Measure Of Mixedness Research Articles

Trade-offs between coherence and mixedness and their evolution under quantum noise channels

Generally quantum processes tend to mix the pure input states and degrade the original coherence. Such processes display exactly the trade-offs between the coherence and the mixedness. In this paper, we present trade-offs between the coherence measures (l2 norm of coherence, coherence via skew information, geometric measure of coherence) and the mixedness (normalized linear entropy mixedness, geometric measure of mixedness, mixedness via skew information). The evolution of these trade-offs are also investigated under four quantum noise channels: the bit flip, phase flip, depolarizing and amplitude damping channels.

Negativity vs. purity and entropy in witnessing entanglement

In this paper, we explore the value of measures of mixedness in witnessing entanglement. While all measures of mixedness may be used to witness entanglement, we show that all such entangled states must have a negative partial transpose (NPT). Where the experimental resources needed to determine this negativity scale poorly at high dimension, we compare different measures of mixedness over both Haar-uniform and uniform-purity ensembles of joint quantum states at varying dimension to gauge their relative success at witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly (albeit not exclusively) more successful at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses.

Complete Complementarity Relations in System–environment Decoherent Dynamics

We investigate the system-environment information flow from the point of view ofcomplete complementarity relations. We consider some commonly used noisy quantum channels:Amplitude damping, phase damping, bit flip, bit-phase flip, phase flip, depolarizing, and correlatedamplitude damping. By starting with an entangled bipartite pure quantum state, with the linearentropy being the quantifier of entanglement, we study how entanglement is redistributed and turnedinto general correlations between the degrees of freedom of the whole system. For instance, it ispossible to express the entanglement entropy in terms of the multipartite quantum coherence or interms of the correlated quantum coherence of the different partitions of the system. In addition,we notice that for the depolarizing and bit-phase flip channels the wave and particle aspects candecrease or increase together. Besides, by considering the environment as part of a pure quantumsystem, the linear entropy is shown to be not just a measure of mixedness of a particular subsystem,but a correlation measure of the subsystem with rest of the world.

Extracting a mixing parameter from 2D radiographic imaging of variable-density turbulent flow

We extract a suitably averaged fluctuating density from the two-dimensional radiographic image of a flow. The X-ray attenuation is given by the Beer–Lambert law which exponentially damps the incident beam intensity by a factor proportional to the density, opacity and thickness of the target. By making reasonable assumptions for the mean density, opacity and effective thickness of the target flow, we estimate the density fluctuation contribution to the attenuation. The extracted density fluctuations averaged across the thickness of the flow in the direction of the beam may be used to form the density–specific-volume correlation b. In a statistical description of variable-density turbulence, b quantifies the degree of mixedness. The ability to extract a measure of mixedness from experimental data would be a powerful tool that could be used in the validation of mix models. The scheme proposed is tested for DNS data computed for variable density buoyancy-driven mixing. We quantify the deficits in the extracted value of b due to target thickness, Atwood number and modeled signal noise. This analysis justifies using the proposed scheme to infer the mix parameter from thin targets at moderate to low Atwood numbers. To illustrate how the scheme might be used in a practical problem, we demonstrate its application to a radiographic image of counter-shear flow obtained from experiments at the National Ignition Facility.

Bounds on mixedness and entanglement of quantum teleportation resources

For a standard teleportation protocol, rank dependent upper bounds on Von Neumann entropy and linear entropy beyond which the states of respective ranks cease to be useful for quantum teleportation are derived analytically for a general d×d bipartite systems. For two qubit mixed states, we obtain rank dependent lower bound on concurrence below which the state is useless for quantum teleportation. For two qubit system, we construct mixed states of different ranks that exhibit theoretical rank dependent upper bounds on the measures of mixedness and lower bounds on the concurrence.

Frobenius-norm-based measures of quantum coherence and asymmetry.

We formulate the Frobenius-norm-based measures for quantum coherence and asymmetry respectively. In contrast to the resource theory of coherence and asymmetry, we construct a natural measure of quantum coherence inspired from optical coherence theory while the group theoretical approach is employed to quantify the asymmetry of quantum states. Besides their simple structures and explicit physical meanings, we observe that these quantities are intimately related to the purity (or linear entropy) of the corresponding quantum states. Remarkably, we demonstrate that the proposed coherence quantifier is not only a measure of mixedness, but also an intrinsic (basis-independent) quantification of quantum coherence contained in quantum states, which can also be viewed as a normalized version of Brukner-Zeilinger invariant information. In our context, the asymmetry of N-qubit quantum systems is considered under local independent and collective transformations. In- triguingly, it is illustrated that the collective effect has a significant impact on the asymmetry measure, and quantum correlation between subsystems plays a non-negligible role in this circumstance.

Mixedness and entanglement in the presence of localized closed timelike curves

We examine mixedness and entanglement of the chronology-respecting (CR) system with assuming that quantum mechanical closed timelike curves (CTCs) exist in nature and by introducing the qubit system and applying the general controlled operations between CR and CTC systems. We use the magnitude of Bloch vector as a measure of mixedness. While Deutschian-CTC (D-CTC) either preserves or decreases the magnitude, postselected-CTC (P-CTC) can increases it. Nonintuitively, even the completely mixed CR-qubit can be converted into a pure state after CTC-qubit travels around the P-CTC. It is also shown that while D-CTC cannot increase the entanglement of CR system, P-CTC can increase it. Surprisingly, any partially entangled state can be maximally entangled pure state if P-CTC exists. Thus, distillation of P-CTC-assisted entanglement can be easily achieved without preparing the multiple copies of the partially entangled state.

Geometric approach to the distribution of quantum states in bipartite physical systems

Any set of pure states living in an given Hilbert space possesses a natural and unique metric --the Haar measure-- on the group $U(N)$ of unitary matrices. However, there is no specific measure induced on the set of eigenvalues $\Delta$ of any density matrix $\rho$. Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on $\Delta$. In the present work we shall employ a simple measure on $\Delta$ that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio $R=1/Tr(\rho^2)$ and the concomitant maximum eigenvalue $\lambda_m$. The cases studied will be the qubit-qubit system and the qubit-qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the $R$-domain and the $\lambda_m$-domain.

An analysis of the entropy of mixing for granular materials

Many existing definitions of the entropy of mixing for granular materials involve a “total entropy” that is calculated from the “local entropies” of all cells in the domain of the mixture. The total entropy has been used as a measure of mixedness by many authors, but they have virtually never presented the values of the individual local entropies. For comparison purposes, we introduced a parallel definition of the entropy of mixing by considering an alternative “total entropy” that is based on the “per-species entropies” of all types of particles in the mixture. Using a simple mixing model in continuous variables, we showed that the contributions of the individual “local entropies” and the “per-species entropies” to their respective totals show different trends during a specific, idealised mixing process: while the total entropy constantly increases following either definition, only the local entropies show non-monotonic changes; on the other hand, only the per-species entropies reach the maximum possible value of the Shannon entropy at the steady state of mixing. We rationalised these differences by considering the changes in the probabilities associated with each individual entropic term, in the context of the properties of the Shannon entropy, and confirmed these ideas by studying the two-dimensional phase-space trajectories of the individual entropic terms for the mixing process considered.

Maximal mixing by incompressible fluid flows

We consider a model for mixing binary viscous fluids under an incompressible flow. We prove the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H−1 norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

Twist and teleportation analogy of the black hole final state

Mathematical connection between the quantum teleportation, the most unique feature of quantum information processing, and the black hole final state is studied taking into account the non trivial spacetime geometry. We use the twist operatation for the generalized entanglement measurement and the final state boundary conditions to obtain transfer theorems for the black hole evaporation. This would enable us to put together the universal quantum teleportation and the black hole evaporation in the unified mathematical footing. For a renormalized post selected final state of outgoing Hawking radiation, we found that the measure of mixedness is preserved only in the special case of final-state boundary condition in the micro-canonical form, which resmebles perfect teleportation channel.

The black hole final state for the Dirac fields in Schwarzschild spacetime

We show that the internal stationary state of a black hole for massless Dirac fields can be represented by an entangled state of collapsing matter and infalling Hawking radiation. This implies that the Horowitz-Maldacena conjecture for the black hole final state originally proposed for the massless scalar fields is also applicable to fermionic fields as well. For an initially mixed state we find that the measure of mixedness is expected to decrease under evaporation.

Decoherence of tripartite states—a trapped ion coupled to an optical cavity

We investigate the decoherence process of a three-qubit system obtained by manipulating the state of a trapped two-level ion coupled to an optical cavity. Interaction of the ion with a resonant laser and the cavity field tuned to red sideband of ionic vibrational motion generates tripartite entanglement of the internal state of the ion, vibrational state of ionic centre of mass motion and the cavity-field state. Non-dissipative decoherence occurs due to entanglement of the system with the environment, modelled as a set of non-interacting harmonic oscillators. Analytic expressions for the state operator of tripartite composite system, the probability of generating maximally entangled GHZ state and the population inversion have been obtained. Coupling to environment results in exponential decay of off-diagonal matrix elements of the state operator with time as well as a phase decoherence of the component states. Numerical calculations to examine the time evolution of GHZ state generation probability and population inversion for different values of system environment coupling strengths are performed. Using negativity as an entanglement measure and linear entropy as a measure of mixedness, the entanglement dynamics of the tripartite system in the presence of decoherence sources has been analysed. The maximum tripartite entanglement is found to decrease with increase in the strength of system–environment coupling. The negativity as well as the linear entropy as entanglement measures gives qualitatively similar results, uniquely identifying maximally entangled and separable states of the system. For large values of system–environment coupling strength, the mixed states of the composite system lying at the boundary of an entangled–separable region are reached. For these states, whereas the negativity measures only quantum correlations, the linear entropy measures classical correlations as well.

Dynamics of Tripartite Entanglement

Maximally entangled states are of utmost importance to quantum communication, dense coding, and quantum teleportation. With a trapped ion placed inside a high finesse optical cavity, interacting with field of an external laser and quantized cavity field, a scheme to generate a maximally entangled three qubit GHZ state, is proposed. The dynamics of tripartite entanglement is investigated, using negativity as an entanglement measure and linear entropy as a measure of mixedness of a state. It is found that (a) the number of modes available to the subsystem determines the maximum entanglement of a subsystem, (b) at entanglement maxima and minima, linear entropy and negativity uniquely determine the nature of state, but the two measures do not induce the same ordering of states, and (c) for a special choice of system parameters maximally entangled tripartite two mode GHZ state is generated. The scheme presented for GHZ state generation is a single step process and is reduction-free.

Maximal entanglement versus entropy for mixed quantum states

Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy.

On Kites, Comets, and Stars. Sums of Eigenvector Coefficients in (Molecular) Graphs

Two graph invariants were encountered that form the link between (molecular) walk counts and eigenvalues of graph adjacency matrices. In particular, the absolute value of the sum of coefficients of the first or principal (normalized) eigenvector, s1, and the analogous quantity sn, pertaining to the last eigenvector, appear in equations describing some limits (for infinitely long walks) of relative frequencies of several walk counts. Quantity s1 is interpreted as a measure of mixedness of a graph, and sn, which plays a role for bipartite graphs only, is interpreted as a measure of the imbalance of a bipartite graph. Consequently, sn is maximal for star graphs, while the minimal value of sn is zero. Mixedness s1 is maximal for regular graphs. Minimal values of s1 were found by exhaustive computer search within the sample of all simple connected undirected n-vertex graphs, n≤10: They are encountered among graphs called kites. Within the special sample of tree graphs (searched for n≤20) so-called double snakes have maximal s1, while the trees with minimal s1 are so-called comets. The behaviour of stars and double snakes can be described by exact equations, while approximate equations for s1 of kites and comets could be derived that are fully compatible with and allow to predict some pecularities of the results of the computer search. Finally, the discriminating power of s1, determined within trees and 4-trees (alkanes), was found to be high.

Quantification of Mixing and Mixing Rate from Experimental Observations

A new measure of mixedness, based on entropy considerations, and a related mixing rate are introduced. It is argued that the time rate of change of the mixedness is proportional to the mixing rate. The new measures are applied to experimental observations on an axisymmetric jet and an array of jets in crossflow. The concentration field of an axisymmetric jet is measured at 10 downstream locations by optical imaging of Rayleigh scattering from a laser sheet. Mixedness and mixing rate are calculated for each of the locations. In agreement with theory, the mixedness of the self-similar jet is constant along the length of the jet. The measures allow us to locate the instantaneous realization that is most typical in terms of mixedness and mixing rate. Also analyzed is the mixing of a row of jets injecting fluid into a crossflow. The jets are sufficiently close that there is significant interaction between neighboring jets. The mixedness and mixing rate were calculated. Well-mixed regions have a low mixing rate, whereas poorly mixed regions tend to mix most rapidly. The mixedness and mixing rate allow the assessment of the effectiveness of different orifice shapes in promoting rapid mixing