Qutrit Systems Research Articles

A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems.

The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposal for quasiunitary transforms of qubits, in view of the unitary transform of the total qutrit system. The experimental realization is also addressed. The probability representation of the evolution and its information-entropic characteristics are considered.

Landau-Majorana-Stückelberg-Zener dynamics driven by coupling for two interacting qutrit systems

A time-dependent two interacting spin-qutrit model is analysed and solved. The two interacting qutrits are subjected to a longitudinal field linearly varying over time as in the Landau-Majorana-St\"uckelberg-Zener (LMSZ) scenario. Although a transverse field is absent, we show the occurrence of LMSZ transitions assisted by the coupling between the two spin-qutrits. Such a physical effects permits to estimate experimentally the coupling strength between the spins and allows the generation of entangled states of the two qutrits by appropriately setting the slope of the ramp. Furthermore, the possibility of local and non-local control as well as the existence of dark states of the two qutrits have been brought to light. Effects stemming from a noisy surrounding environment are also taken into account by introducing a random fluctuating field component as well as non-Hermitian terms in the Hamiltonian model.

Suppressing the information losses of accelerated qubit–qutrit system

Protecting the accelerated information coherence and suppressing of its decay due to the acceleration is achieved by using different noisy channels. The behavior of three different types of information; Fisher, nonlocal and local information of qubit–qutrit system is discussed. It is shown that, the estimation rate of the acceleration parameter is maximized if both subsystems are accelerated and interacted locally with their noise channels. However, if the largest subsystem is accelerated, then the estimation rate is minimized. The rate of change of the nonlocal information depends on which subsystem is accelerated. The results display that the local information is almost unaltered during the acceleration process, that is, the information is frozen. We show that the phase-flip channel represents the weak channel, where its suppressing of the decay due to the acceleration is smaller than the dephasing and depolarize of channels.

Qubit representation of qudit states: correlations and state reconstruction

A method to establish a qubit decomposition of a general qudit state is presented. This new representation allows a geometrical depiction of any qudit state in the Bloch sphere. Additionally, we show that the nonnegativity conditions of the qudit state imply the existence of quantum correlations between the qubits which compose it. These correlations are used to define new inequalities which the density matrices components must satisfy. The importance of such inequalities in the reconstruction of a qudit state is addressed. As an example of the general procedure the qubit decomposition of a qutrit system is shown, which allows a classification of the qutrit states by fixing their invariants ${\rm Tr}(\hat{\rho}^2)$, ${\rm Tr}(\hat{\rho}^3)$.

Protecting high-dimensional quantum entanglement from the amplitude-phase decoherence sources by weak measurement and reversal

Entangled states with high-dimensional systems have been investigated widely because of its various advantages. This paper investigates the effect of the weak measurement and reversal (WMR) on the protection of the entanglement for an arbitrarily entangled qutrit–qutrit state from the amplitude-phase decoherence sources. By solving the master equation of the qutrit–qutrit system, we numerically get the maximal amount of entanglement. The numerical results show explicitly that the WMR operation indeed helps for protecting entanglement from amplitude-phase decoherence. In particular, WMR can protect entanglement more efficiently in the case of strong decay rates than those with small decay rates. These results shed new light on the protection of high-dimensional quantum entanglement in a complex quantum environment.

Nonasymptotic assisted distillation of quantum coherence

We characterize the operational task of environment-assisted distillation of quantum coherence under different sets of free operations when only a finite supply of copies of a given state is available. We first evaluate the one-shot assisted distillable coherence exactly, and introduce a semidefinite programming bound on it in terms of a smooth entropic quantity. We prove the bound to be tight for all systems in dimensions 2 and 3, which allows us to obtain computable expressions for the one-shot rate of distillation, establish an analytical expression for the best achievable fidelity of assisted distillation for any finite number of copies, and fully solve the problem of asymptotic zero-error assisted distillation for qubit and qutrit systems. Our characterization shows that all relevant sets of free operations in the resource theory of coherence have exactly the same power in the task of one-shot assisted coherence distillation, and furthermore resolves a conjecture regarding the additivity of coherence of assistance in dimension 3.

Violation of the Lüders bound of macrorealist and noncontextual inequalities

In a recent Letter [PRL, 113, 050401 (2014)], it is shown that the quantum violation of a three-time Leggett-Garg inequality (LGI) for a dichotomic qutrit system can exceed the L$\ddot{u}$ders bound. This is obtained by using a degeneracy breaking projective measurement rule which the authors termed as von Neumann rule. Such violation can even approach the algebraic maximum in the asymptotic limit of system size. In this paper, we question the implication of such violation of L$\ddot{u}$ders bound and its conceptual relevance in LG scenario. We note an important fact that the basis for implementing the proposed von Neumann rule for a degenerate observable is non-unique and show that the violation of L$\ddot{u}$ders bound is crucially dependent on the choice of basis. Further, we demonstrate the violation of L$\ddot{u}$ders bound of the simplest non-contextual inequality (NCI) which is in contrast to the reasoning provided in the aforementioned Letter. This result further raises the doubts regarding the validity of the proposed rule as a viable projective measurement. We discuss the relevance of such results with respect to the usual quantum violation of LGI and NCI.

Improved quantum entropic uncertainty relations

We study entropic uncertainty relations by using stepwise linear functions and quadratic functions. Two kinds of improved uncertainty lower bounds are constructed: the state-independent one based on the lower bound of Shannon entropy and the tighter state-dependent one based on the majorization techniques. The analytical results for qubit and qutrit systems with two or three measurement settings are explicitly derived, with detailed examples showing that they outperform the existing bounds. The case with the presence of quantum memory is also investigated.

Quantum filtering of accelerated qubit–qutrit system

The possibility of immunizing and improving the entanglement of accelerated systems via local filtering is discussed. The maximum bounds of entanglement depend on the dimensions of the accelerated and the filtered subsystems. If the small dimensional subsystem is accelerated and the large dimensional subsystem is filtered, one can get a long-lived entanglement. Moreover, if the larger subsystem is accelerated, then by filtering any subsystem, the upper bounds of entanglement of the filtered state are larger than that depicted for then non-filtered states. For any accelerated subsystem, the entanglement always increases as the filter strength of the large dimensional subsystem increases.

Coherence makes quantum systems ‘magical’

Two primary facets of quantum technological advancement that hold great promise are quantum communication and quantum computation. For quantum communication, the canonical resource is entanglement. For quantum gate implementation, the resource is ‘magic’ in an auxiliary system. It has already been shown that quantum coherence is the fundamental resource for the creation of entanglement. We argue on the similar spirit that quantum coherence is the fundamental resource when it comes to the creation of ‘magic’. This unifies the two strands of modern development in quantum technology under the common underpinning of existence of quantum superposition, quantified by the coherence in quantum theory. We also obtain a coherence monotone from the discrete Wigner distribution. We further study the interplay between quantum coherence and ‘magic’ in a qutrit system and that between quantum entanglement and ‘magic’ in a qutrit-qubit setting.

Exploration of the entropic uncertainty relation for a qutrit system under decoherence

In this paper, we have performed a detailed investigation of the dynamics of the entropic uncertainty relation for a qutrit with and without a memory system under an amplitude damping channel at finite temperature. We focus mainly on how the temperature affects the entropic uncertainty relation. The results show that the dynamics of the entropic uncertainty exhibits a non-monotonic phenomenon. As r increases (r is usually a function of temperature), the amount of the entropic uncertainty will first increase and then subsequently decrease. Moreover, based on measurement reversal from a weak measurement, we have also investigated the reduction of the entropic uncertainty for a qutrit system under decoherence. It has been found that the measurement reversal technique can always reduce the entropic uncertainty effectively at finite temperature, while the prior weak measurement technique is valid only at a lower temperature but fails at a higher channel temperature. Finally, a phenomenological explanation of our results is also presented.

Free and bound entanglement dynamics in qutrit systems under Markov and non-Markov classical noise

We investigate in detail the dynamics of decoherence, free and bound entanglements, and the conversion from one to another (quantum state transitions), in a two non-interacting qutrits system initially entangled and subject to independents or a common classical noise. Both Markovian and non-Markovian environments are considered. Furthermore, isotropic and bound entangled states for qutrits systems are considered as initial states. We show the efficiency of the formers over the latters against decoherence, and in preserving quantum entanglement. The loss of coherence increases monotonically with time up to a saturation value depending upon the initial state parameter and is stronger in a collective Markov environment. For the non-Markov regime the presence or absence of entanglement revival and entanglement sudden death phenomena is deduced depending on both the peculiar characteristics of the noise, the physical setup and the initial state of the system. We demonstrate distillability sudden death for conveniently selected parameters in bound entangled states; meanwhile, it is completely absent for isotropic states, where entanglement sudden death is avoided for dynamic noise independently of the noise regime and the physical setup. Our results indicate that distillability sudden death under the Markov/non-Markov noise considered can be avoided depending upon the physical setup.

Demonstration of quantum permutation parity determine algorithm in a superconducting qutrit**Project supported by the National Key Basic Research and Development Program of China (Grant No. 2016YFA0301802) and the National Natural Science Foundation of China (Grant Nos. 11504165, 11474152, and 61521001).

A quantum algorithm provides a new way in solving certain computing problems and usually faster than classical algorithms. Here we report an implementation of a quantum algorithm to determine the parity of permutation in a single three-dimensional (3D) superconducting transmon qutrit system. The experiment shows the capacity to speed up in a qutrit, which can also be extended to a multi-level system for solving high-dimensional permutation parity determination problem.

Disentanglement and quantum states transitions dynamics in spin-qutrit systems: dephasing random telegraph noise and the relevance of the initial state

Using negativity and realignment criterion as quantifiers of free and bound entanglements respectively, we present in details the analytical study of the entanglements and quantum states transitions dynamics in a two-qutrit system driven by dephasing random telegraph noise channel(s). Both collective and independent system–environment couplings as well as the Markovian and the non-Markovian regimes of the noise channel(s) are considered. Two non-equivalent initial states and their locally equivalent through a local unitary operation (LUO) are also considered. We demonstrate a stronger entanglement under independent Markovian environments than with a collective one; meanwhile, for the non-Markovian regime, entanglement is stronger under a collective environment than with independent ones. States transitions as well as the (re)activation of bound entanglement (for initially free entangled states) can be found for a specific class of initial states, but can, however, be avoided by means of a LUO on the initial state. While unavoidable disentanglement occurs for independents coupling, we demonstrate the possibility of indefinite free entanglement survival in the qutrit system under a common environment by converting the initial entangled state using the local unitary operation.

Quantum correlations and coherence dynamics in qutrit–qutrit systems under mixed classical environmental noises

We investigate the dynamics of entanglement, quantum discord (QD) and state coherence in a bipartite and noninteracting spin-qutrits system under mixed classical noises. Specifically, the collective effects of static noise (SN) and random telegraphic noise (RTN) each being coupled with a marginal system, are analyzed. While the static noise models a non-Markovian environment, the dynamic noise can model both a Markovian or a non-Markovian environment, and both dynamics are studied. We show that quantum correlations and coherence may survive the noise degrading effects at sufficiently long time when the Markovian regime of the RTN is considered. Meanwhile, the opposite is found in the non-Markovian regime, wherein the nonmonotonic dynamics of quantum features avoid sudden death phenomena. However, the static noise is more fatal to the survival of quantum correlations and quantum state coherence as compared to the RTN.

Bipartite qutrit local realist inequalities and the robustness of their quantum mechanical violation

Distinct from the type of local realist inequality (known as the Collins–Gisin–Linden–Massar–Popescu or CGLMP inequality) usually used for bipartite qutrit systems, we formulate a new set of local realist inequalities for bipartite qutrits by generalizing Wigner's argument that was originally formulated for the bipartite qubit singlet state. This treatment assumes existence of the overall joint probability distributions in the underlying stochastic hidden variable space for the measurement outcomes pertaining to the relevant trichotomic observables, satisfying the locality condition and yielding the measurable marginal probabilities. Such generalized Wigner inequalities (GWI) do not reduce to Bell–CHSH type inequalities by clubbing any two outcomes, and are violated by quantum mechanics (QM) for both the bipartite qutrit isotropic and singlet states using trichotomic observables defined by six-port beam splitter as well as by the spin-1 component observables. The efficacy of GWI is then probed in these cases by comparing the QM violation of GWI with that obtained for the CGLMP inequality. This comparison is done by incorporating white noise in the singlet and isotropic qutrit states. It is found that for the six-port beam splitter observables, QM violation of GWI is more robust than that of the CGLMP inequality for singlet qutrit states, while for isotropic qutrit states, QM violation of the CGLMP inequality is more robust. On the other hand, for the spin-1 component observables, QM violation of GWI is more robust for both the types of states considered.

Entanglement monogamy in three qutrit systems

By introducing an arbitrary-dimensional multipartite entanglement measure, which is defined in terms of the reduced density matrices corresponding to all possible two partitions of the entire system, we prove that multipartite entanglement cannot be freely shared among the parties in both n-qubit systems and three-qutrit systems. Furthermore, our result implies that the satisfaction of the entanglement monogamy is related to the number of particles in the quantum system. As an application of three-qutrit monogamy inequality, we give a condition for the separability of a class of two-qutrit mixed states in a 3 ⊗ 3 system.

Model dynamics for quantum computing

A model master equation suitable for quantum computing dynamics is presented. In an ideal quantum computer (QC), a system of qubits evolves in time unitarily and, by virtue of their entanglement, interfere quantum mechanically to solve otherwise intractable problems. In the real situation, a QC is subject to decoherence and attenuation effects due to interaction with an environment and with possible short-term random disturbances and gate deficiencies. The stability of a QC under such attacks is a key issue for the development of realistic devices. We assume that the influence of the environment can be incorporated by a master equation that includes unitary evolution with gates, supplemented by a Lindblad term. Lindblad operators of various types are explored; namely, steady, pulsed, gate friction, and measurement operators. In the master equation, we use the Lindblad term to describe short time intrusions by random Lindblad pulses. The phenomenological master equation is then extended to include a nonlinear Beretta term that describes the evolution of a closed system with increasing entropy. An external Bath environment is stipulated by a fixed temperature in two different ways. Here we explore the case of a simple one-qubit system in preparation for generalization to multi-qubit, qutrit and hybrid qubit–qutrit systems. This model master equation can be used to test the stability of memory and the efficacy of quantum gates. The properties of such hybrid master equations are explored, with emphasis on the role of thermal equilibrium and entropy constraints. Several significant properties of time-dependent qubit evolution are revealed by this simple study.

Measurement-induced qudit geometric discord

We study measurement-induced geometric discord based on the trace norm and generalize some properties known for qutrits to qudits. Previous preliminary results for bipartite qutrit systems (i.e. d = 3 systems) are here strictly proved for arbitrary d. The present study supports the observations, coming also from other approaches, that systems with show similar behaviour when quantum correlations are concerned, but there is pronounced difference between d = 2 and d = 3. Qubit systems are exceptionally simple. The underlying geometry of state spaces and related Lie groups is responsible for this.

Thermal entanglement and lower bound of the geometric discord for a two-qutrit system with linear coupling and nonuniform external magnetic field

The effects of temperature and linear coupling constant on the lower bound of the geometric discord and negativity of a qutrit–qutrit system in Heisenberg model with (and without) parallel and antiparallel external magnetic fields have been investigated. We show that the lower bound of the geometric discord and negativity are about zero for negative linear coupling constant in parallel magnetic fields, while they are nonzero in the finite antiparallel magnetic fields. For negative linear coupling constant, as temperature increases, both measures become zero faster than in the case of positive linear coupling constant in antiparallel magnetic fields.