Trace Distance Research Articles

Quantum coherence measure based on affinity

Quantification of quantum coherence is a fundamental and key issue in the field of resource theory and quantum information processing. Exploiting an affinity-based metric, in this article, we define a faithful measure of quantum coherence. It is shown that the affinity-based coherence is bounded by that based on fidelity and trace distance. The measure of quantumness in terms of the difference between bipartite coherence and corresponding product state coherence is also identified. Finally, we interpret the operational meaning of the affinity-based coherence as a lower bound of interferometric power of the quantum state.

Enhancement of the Quantum Coherence of the Rydberg Atoms System by Weak Measurement

AbstractThe influence of weak measurement along with quantum measurement reversal on the Rydberg atoms system is investigated, and is found to have an important practical significance as both weak measurement along with quantum measurement reversal and Rydberg atoms system can be experimentally realized. The quantum coherence and the entropic uncertainty relation of the system are explored, and it is found that the quantum coherence of the system is enhanced and the values of uncertainty and the uncertainty bound are reduced under some particular weak measurements along with quantum measurement reversals. This indicates that the decoherence effect of the system can be weakened by weak measurement along with quantum measurement reversal. Furthermore, the trace distance which is a reflection of the entanglement is also examined, and it is shown that weak measurement along with quantum measurement reversal can improve the entanglement of the system.

Intrinsic decoherence effects on measurement-induced nonlocality

By considering an exactly solvable model of a two interacting spin-1/2 qubits described by the Heisenberg anisotropic interaction in the presence of intrinsic decoherence, we study the dynamics of entanglement quantified by the concurrence and measurement-induced nonlocality (MIN) based on Hilbert-Schmidt norm and trace distance with different initial conditions. We highlight the relationship between the entanglement and MIN for the pure initial state. For an initial separable state, it is found that the robustness and the generation of the quantum correlations depend on the physical parameters. While considering the entangled state as an initial state, the results show that despite the phase decoherence all the correlations reach their steady state values after exhibiting some oscillations. Finally, we show that the existence of quantum correlation captured by MIN in the unentangled state.

Application of image processing methods for the characterization of selected features and wear analysis in surface topography measurements

In this paper, the different methods of processing of surface topography image (STI) were proposed for the characterization of selected features from the surface texture. The proposed analysis was performed to improve the functionality of the image assessment in the characterization of material contact properties (e.g. wear-resistant, lubricant retention, sealing or friction) according to the results of surface topography measurement. Various type of surfaces from manufactured details was considered, e.g. turned, ground, honed, laser-textured or isotropic. They were measured by the white light interferometer. Results were analyzed with areal surface topography maps (image) characterization. It was found that wavelet filtering approaches might be valuable in assessments of the selected surface texture features, such as dimple (size or distributions) or treatment trace distances, received by an application of STI processing methods. Additionally, proposed in the paper, the image digital extension method might be crucial in the definition of feature size (depth, width), especially when they are located on the edge of analyzed (measured) detail. Moreover, STI might be advantageous in the reduction of some surface texture measurement errors (a selected type of noise).

Modulus of continuity of the quantum f-entropy with respect to the trace distance

A well-known result due to Fannes is a certain upper bound on the modulus of continuity of the von Neumann entropy with respect to the trace distance between density matrices; this distance is the maximum probability of distinguishing between the corresponding quantum states. Much more recently, Audenaert obtained an exact expression of this modulus of continuity. In the present note, Audenaert's result is extended to a broad class of entropy functions indexed by arbitrary continuous convex functions $f$ in place of the Shannon--von Neumann function $x\mapsto x\log_2x$. The proof is based on the Schur majorization.

Quantum Coherence and Transfer of Quantum Information with a Kerr Medium Under Decoherence

We investigate the dynamics of quantum coherence of a system consisting of two interacting atoms coupling with a common cavity which is filled with a nonlinear Kerr-like medium, in the presence of intrinsic decoherence. It is shown that the stationary quantum coherence can arise in the system as the time approach to infinite. In particular, the amount of stationary quantum coherence can be enhanced by adjusting the Kerr interaction and dipole-dipole interaction. Furthermore, we also explore the influence of Kerr interaction and dipole-dipole interaction on the transfer of quantum information by a witness of the trace distance and find that the transferred information and stationary quantum information can be controlled by the Kerr interaction and dipole-dipole interaction.

Euler-Milstein and relevant approaches for high-precision stochastic simulation of quantum trajectories

For quantum diffusive measurements, the system’s dynamics can be described by the Itô stochastic master equation, which works well for an infinitesimal time resolution. However, in practical quantum experiments, one cannot make a time step to be infinitesimal, as it can introduce correlation of noise in time, making the Markovian assumption invalid. On the other hand, increasing a time step can cause errors from non-commuting operations describing the system’s dynamics. We therefore consider implementing the Euler-Milstein and relevant approaches, namely the Itô map, the high-order completely positive map, and the quantum Bayesian, to simulate quajntum trajectory for a quantum system under diffusive continuous measurements. In particular, we numerically simulate trajectories for a qubit measurement in z basis. We show the comparison of individual trajectories and their averaged trajectories among these approaches. We find that the high-order completely positive map approach yields the most accurate averaged quantum trajectory. Furthermore, we also investigate the trace distance from true stochastic quantum trajectories, comparing the four approaches using the numerical simulation. We show that, for a realistic time resolution (as in a superconducting qubit, experiment), the high-order map does give the most accurate estimate of the qubit trajectories.

Quantum-channel capacity of distributing orbital-angular-momentum states for underwater optical quantum communication.

We employ non-diffractive Bessel-Gaussian beams to investigate the effect of oceanic turbulence on quantum communication protocols via behaviors of quantum-channel capacity and trace distance, based on the analytical expression of the phase structure function of an orbital-angular-momentum (OAM) beam in underwater wireless optical communication. Our results show that turbulence conditions with a larger inner-scale and outer-scale factors, higher dissipation rate of kinetic energy, lower dissipation rate of the mean-squared temperature, and smaller temperature-salinity contribution ratio are beneficial to quantum communication performance. Moreover, we show that the distribution protocol may be improved by distributing quantum superposition states instead of OAM eigenstates. We believe our work provides the first theoretical exploration of quantum-channel capacity in underwater OAM quantum communication.

Optimal extensions of resource measures and their applications

We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e. an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a new technique to extend pure state entanglement measures to mixed bipartite states.

Dynamics of trace distance and Bures correlations in a three-qubit XY chain: Intrinsic noise model

In this paper, we analyze the robustness of the quantum correlations of the nearest neighbor and the next-to-neighbor qubits in the intrinsic noise model, that describes the dynamics of the decoherence for a system formed by 3-qubit Heisenberg XY chain. The effects of the uniformity and the inhomogeneity of the applied magnetic field are considered. The generated mixedness entropy, the robustness of the trace distance discord, measurement-induced non-locality and Bures distance correlations are examined against the spin interaction coupling, the uniformity and the inhomogeneity of the applied external magnetic field. In the absence of the intrinsic noise, the inhomogeneity of the applied magnetic field enhances the quantum correlations and the generated mixedness entropy. In the presence of the intrinsic noise for the nearest neighbor qubits, inhomogeneity effect of the magnetic field behaves as an additional decoherence resource. • Quantum correlations of the nearest neighbor and the next-to-neighbor qubits in the intrinsic noise model. • The effects of the uniformity and the inhomogeneity of the applied magnetic field are considered. • In the absence of the intrinsic noise, the inhomogeneity of the applied magnetic field enhances the quantum correlations. • The intrinsic noise for the nearest neighbor qubits, inhomogeneity of the magnetic field behaves as an additional damping.

Effects of Oceanic Turbulence on Orbital Angular Momenta of Optical Communications

The propagation properties of Laguerre-Gaussian beams in oceanic turbulence are investigated for both single-photon and biphoton cases. For single-photon communication, the channel capacity and trace distance are employed, both of which effectively reveal the communication performance via different viewpoints. For the biphoton case, we consider distributions of quantum resources including entanglement and quantum coherence. Turbulence conditions with a larger inner-scale and anisotropic factors, higher dissipation rate of kinetic energy, lower dissipation rate of the mean-squared temperature, and lower temperature-salinity contribution ratio combined with longer wavelength and an appropriate range of optimal beam width are beneficial to communication performances. Our results provide theoretical significance to improve the orbital-angular-momentum communication via oceanic turbulence.

Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels

In the first part of this thesis, we present a general technique for establishing local and uniform continuity bounds for Schur concave functions. Our technique uses a particular relationship between and the trace distance between quantum states. Namely, the pre-order attains a minimum over $\epsilon$-balls in this distance. By tracing the path of the majorization-minimizer as a function of the distance $\epsilon$, we obtain the path of majorization flow. This yields a new proof of the Audenaert-Fannes continuity bound for the von Neumann entropy in a universal framework which extends to the other functions, including the $\alpha$-Renyi entropy, for which we obtain novel bounds in the case $\alpha > 1$. We apply this technique to other Schur concave functions, such as the number of connected components of a certain random graph model, and the number of distinct realizations of a random variable. In the second part, we consider repeated interaction systems, in which a system of interest interacts with a sequence of probes one at a time. We characterize which repeated interaction systems break any initially-present entanglement between the system and an untouched reference after finitely many steps. Additionally, when the probes and their interactions with the system are slowly-varying (i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality between the entropy change of the system and the energy change of the probes, in the limit in which the number of steps tends to infinity and both the difference between consecutive probes and the difference between their interactions vanishes. This analysis proceeds at a fine-grained level by means of a two-time measurement protocol, in which the energy of the probes is measured before and after each interaction.

Hybrid quantum network design against unauthorized secret-key generation, and its memory cost

A significant number of servers that constitute the Internet are to provide private data via private communication channels to mutually anonymous registered users. Such are the servers of banks, hospitals that provide cloud storage and many others. Replacing communication channels by maximally entangled states is a promising idea for the quantum-secured Internet (QI). While it is an important idea for large distances secure communication, for the case of the mentioned class of servers pure entanglement based solution is not only unnecessary but also opens a threat. A crack stimulating a node to generate secure connections via entanglement swapping between two hackers can cause uncontrolled consumption of resources. Turning into positive a recently proven no-go result by S. B\"auml et al. [15], we propose a natural countermeasure against this threat. The solution bases on connections between hub-nodes and end-users realized with states that contain secure key but do not allow for swapping of this key.We then focus on the study of the quantum memory cost of such a scheme and prove a fundamental lower bound on its memory overhead. In particular, we show that to avoid the possibility of entanglement swapping, it is necessary to store at least twice as much memory than it is the case in standard quantum-repeater-based network design. For schemes employing either states with positive partial transposition that approximates certain privates states or private states hardly distinguishable from their attacked versions, we derive much tighter lower bounds on required memory. Our considerations yield upper bounds on a two-way repeater rate for states with positive partial transposition (PPT), which approximates strictly irreducible private states. As a byproduct, we provide a lower bound on the trace distance between PPT and private states, shown previously only for private bits.

Rényi entropy and subsystem distances in finite size and thermal states in critical XY chains

We study the Rényi entropy and subsystem distances on one interval for the finite size and thermal states in the critical XY chains, focusing on the critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, Rényi entropy, subsystem trace distance, Schatten two-distance, and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, the two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we compare the spin chain numerical results with the analytical results in conformal field theories and find perfect matches in the continuum limit.

Quantum phase transitions and scaling behaviors of extended 1D compass spin-chain model

We use multipartite entanglement and trace distance to detect the quantum phase transitions of the extended one-dimensional compass spin-chain model by applying the density matrix renormalization group method which is represented by the matrix product state. It is shown that singular behaviors of the first-order derivative of the multipartite entanglement and trace distance occur at the critical point of the system. The scaling behaviors of trace distance and multipartite entanglement are also discussed, and we show that the universal finite-size scaling law is valid for the multipartite entanglement and trace distance around the critical point. Moreover, we explore the quantum coherence for this model and find that the first-order derivative of the quantum coherence also displays discontinuity and exhibits singular critical behaviors that are the same as the trace distance and multipartite entanglement.

Steady-state quantum transport through an anharmonic oscillator strongly coupled to two heat reservoirs

We investigate the transport properties of an anharmonic oscillator, modeled by a single-site Bose-Hubbard model, coupled to two different thermal baths using the numerically exact thermofield based chain-mapping matrix product states (TCMPS) approach. We compare the effectiveness of TCMPS to probe the nonequilibrium dynamics of strongly interacting system irrespective of the system-bath coupling against the global master equation approach in Gorini-Kossakowski-Sudarshan-Lindblad form. We discuss the effect of on-site interactions, temperature bias as well as the system-bath couplings on the steady-state transport properties. Last, we also show evidence of non-Markovian dynamics by studying the nonmonotonicity of the time evolution of the trace distance between two different initial states.

Multi-model inference analysis of toxicological responses and levels of heavy metals in soft tissue of land snail Cornu aspersum caged in proximity to an industrial setting

The use of indicator species has become a standard monitoring approach in environmental risk assessment; however, pollutant bioavailability and its geographical distribution as well as species-sensitiveness could be affected by different factors. The present study investigated the potential use of the land snail Cornu aspersum as bioindicator of industrial air pollution by using cytological and biochemical responses and trace element levels in individuals caged in proximity to an industrial setting. Eleven sites were selected based on wind direction and distance. Snails were placed in holed plastic cages, transplanted to each site and maintained for 30 days fed at libitum and under constant humidity. Several oxidative stress responses including catalase (CAT) and glutathione-s-transferase (GST) activities, levels of malondialdehyde (MDA) and total metallothioneins (MTs) content were determined in snail hepatopancreas while lysosomal membrane stability (LMS) and loss of DNA integrity through micronuclei frequency (MN) were assessed in haemocytes. Trace elements were also analysed in snail whole soft tissues. Multi-model inference used to predict snail’s biological responses in relation to trace elements levels and distance from the industrial setting was able to disentangle the relative contributions of different influencing predictors and could be successfully applied in environmental risk assessment by using land snail as bioindicator species.

Black hole interiors via spin models

To model the interior of a black hole, a study is made of a spin system with long-range random four-spin couplings that exhibits quantum chaos. The black hole limit corresponds to a system where the microstates are approximately degenerate and equally likely, corresponding to the high temperature limit of the spin system. At the leading level of approximation, reconstruction of bulk physics implies that local probes of the black hole should exhibit free propagation and unitary local evolution. We test the conjecture that a particular mean field Hamiltonian provides such a local bulk Hamiltonian by numerically solving the exact Schrodinger equation and comparing the time evolution to the approximate mean field time values. We find excellent agreement between the two time evolutions for timescales smaller than the scrambling time. In earlier work, it was shown bulk evolution along comparable timeslices is spoiled by the presence of the curvature singularity, thus the matching found in the present work provides evidence of the success of this approach to interior holography. The numerical solutions also provide a useful testing ground for various measures of quantum chaos and global scrambling. A number of different observables, such as entanglement entropy, out-of-time-order correlators, and trace distance are used to study these effects. This leads to a suitable definition of scrambling time, and evidence is presented showing a logarithmic variation with the system size.

Guessing probability in quantum key distribution

On the basis of the existing trace distance result, we present a simple and efficient method to tighten the upper bound of the guessing probability. The guessing probability of the final key k can be upper bounded by the guessing probability of another key {bf{k}}^{prime}, if {bf{k}}^{prime} can be mapped from the final key k. Compared with the known methods, our result is more tightened by thousands of orders of magnitude. For example, given a 10−9-secure key from the sifted key, the upper bound of the guessing probability obtained using our method is 2 × 10−3277. This value is smaller than the existing result 10−9 by more than 3000 orders of magnitude. Our result shows that from the perspective of guessing probability, the performance of the existing trace distance security is actually much better than what was assumed in the past.

Measurement-induced geometric measures of correlations based on the trace distance for two-qubit X states

We apply the modified Brodutch and Modi method of constructing geometric measures of correlations to obtain analytical expressions for measurement-induced geometric classical and quantum correlations based on the trace distance for two-qubit X states. Moreover, we study continuity of the classical and quantum correlations for these states. In particular, we show that these correlations may not be continuous.