of the ordered variable resulting from a uniform continuous probability law is worked out. For order statistics, this method offers important insights into the characteristics and behavior of the dynamic Tsallis entropy, which is associated with past events. In addition, we investigate how to find a bound for the new dynamic information measure related to the lifetime unit under various conditions and whether it is monotonic with respect to the time when the device is idle. By exploring these properties and also investigating the monotonic behavior of the new dynamic information measure, we contribute to a broader understanding of order statistics and related entropy quantities.

Generation of 1 GB full entropy random numbers with the enhanced-NRBG method

Random numbers have significant applications in fundamental science, high-level scientific research, cryptography and several other areas where there is a pressing need for high-quality random numbers. We present an experimental demonstration of a non-deterministic random bit generator from a quantum entropy source and a deterministic random bit generator mechanism to provide high quality random numbers of 1 GB data size. Quantum entropy is realized by a series of quantum chips based on radioactive isotope Americium-241. The extracted raw random numbers are further post-processed to generate a high-entropy seed for the hash based deterministic random bit generator. We discuss the implementation of randomness extraction algorithm and Hash-DRBG algorithm in detail. The random numbers pass all the randomness measures provided in standard statistical test suites.

The relation between susceptibility and quantum entropy in the ladder-plus-Y double quantum dot system under spontaneously generated coherence

The relation between susceptibility and quantum entropy in the ladder-plus-Y double quantum dot system under spontaneously generated coherence

Perturbation theory for the logarithm of a positive operator

In various contexts in mathematical physics, such as out-of-equilibrium physics and the asymptotic information theory of many-body quantum systems, one needs to compute the logarithm of a positive unbounded operator. Examples include the von Neumann entropy of a density matrix and the flow of operators with the modular Hamiltonian in the Tomita-Takesaki theory. Often, one encounters the situation where the operator under consideration, which we denote by ∆, can be related by a perturbative series to another operator ∆0, whose logarithm is known. We set up a perturbation theory for the logarithm log ∆. It turns out that the terms in the series possess a remarkable algebraic structure, which enables us to write them in the form of nested commutators plus some “contact terms”.

Residual Tsallis Entropy and Record Values: Some New Insights

Recently, the uncertainty aspects of record values have been increasingly studied in the literature. In this paper, we study the residual Tsallis entropy of upper record values coming from random samples. In the continuous case, we define the Tsallis entropy quantity for the residual lifetime of upper record values in general distributions as the residual Tsallis entropy of upper record values coming from a uniform distribution. We also obtain a lower bound on the residual Tsallis entropy of upper data set values originating from an arbitrary continuous probability distribution. We also discuss the monotonic property of the residual Tsallis entropy of upper data sets.

Von Neumann algebra description of inflationary cosmology

We study the von Neumann algebra description of the inflationary quasi-de Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero energy flux across the horizon, which can be identified with the expectation value of the static time translation generator. Moreover, as a dS isometry associated with the static time translation is spontaneously broken, the fluctuation in time is accumulated, which induces the fluctuation in the energy flux. When the inflationary period is given by (ϵHH)-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\epsilon _H H)^{-1}$$\\end{document} where ϵH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon _H$$\\end{document} is the slow-roll parameter measuring the increasing rate of the Hubble radius, both the energy flux and its fluctuation diverge in the G→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G \\rightarrow 0$$\\end{document} limit. Taking the fluctuation in the energy flux and that in the observer’s energy into account, we argue that the inflationary quasi-dS space is described by Type II∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_\\infty $$\\end{document} algebra. As the entropy is not bounded from above, this is different from Type II1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_1$$\\end{document} description of perfect dS space in which the entropy is maximized by the maximal entanglement. We also show that our result is consistent with the observation that the von Neumann entropy for the density matrix reflecting the fluctuations above is interpreted as the generalized entropy.

Enhancing the Security of the BB84 Quantum Key Distribution Protocol against Detector-Blinding Attacks via the Use of an Active Quantum Entropy Source in the Receiving Station.

True randomness is necessary for the security of any cryptographic protocol, including quantum key distribution (QKD). In QKD transceivers, randomness is supplied by one or more local, private entropy sources of quantum origin which can be either passive (e.g., a beam splitter) or active (e.g., an electronic quantum random number generator). In order to better understand the role of randomness in QKD, I revisit the well-known "detector blinding" attack on the BB84 QKD protocol, which utilizes strong light to achieve undetectable and complete recovery of the secret key. I present two findings. First, I show that the detector-blinding attack was in fact an attack on the receiver's local entropy source. Second, based on this insight, I propose a modified receiver station and a statistical criterion which together enable the robust detection of any bright-light attack and thus restore security.

Testing the Minimum System Entropy and the Quantum of Entropy.

Experimental and theoretical results about entropy limits for macroscopic and single-particle systems are reviewed. All experiments confirm the minimum system entropy S⩾kln2. We clarify in which cases it is possible to speak about a minimum system entropykln2 and in which cases about a quantum of entropy. Conceptual tensions with the third law of thermodynamics, with the additivity of entropy, with statistical calculations, and with entropy production are resolved. Black hole entropy is surveyed. Claims for smaller system entropy values are shown to contradict the requirement of observability, which, as possibly argued for the first time here, also implies the minimum system entropy kln2. The uncertainty relations involving the Boltzmann constant and the possibility of deriving thermodynamics from the existence of minimum system entropy enable one to speak about a general principle that is valid across nature.

Experimental certification of more than one bit of quantum randomness in the two inputs and two outputs scenario

One of the striking properties of quantum mechanics is the occurrence of the Bell-type non-locality. They are a fundamental feature of the theory that allows two parties that share an entangled quantum system to observe correlations stronger than possible in classical physics. In addition to their theoretical significance, non-local correlations have practical applications, such as device-independent randomness generation, providing private unpredictable numbers even when they are obtained using devices delivered by an untrusted vendor. Thus, determining the quantity of certifiable randomness that can be produced using a specific set of non-local correlations is of significant interest. In this paper, we present an experimental realization of recent Bell-type operators designed to provide private random numbers that are secure against adversaries with quantum resources. We use semi-definite programming to provide lower bounds on the generated randomness in terms of both min-entropy and von Neumann entropy in a device-independent scenario. We compare experimental setups providing Bell violations close to the Tsirelson’s bound with lower rates of events, with setups having slightly worse levels of violation but higher event rates. Our results demonstrate the first experiment that certifies close to two bits of randomness from binary measurements of two parties. Apart from single-round certification, we provide an analysis of finite-key protocol for quantum randomness expansion using the Entropy Accumulation theorem and show its advantages compared to existing solutions.