We show that entanglement can always arise in the interaction of an arbitrarily large system in any mixed state with a single qubit in a pure state. This small initial purity is enough to enforce entanglement even when the total entropy is close to maximum. We demonstrate this feature using the Jaynes-Cummings interaction of a two level atom in a pure state with a field in a thermal state at an arbitrarily high temperature. We find the time and temperature variation of a lower bound on the amount of entanglement produced and study the classical correlations quantified by the mutual information.
Analytical solution and entanglement swapping of a double Jaynes-Cummings model in non-Markovian environments are investigated by the timeconvolutionless master equation method. We obtain the analytical solution of this model and discuss in detail the influence of atom-cavity coupling, non-Markovian effect and initial state purity on entanglement dynamics. The results show that, in the non-Markovian environments, the entanglement between two cavities can be swapped to other bipartite subsystems by interaction between an atom and its own cavity. Due to the dissipation of environment, the entang…
We study production of entanglement with highly-mixed states. We find that entanglement between highly mixed states can be generated via a direct unitary interaction even when both states have purities arbitrarily close to zero. This indicates that purity of a subsystem is not required for entanglement generation. Our result is in contrast to previous studies where the importance of the subsystem purity was emphasized.
We study the stability of unitary quantum dynamics of composite systems (for example: central system + environment) with respect to weak interaction between the two parts. Unified theoretical formalism is applied to study different physical situations: (i) coherence of a forward evolution as measured by purity of the reduced density matrix, (ii) stability of time evolution with respect to small coupling between subsystems, and (iii) Loschmidt echo measuring dynamical irreversibility. Stability has been measured either by fidelity of pure states of a composite system, or by the so-called reduce…
Correlations obtained from sequences of measurements have been employed to distinguish among different physical theories or to witness the dimension of a system. In this work we show that they can also be used to establish semi-device independent lower bounds on the purity of the initial quantum state or even on one of the post-measurement states. For single systems this provides information on the quality of the preparation procedures of pure states or the implementation of measurements with anticipated pure post-measurement states. For joint systems one can combine our bound with results fro…
We establish contact between the delocalization properties of pure quantum states, as quantified by their number of principal components, and the average generalized entanglement properties, as quantified by purity measures relative to different observable sets. We find that correlations between products of state vector components with respect to Hamming distance play an important role in the structure of subsystem-based purity measures. In particular, we derive general conditions under which the amount of global multipartite entanglement relates to the inverse participation ratio averaged ove…
The entanglement, purity and energy of two isolated two-level atoms which are initially prepared in Bell state and each interacts with a thermal cavity field are investigated by considering the atomic motion and the field-mode structure. We achieve the analytical solutions of the atomic qubits by using the algebraic dynamical approach and the influences of the field-mode structure parameter, the strength of the thermal field and the detuning on the entanglement, purity and energy are discussed. We also investigate the state evolution of the atomic qubits based on the entanglement-purity-energy…
The Bures-Hall ensemble is a unique measure of density matrices that satisfies various distinguished properties in quantum information processing. In this work, we study the statistical behavior of entanglement over the Bures-Hall ensemble as measured by the simplest form of an entanglement entropy - the quantum purity. The main results of this work are the exact second and third moment expressions of quantum purity valid for any subsystem dimensions, where the corresponding results in the literature are limited to the scenario of equal subsystem dimensions. In obtaining the results, we have d…
The dynamics of two coupled spins-1/2 interacting with a spin-bath via the quantum Heisenberg XY coupling is studied. The pair of central spins served as a quantum open subsystem are initially prepared in two types of states: the product states and the Bell states. The bath, which consists of $N$ (in the thermodynamic limit $N\to\infty$) mutually coupled spins-1/2, is in a thermal state at the beginning. By the Holstein-Primakoff transformation, the model can be treated effectively as two spin qubits embedded in a single mode cavity. The time-evolution of the purity, z-component summation and …
Based on the complementarity relation between entanglement of a composite system and the purity of a subsystem, we propose a simple method to measure the amount of entanglement. The method can be applied to a bipartite system in a pure state of any arbitrary dimension. It requires only single qudit rotations and straightforward probability measurements performed on one of the subsystems, and can thus be easily implemented experimentally using linear optical devices.
We study dynamical generation of entanglement in bipartite quantum systems, characterized by purity (or linear entropy), and caused by the coupling between the two subsystems. Explicit semiclassical theory of purity decay is derived for integrable classical dynamics of the uncoupled system, and for localized (general Gaussian wave-packet) initial states. Purity decays as an algebraic function of time times strength of perturbation, independently of the Planck's constant.
Some features of the global entanglement of a composed quantum system can be quantified in terms of the purity of a balanced bipartition, made up of half of its subsystems. For the given bipartition, purity can always be minimized by taking a suitable (pure) state. When many bipartitions are considered, the requirement that purity be minimal for all bipartitions can engender conflicts and frustration arises. This unearths an interesting link between frustration and multipartite entanglement, defined as the average purity over all (balanced) bipartitions.
It is well-known that pure quantum states are typically almost maximally entangled, and thus have close to maximally mixed subsystems. We consider whether this is true for probabilistic theories more generally, and not just for quantum theory. We derive a formula for the expected purity of a subsystem in any probabilistic theory for which this quantity is well-defined. It applies to typical entanglement in pure quantum states, coin tossing in classical probability theory, and randomization in post-quantum theories; a simple generalization yields the typical entanglement in (anti)symmetric quan…
A perturbative treatment of reduced density operators of quantum subsystems is implemented in the same spirit as Fermi Golden Rule for scattering. Analytic expressions for linear entropy (a measure of purity loss, and in some cases of coherence loss) and for subsystem's energy variations (dissipation) are given. Application to electromagnetic field superposition states in a dissipative cavity is performed. Evaluation of typical field and reservoir time scales ($τ_{F}$ and $τ_{R}$) show that even for small temperatures they are very different. We also indicate the condition on the cutoff temper…
Minimal Subsystems for Characterizing Genuine Multipartite Entanglement
Department of Physics, Indian Institute of Technology Bombay（印度理工学院孟买分校物理系）；Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学联网量子设备单元）；Center of Excellence in Quantum Information, Computation, Science and Technology, Indian Institute of Technology Bombay（印度理工学院孟买分校量子信息、计算科学与技术卓越中心）
We solve the dynamics of an open quantum system where N strongly driven two-level atoms are equally coupled on resonance to a dissipative cavity mode. Analytical results are derived on decoherence, entanglement, purity, atomic correlations and cavity field mean photon number. We predict decoherence-free subspaces for the whole system and the N-qubit subsystem, the monitoring of quantum coherence and purity decay by atomic populations measurements, the conditional generation of atomic multi-partite entangled states and of cavity cat-like states. We show that the dynamics of atoms prepared in st…
We identify subsystem fluctuations (variances) that measure entanglement in an arbitrary bipartite pure state. These fluctuations are of observables that generalize the notion of polarization to an arbitrary N-level subsystem. We express this polarization fluctuation in terms of subsystem purity and other entanglement measures. The derived fluctuation-entanglement relation is evaluated for the ground states of a one-dimensional free fermion gas and the Affleck-Kennedy-Lieb-Tasaki spin chain. Our results provide a framework for experimentally measuring entanglement using Stern-Gerlach-type stat…
We study the short-time evolution of the bipartite entanglement in quantum lattice systems with local interactions in terms of the purity of the reduced density matrix. A lower bound for the purity is derived in terms of the eigenvalue spread of the interaction Hamiltonian between the partitions. Starting from an initially separable state the purity decreases as $1 - (t/τ)^2$, i.e. quadratically in time, with a characteristic time scale $τ$ that is inversly proportional to the boundary size of the subsystem, i.e., as an area-law. For larger times an exponential lower bound is derived correspon…
A Breakdown Case Study of the Lindblad Approach via Entanglement and Purity
Università di Salerno, and INFN — Gruppo Collegato di Salerno（萨莱诺大学，意大利国家核物理研究所萨莱诺协作组）；University of Camerino（卡梅里诺大学）；Institut Ruđer Bošković（鲁德·博什科维奇研究所）
The entanglement and localization in eigenstates of strongly chaotic subsystems are studied as a function of their interaction strength. Excellent measures for this purpose are the von-Neumann entropy, Havrda-Charv{\' a}t-Tsallis entropies, and the averaged inverse participation ratio. All the entropies are shown to follow a remarkably simple exponential form, which describes a universal and rapid transition to nearly maximal entanglement for increasing interaction strength. An unexpectedly exact relationship between the subsystem averaged inverse participation ratio and purity is derived that…
In this thesis we study the behavior of bipartite entanglement of a large quantum system, by analyzing the distribution of the Schmidt coefficients of the reduced density matrix. Applying the general methods of classical statistical mechanics, we develop a canonical approach for the study of the distribution of these coefficients for a fixed value of the average entanglement. We introduce a partition function depending on a fictitious temperature, which localizes the measure on the set of states with higher and lower entanglement, if compared to typical (random) states with respect to the Haar…
We study entanglement via the subsystem purity relative to bipartitions of arbitrary excited states in (1+1)-dimensional conformal field theory, equivalent to the scaling limit of one dimensional quantum critical systems. We compute the exact subpurity as a function of the relative subsystem size for numerous excited states in the Ising and three-state Potts models. We find that it decays exponentially when the system and the subsystem sizes are comparable until a saturation limit is reached near half-partitioning, signaling that excited states are maximally entangled. The exponential behavior…
Competition between unitary dynamics that scrambles quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we study this phenomenon in an analytically tractable all-to-all Brownian hybrid circuit model composed of qubits. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables…
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal ({\em i.e.}referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states …
We present a framework, compliant with the general canonical principle of statistical mechanics, to define measures on the set of pure Gaussian states of continuous variable systems. Within such a framework, we define two specific measures, referred to as `micro-canonical' and `canonical', and apply them to study systematically the statistical properties of the bipartite entanglement of n-mode pure Gaussian states (as quantified by the entropy of a subsystem). We rigorously prove the "concentration of measure" around a finite average, occurring for the entanglement in the thermodynamical limit…
Let a pure state ψbe chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix ρof an N-dimensional subsystem. The bipartite entanglement properties of ψare encoded in the spectrum of ρ. By means of a saddle point method and using a "Coulomb gas" model for the eigenvalues, we obtain the typical spectrum of reduced density matrices. We consider the cases of an unbiased ensemble of pure states and of a fixed value of the purity. We finally obtain the eigenvalue distribution by using a statistical mechanics approach based on the introduction of a partition functi…
Characterizing and quantifying quantum correlations in states of many-particle systems is at the core of a full understanding of phase transitions in matter. In this work, we continue our investigation of the notion of generalized entanglement [Barnum et al. Phys. Rev. A 68, 032308 (2003)] by focusing on a simple Lie-algebraic measure of purity of a quantum state relative to an observable set. For the algebra of local observables on multi-qubit systems, the resulting local purity measure is equivalent to a recently introduced global entanglement measure [Meyer and Wallach, J. Math. Phys. 43, 4…
Entanglement engineering in magnomechanical system via cross-Kerr interaction and mechanical parametric amplification
University of Yaounde I（雅温得第一大学）；University of Ngaoundere（恩冈德雷大学）；Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University（斯泰伦博斯大学瓦伦伯格研究中心高级研究院）；Umm Al-Qura University（乌姆古拉大学）；Princess Nourah bint Abdulrahman University（公主努拉·宾特·阿卜杜勒·拉赫曼大学）；University of Bisha（比沙大学）
We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary $N \leq M$, a general relation between the $n$-point densities and the cross-moments of the eigenvalues of the reduced density matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level de…
