Triangle Geometry for Qutrit States in the Probability Representation

We express the matrix elements of the density matrix of the qutrit state in terms of probabilities associated with artificial qubit states. We show that the quantum statistics of qubit states and observables is formally equivalent to the statistics of classical systems with three random vector variables and three classical probability distributions obeying special constrains found in this study. The Bloch spheres geometry of qubit states is mapped onto triangle geometry of qubits. We investigate the triada of Malevich’s squares describing the qubit states in quantum suprematism picture and the inequalities for the areas of the squares for qutrit (spin-1 system). We expressed quantum channels for qutrit states in terms of a linear transform of the probabilities determining the qutrit-state density matrix.     

Generalized Qubit Portrait of the Qutrit-State Density Matrix

We obtain new inequalities for tomographic probability distributions and density matrices of qutrit states by generalization of the qubit-portrait method. We propose an approach based on the quditportrait method of obtaining new entropic inequalities. Our approach can be applied to the case of arbitrary nonnegative hermitian matrices, including the density matrices of multipartite qudit states.     

Contextuality and the probability representation of quantum states

We consider the notions of contextuality and noncontextuality within the framework of the probability representation of quantum states. We present an example of qutrit states and violation of the noncontextuality inequalities using the spin tomogram and tomographic symbols of the observables.     

Qubit portrait of qudit states and Bell inequalities

A linear map of qudit tomogram onto qubit tomogram (qubit portrait) is proposed as a characteristics of the qudit state. In view of the qubit-portrait method, the Bell inequalities for two qubits and two qutrits are discussed within the framework of the probability-representation of quantum mechanics. A semigroup of stochastic matrices is associated with tomographic-probability distributions of qubit and qutrit states. Bell-like inequalities are studied using the semigroup of stochastic matrices. The qudit-qubit map of tomographic probability distributions is discussed as an ansatz to provide a necessary condition for the separability of quantum states.     

Hidden Correlations and Entanglement in Single-Qudit States<Superscript>†</Superscript>

We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.     

Fidelity and entanglement breaking properties of qutrit channels

We investigate fidelity and entanglement breaking properties of quantum qutrit channels. We focus on channel fidelity evaluated for pure initial states and entanglement fidelity for purified mixed states (or pure entangled qutrit states) and use negativity as an entanglement measure for qutrits. We analyze properties of qutrit gates and channels based on affine transformations of qutrit Bloch vectors. We employ channel complete positivity constraints into the discussion of fidelity and entanglement behaviour.     

Control and tomography of a three level superconducting artificial atom

A number of superconducting qubits, such as the transmon or the phase qubit, have an energy level structure with small anharmonicity. This allows for convenient access of higher excited states with similar frequencies. However, special care has to be taken to avoid unwanted higher-level populations when using short control pulses. Here we demonstrate the preparation of arbitrary three level superposition states using optimal control techniques in a transmon. Performing dispersive readout, we extract the populations of all three levels of the qutrit and study the coherence of its excited states. Finally we demonstrate full quantum state tomography of the prepared qutrit states and evaluate the fidelities of a set of states, finding on average 95%.     

Simulation of a quantum NOT gate for a single qutrit system

A three-level system based an a three-level atom interacting with a detuned cavity is considered. Because of the fact that the three-level atom defines a total normalized state composed of superposition of three different single-level states, it is assumed that such a system implements a qutrit. In order to achieve a quantum NOT gate for a single qutrit, the respective Schrödinger equation is solved numerically within a two-photon rotating wave approximation. For small values of one-photon detuning, there appear decoherence effects. Meanwhile, for large values of one-photon detuning, an ideal quantum NOT gate for a single qutrit is achieved. An expression for the execution time of the quantum NOT gate for a single qutrit as a function of the one-photon detuning is found.     

Transferring quantum entangled states between multiple single-photon-state qubits and coherent-state qubits in circuit QED

We present a way to transfer maximally- or partially-entangled states of n single-photon-state (SPS) qubits onto ncoherent-state (CS) qubits, by employing 2nmicrowave cavities coupled to a superconducting flux qutrit. The two logic states of a SPS qubit here are represented by the vacuum state and the single-photon state of a cavity, while the two logic states of a CS qubit are encoded with two coherent states of a cavity. Because of using only one superconducting qutrit as the coupler, the circuit architecture is significantly simplified. The operation time for the state transfer does not increase with the increasing of the number of qubits. When the dissipation of the system is negligible, the quantum state can be transferred in a deterministic way since no measurement is required. Furthermore, the higher-energy intermediate level of the coupler qutrit is not excited during the entire operation and thus decoherence from the qutrit is greatly suppressed. As a specific example, we numerically demonstrate that the high-fidelity transfer of a Bell state of two SPS qubits onto two CS qubits is achievable within the present-day circuit QED technology. Finally, it is worthy to note that when the dissipation is negligible, entangled states of n CS qubits can be transferred back onto n SPS qubits by performing reverse operations. This proposal is quite general and can be extended to accomplish the same task, by employing a natural or artificial atom to couple 2nmicrowave or optical cavities.     

Weighted Information and Weighted Entropic Inequalities for Qutrit States

We review the notion of weighted quantum entropy and consider the weighted quantum entropy for bipartite and noncomposite quantum systems. We extend the subadditivity condition, the inequality known for the weighted entropy information, to the case of indivisible qudit system, such as a qutrit. We discuss the new inequality for the qutrit density matrix for different weights and states, as well as the role of weighted entropy with respect to nonlinear quantum channels.     

Notes on l2 Norm of Coherence

International Journal of Theoretical Physics - We investigate the l2 norm of coherence in this paper. We first calculate the l2 norms of coherence for any qubit states and three qutrit states, and...     

Transferring entangled states of photonic cat-state qubits in circuit QED

We propose a method for transferring quantum entangled states of two photonic cat-state qubits(cqubits)from two microwave cavities to the other two microwave cavities.This proposal is realized by using four microwave cavities coupled to a superconducting flux qutrit.Because of using four cavities with different frequencies,the inter-cavity crosstalk is significantly reduced.Since only one coupler qutrit is used,the circuit resource is minimized.The entanglement transfer is completed with a singlestep operation only,thus this proposal is quite simple.The third energy level of the coupler qutrit is not populated during the state transfer,therefore decoherence from the higher energy level is greatly suppressed.Our numerical simulations show that high-fidelity transfer of two-cqubit entangled states from two transmission line resonators to the other two transmission line resonators is feasible with current circuit QED technology.This proposal is universal and can be applied to accomplish the same task in a wide range of physical systems,such as four microwave or optical cavities,which are coupled to a natural or artificial three-level atom.     

State-independent proof of Kochen-Specker theorem with 13 rays

Quantum contextuality, as proved by Kochen and Specker, and also by Bell, should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified. However, for the simplest system capable of exhibiting contextuality, a qutrit, the quantum contextuality is verified only state dependently in experiment because too many (at least 31) observables are involved in all the known state-independent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all noncontextual realistic models while being violated by all qutrit states. Thus our inequality facilitates a state-independent test of the quantum contextuality for an indivisible quantum system. We also provide a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a magic cube.     

Speeding up the Generation of Entangled State between a Superconducting Qubit and Cavity Photons via Counterdiabatic Driving

Optimal generation of entangled states is of critical significance for robust quantum information processing. An effective scheme is presented for speeding up the generation of an entangled state between a superconducting qubit and microwave photons via counterdiabatic driving. At a magic bias point, the first three levels of a charge-phase quantum circuit constitute an effective qutrit. An entangled state based on adiabatic population transfer is first achieved. By the technique of shortcuts to adiabaticity, a counterdiabatic driving is applied to the qutrit, which then accelerates the entanglement generation significantly. Moreover, with the accessible decoherence rates, the rapid operations in a shortcut way are highly robust when compared with adiabatic manipulations. The scheme could offer a promising approach toward optimal preparation of entangled states with superconducting artificial atoms in circuit quantum electrodynamics, experimentally.     

Splitting a qudit state via Greenberger-Horne-Zeilinger states of qubits

We present a tripartite quantum information splitting scheme which splits a qutrit state via two GHZ states. The scheme is then generalized to splitting a qudit state among any number of receivers. We show that this scheme is also applicable to splitting any multi-qudit entangled states.     

两态和三态多体纯态的度量和分类

基于多体纯态纠缠的度量方法，研究了两态两体和三体以及三态两体纯态纠缠的分类和度量.结果表明，在随机局域操作与经典通信(SLOCC)等价的意义下，纯态的纠缠方式可用所定义纠缠度的不同极大值来表征.通过仔细的计算和分析发现，两态三体和三态两体纯态各有三种SLOCC不等价的基本纠缠方式.最后将三态两体纯态纠缠度的计算与新近提出的熵积纠缠度方案进行了比较，并进行了讨论. 关键词： 纯态纠缠 极值纠缠 随机局域操作和经典通信     

Quantum contextuality for rational vectors

The Kochen-Specker theorem states that noncontextual hidden variable models are inconsistent with the quantum predictions for every yes-no question on a qutrit, corresponding to every projector in three dimensions. It has been suggested [D.A. Meyer, Phys. Rev. Lett. 83 (1999) 3751] that the inconsistency would disappear when restricting to projectors on unit vectors with rational components; that noncontextual hidden variables could reproduce the quantum predictions for rational vectors. Here we show that a qutrit state with rational components violates an inequality valid for noncontextual hidden-variable models [A.A. Klyachko et al., Phys. Rev. Lett. 101 (2008) 020403] using rational projectors. This shows that the inconsistency remains even when using only rational vectors.     

利用偶极封锁制备多粒子三维纠缠态

选择碱金属原子的三个稳定基态为qutrit,其里德堡态为辅助态,借助调节经典脉冲所实现的单量子比特操作和偶极封锁机制分别制备了两原子三维最大纠缠态和三原子三维最大纠缠态。     

Transfer of quantum entangled states between superconducting qubits and microwave field qubits

Transferring entangled states between matter qubits and microwave-field (or optical-field) qubits is of fundamental interest in quantum mechanics and necessary in hybrid quantum information processing and quantum communication. We here propose a way for transferring entangled states between superconducting qubits (matter qubits) and microwave-field qubits. This proposal is realized by a system consisting of multiple superconducting qutrits and microwave cavities. Here, „qutrit” refers to a three-level quantum system with the two lowest levels encoding a qubit while the third level acting as an auxiliary state. In contrast, the microwave-field qubits are encoded with coherent states of microwave cavities. Because the third energy level of each qutrit is not populated during the operation, decoherence from the higher energy levels is greatly suppressed. The entangled states can be deterministically transferred because measurement on the states is not needed. The operation time is independent of the number of superconducting qubits or microwave-field qubits. In addition, the architecture of the circuit system is quite simple because only a coupler qutrit and an auxiliary cavity are required. As an example, our numerical simulations show that high-fidelity transfer of entangled states from two superconducting qubits to two microwave-field qubits is feasible with present circuit QED technology. This proposal is quite general and can be extended to transfer entangled states between other matter qubits (e.g., atoms, quantum dots, and NV centers) and microwave- or optical-field qubits encoded with coherent states.     

The Negativity‐to‐Violation Map between Wigner Function and Quantum Contextuality Inequality for a Single Qudit

The relation between discrete Wigner function and quantum contextuality based on graph theory has been studied, following the work in [Nature 510,351(2014)]. To do this, non‐stabilizer projectors have been introduced to a series of non‐contextuality graphs based on stabilizer projectors for a single qudit with odd prime dimension. It has been found that, for a phase space point defined by Wootters, there exists a given set of states for an odd‐prime qudit where the negative discrete Wigner function on the phase space point means its quantum contextuality under measurements on the graphs designed by a specific method. To implement this method, a subset of non‐stabilizer projectors has been found. In the union of the set of states for all phase space points, there exists a negativity‐to‐violation map between Wigner function and quantum contextuality inequality. The robustness of the equivalence under depolarizing noise has been analyzed and discussed. For demonstration purposes, the graphs with different independence numbers and the corresponding set of states have been established on a single qutrit. Different to the cited work, this method involves only a single qudit, then is experimentally feasible for a qutrit.