Negativity fonts, multiqubit invariants and four qubit maximally entangled states

Recently, we introduced negativity fonts as the basic units of multipartite entanglement in pure states. We show that the relation between global negativity of partial transpose of N?qubit state and linear entropy of reduced single qubit state yields an expression for global negativity in terms of determinants of negativity fonts. Transformation equations for determinants of negativity fonts under local unitaries (LU??s) are useful to construct LU invariants such as degree four and degree six invariants for four qubit states. The difference of squared negativity and N?tangle is an N qubit invariant which contains information on entanglement of the state caused by quantum coherences that are not annihilated by removing a single qubit. Four qubit invariants that detect the entanglement of specific parts in a four qubit state are expressed in terms of three qubit subsystem invariants. Numerical values of invariants bring out distinct features of several four qubit states which have been proposed to be the maximally entangled four qubit states.     

Entanglement sudden death and birth in qubit–qutrit systems under Dzyaloshinskii–Moriya interaction

We study entanglement dynamics of qubit–qutrit pair under Dzyaloshinskii–Moriya (DM) interaction. The qubit–qutrit pair acts as a closed system and one external qubit serve as the environment for the pair. The external qubit interact with qubit of closed system via DM interaction. This interaction frequently kills the entanglement between qubit–qutrit pair, which is also periodically recovered. On the other hand two parameter class of state of qubit–qutrit pair also affected by DM interaction and one parameter class of state remains unaffected. The frequency of occurrence of entanglement sudden death and entanglement sudden birth in two parameter class of state is half than qubit–qutrit pure state. We used our quantification of entanglement as negativity measure.     

Robustness of Greenberger$$\textendash $$Horne$$\textendash $$Zeilinger and W states against Dzyaloshinskii-Moriya interaction

In this article, the robustness of tripartite Greenberger–Horne–Zeilinger (GHZ) and W states is investigated against Dzyaloshinskii-Moriya (i.e. DM) interaction. We consider a closed system of three qubits and an environmental qubit. The environmental qubit interacts with any one of the three qubits through DM interaction. The tripartite system is initially prepared in GHZ and W states, respectively. The composite four qubits system evolve with unitary dynamics. We detach the environmental qubit by tracing out from four qubits, and profound impact of DM interaction is studied on the initial entanglement of the system. As a result, we find that the bipartite partitions of W states suffer from entanglement sudden death (i.e. ESD), while tripartite entanglement does not. On the other hand, bipartite partitions and tripartite entanglement in GHZ states do not feel any influence of DM interaction. So, we find that GHZ states have robust character than W states. In this work, we consider generalised GHZ and W states, and three \(\pi \) is used as an entanglement measure. This study can be useful in quantum information processing where unwanted DM interaction takes place.     

Geometric phase as a determinant of a qubit�C environment coupling

We investigate the qubit geometric phase and its properties in dependence on the mechanism for decoherence of a qubit weakly coupled to its environment. We consider two sources of decoherence: dephasing coupling (without exchange of energy with environment) and dissipative coupling (with exchange of energy). Reduced dynamics of the qubit is studied in terms of the rigorous Davies Markovian quantum master equation, both at zero and non–zero temperature. For pure dephasing coupling, the geometric phase varies monotonically with respect to the polar angle (in the Bloch sphere representation) parameterizing an initial state of the qubit. Moreover, it is antisymmetric about some points on the geometric phase-polar angle plane. This is in distinct contrast to the case of dissipative coupling for which the variation of the geometric phase with respect to the polar angle typically is non-monotonic, displaying local extrema and is not antisymmetric. Sensitivity of the geometric phase to details of the decoherence source can make it a tool for testing the nature of the qubit–environment interaction.     

Groverian Entanglement Measure and Evolution of Entanglement in Search Algorithm for <Emphasis Type="Italic">n</Emphasis>(= 3, 5)-Qubit Systems with Real Coefficients

Evolution of entanglement with the processing of quantum algorithms affects the outcome of the algorithm. Particularly, the performance of Grover’s search algorithm gets worsened if the initial state of the algorithm is an entangled one. The success probability of search can be seen as an operational measure of entanglement. This paper demonstrates an entanglement measure based on the performance of Grover’s search algorithm for three and five qubit systems. We also show that although the overall pattern shows growth of entanglement, its rise to a maximum and then consequent decay, the presence of local fluctuation within each iterative step is likely.      

Quantum teleportation between a single-rail single-photon qubit and a coherent-state qubit using hybrid entanglement under decoherence effects

We study quantum teleportation between two different types of optical qubits using hybrid entanglement as a quantum channel under decoherence effects. One type of qubit employs the vacuum and single-photon states for the basis, called a single-rail single-photon qubit, and the other utilizes coherent states of opposite phases. We find that teleportation from a single-rail single-photon qubit to a coherent-state qubit is better than the opposite direction in terms of fidelity and success probability. We compare our results with those using a different type of hybrid entanglement between a polarized single-photon qubit and a coherent state.     

A Characterization of Global Entanglement

We define a set of 2 ⁿ⁻¹−1 entanglement monotones for n qubits and give a single measure of entanglement in terms of these. This measure is zero except on globally entangled (fully inseparable) states. This measure is compared to the Meyer–Wallach measure for two, three, and four qubits. We determine the four-qubit state, symmetric under exchange of qubit labels, which maximizes this measure. It is also shown how the elementary monotones may be computed as a function of observable quantities. We compute the magnitude of our measure for the ground state of the four-qubit superconducting experimental system investigated in [M. Grajcar et al., Phys. Rev. Lett. 96, 047006 (2006)], and thus confirm the presence of global entanglement in the ground state.      

Quantum information transmission

We present a scheme of quantum information transmission, which transmits the quantum information contained in a single qubit via the quantum correlation shared by two parties (a two-qubit channel), whose quantum discord is non-zero. We demonstrate that quantum correlation, which may have no entanglement, is sufficient to transmit the information needed to reconstruct a quantum state. When the correlation matrix of the two-qubit channel is of full rank (rank three), the information of the qubit (in either a mixed state or a pure state) can be transmitted. The quantum discord of a channel with rank larger than or equal to three is always non-zero. Therefore, non-zero quantum discord is also necessary for our quantum information transmission protocol. The scheme may be useful in remote state tomography and remote state preparation.     

Quantum prisoners’ dilemma under enhanced interrogation

In the quantum version of prisoners’ dilemma, each prisoner is equipped with a single qubit that the interrogator can entangle. We enlarge the available Hilbert space by introducing a third qubit that the interrogator can entangle with the other two. We discuss an enhanced interrogation technique based on tripartite entanglement and analyze Nash equilibria. We show that for tripartite entanglement approaching a W-state, we calculate the Nash equilibria numerically and show that they coincide with the Pareto-optimal choice where both prisoners cooperate. Upon continuous variation between a W-state and a pure bipartite entangled state, the game is shown to have a surprisingly rich structure. The role of bipartite and tripartite entanglement is explored to explain that structure. As an application, we consider an evolutionary game based on our quantum game with a network of agents on a square lattice with periodic boundary conditions and show that the strategy corresponding to Nash equilibrium completely dominates without placing any restrictions on the initial set of strategies.     

VENUS: A Geometrical Representation for Quantum State Visualization

Visualizations have played a crucial role in helping quantum computing users explore quantum states in various quantum computing applications. Among them, Bloch Sphere is the widely-used visualization for showing quantum states, which leverages angles to represent quantum amplitudes. However, it cannot support the visualization of quantum entanglement and superposition, the two essential properties of quantum computing. To address this issue, we propose VENUS, a novel visualization for quantum state representation. By explicitly correlating 2D geometric shapes based on the math foundation of quantum computing characteristics, VENUS effectively represents quantum amplitudes of both the single qubit and two qubits for quantum entanglement. Also, we use multiple coordinated semicircles to naturally encode probability distribution, making the quantum superposition intuitive to analyze. We conducted two well-designed case studies and an in-depth expert interview to evaluate the usefulness and effectiveness of VENUS. The result shows that VENUS can effectively facilitate the exploration of quantum states for the single qubit and two qubits.     

Derivation of the Lewenstein–Sanpera Decomposition via Semidefinite Programming

In many theoretical proposals appeared recently, semidefinite programming was considered as a way to express quantum entanglement. Using semidefinite optimization method, we prove the Lewenstein–Sanpera lemma in a simple elegant manner. Particularly, using this method we obtain Lewenstein–Sanpera decomposition for some examples such as: generic two qubit state in Wootters’s basis, Iso-concurrence state, Bell decomposable state and 2 ⊗ 3 Bell decomposable state.     

Lower bounds of concurrence for <Emphasis Type="Italic">N</Emphasis>-qubit systems and the detection of <Emphasis Type="Italic">k</Emphasis>-nonseparability of multipartite quantum systems

Concurrence, as one of the entanglement measures, is a useful tool to characterize quantum entanglement in various quantum systems. However, the computation of the concurrence involves difficult optimizations and only for the case of two qubits, an exact formula was found. We investigate the concurrence of four-qubit quantum states and derive analytical lower bound of concurrence using the multiqubit monogamy inequality. It is shown that this lower bound is able to improve the existing bounds. This approach can be generalized to arbitrary qubit systems. We present an exact formula of concurrence for some mixed quantum states. For even-qubit states, we derive an improved lower bound of concurrence using a monogamy equality for qubit systems. At the same time, we show that a multipartite state is k-nonseparable if the multipartite concurrence is larger than a constant related to the value of k, the qudit number and the dimension of the subsystems. Our results can be applied to detect the multipartite k-nonseparable states.     

Quantum tasks using six qubit cluster states

The usefulness of the recent experimentally realized six photon cluster state by C. Y. Lu et al. (Nature 3:91, 2007) is investigated for quantum communication protocols like quantum teleportation and quantum information splitting (QIS) and dense coding. We show that the present state can be used for the teleportation of an arbitrary two qubit state deterministically. Later, we devise two distinct protocols for the QIS of an arbitrary two qubit state among two parties. We construct sixteen orthogonal measurement basis on the cluster state, which will lock an arbitrary two qubit state among two parties. The capability of the state for dense coding is investigated and it is shown that one can send five classical bits by sending only three qubits using this state as a shared entangled resource. We finally show that this state can also be utilised in the remote state preparation of an arbitrary two qubit state.     

Heralded noiseless amplification for single-photon entangled state with polarization feature

Heralded noiseless amplification is a promising method to overcome the transmission photon loss in practical noisy quantum channel and can effectively lengthen the quantum communication distance. Single-photon entanglement is an important resource in current quantum communications. Here, we construct two single-photon-assisted heralded noiseless amplification protocols for the single-photon two-mode entangled state and single-photon three-mode W state, respectively, where the single-photon qubit has an arbitrary unknown polarization feature. After the amplification, the fidelity of the single-photon entangled state can be increased, while the polarization feature of the single-photon qubit can be well remained. Both the two protocols only require the linear optical elements, so that they can be realized under current experimental condition. Our protocols may be useful in current and future quantum information processing.     

Experimentally identifying the entanglement class of pure tripartite states

We use concurrence as an entanglement measure and experimentally demonstrate the entanglement classification of arbitrary three-qubit pure states on a nuclear magnetic resonance quantum information processor. Computing the concurrence experimentally under three different bipartitions, for an arbitrary three-qubit pure state, reveals the entanglement class of the state. The experiment involves measuring the expectation values of Pauli operators. This was achieved by mapping the desired expectation values onto the local z magnetization of a single qubit. We tested the entanglement classification protocol on twenty-seven different generic states and successfully detected their entanglement class. Full quantum state tomography was performed to construct experimental tomographs of each state and negativity was calculated from them, to validate the experimental results.     

Quantum Information Storage in the Localized State of a Spin Chain

We describe a chain of qubits with always on exchange interaction in the presence of a spatial inhomogeneity in the qubit level spacing. Similarly to the phenomenon of Anderson localization, this system has a localized eigenstate which can be used to store or trap quantum information. We discuss both the fidelity of storage and the leakage of information from this localized state and show that even a very small defect can be useful. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.     

Quantum steering borders in three-qubit systems

We discuss a family of states describing three-qubit systems in a context of quantum steering phenomena. We show that symmetric steering cannot appear between two qubits—only asymmetric steering can appear in such systems. The main aim of this paper is to discuss the possible relations between the entanglement measures and steering parameter for two-mode mixed state corresponding to the qubit–qubit subsystem. We have derived the conditions determining boundary values of the negativity parametrized by concurrence. We show that two-qubit mixed state cannot be steerable when the negativity of such state is smaller than, or equal to, its boundary value. Finally, we have found ranges of the values of the mixedness measure, parametrized by concurrence and negativity for steerable and unsteerable two-qubit mixed states.     

N-qubit quantum teleportation, information splitting and superdense coding through the composite GHZ–Bell channel

We introduce a general odd qubit entangled system composed of GHZ and Bell pairs and explicate its usefulness for quantum teleportation, information splitting and superdense coding. After demonstrating the superdense coding protocol on the five qubit system, we prove that ‘2N + 1’ classical bits can be sent by sending ‘N + 1’ quantum bits using this channel. It is found that the five-qubit system is also ideal for arbitrary one qubit and two qubit teleportation and quantum information splitting (QIS). For the single qubit QIS, three different protocols are feasible, whereas for the two qubit QIS, only one protocol exists. Protocols for the arbitrary N-qubit state teleportation and quantum information splitting are then illustrated.     

An atom�Cmolecule platform for quantum computing

We propose a combined atom–molecule system for quantum information processing in individual traps, such as provided by optical lattices. In this platform, different species of atoms—one atom carrying a qubit and the other enabling the interaction—are used to store and process quantum information via intermediate molecular states. We show how gates, initialization, and readout operations could be implemented using this approach. In particular, we describe in some detail the implementation of a two-qubit phase gate in which a pair of atoms is transferred into the ground rovibrational state of a polar molecule with a large dipole moment, thus allowing atoms transferred into molecules to interact via their dipole-dipole interaction. We also discuss how the reverse process could be used as a non-destructive readout tool of molecular qubit states. Finally, we generalize these ideas to use a decoherence-free subspace for qubit encoding to minimize the decoherence due to magnetic field fluctuations. In this case, qubits will be encoded into field-insensitive states of two identical atoms, while a third atom of a different species will be used to realize a phase gate.     

Score operators of a qubit with applications

The score operators of a quantum system are the symmetric logarithmic derivatives of the system’s parametrically defined quantum state. Score operators are central to the calculation of the quantum Fisher information (QFI) associated with the state of the system, and the QFI determines the maximum precision with which the state parameters can be estimated. We give a simple, explicit expression for score operators of a qubit and apply this expression in a series of settings. We treat in detail the task of identifying a quantum Pauli channel from the state of its qubit output, and we show that a “balanced” probe state is highly robust for this purpose. The QFI for this task is a matrix, and we study its determinant, for which we establish a Cramér-Rao inequality.