Analytic expressions of discord and geometric discord in Werner derivatives

Werner derivatives are a special kind of mixing states transformed from Werner states by unitary operations (Hiroshima and Ishizaka in Phys Rev A 62:044302, 2000). In this paper, the inherent quantum correlations in Werner derivatives are quantified by two different quantifiers, i.e., quantum discord and geometric discord. Different analytic expressions of the two discords in Werner derivatives are derived out. Some distinct features of the discords and their underlying physics are exposed via discussions and analyses. Moreover, it is found that the amount of quantum correlations quantified by either quantifier in each derivative cannot exceed that in the original Werner state. In other words, no unitary operation can increase quantum correlation in a Werner state as far as the two quantifiers are concerned.     

Geometric quantum discord under noisy environment

In this work, we mainly analyze the dynamics of geometric quantum discord under a common dissipating environment. Our results indicate that geometric quantum discord is generated when the initial state is a product state. The geometric quantum discord increases from zero to a stable value with the increasing time, and the variations of stable values depend on the system size. For different initial product states, geometric quantum discord has some different behaviors in contrast with entanglement. For initial maximally entangled state, it is shown that geometric quantum discord decays with the increasing dissipated time. It is found that for EPR state, entanglement is more robust than geometric quantum discord, which is a sharp contrast to the existing result that quantum discord is more robust than entanglement in noisy environments. However, for GHZ state and W state, geometric quantum discord is more stable than entanglement. By the comparison of quantum discord and entanglement, we find that a common dissipating environment brings complicated effects on quantum correlation, which may deepen our understanding of physical impacts of decohering environment on quantum correlation. In the end, we analyze the effects of collective dephasing noise and rotating noise to a class of two-qubit X states, and we find that quantum correlation is not altered by the collective noises.     

Super-quantum correlation and geometry for Bell-diagonal states with weak measurements

We propose “weak one-way deficit” by weak measurements as the generalization of one-way deficit defined for standard projective measurements. The weak one-way deficit for Werner state is obtained analytically. We find that weak one-way deficit is smaller than the standard one-way deficit, which contrasts with a straightforward expectation based on the known fact that super-quantum discord by weak measurement is always larger than the quantum discord defined by projective measurement. On the other hand, by tuning the weak measurement continuously to the projective measurement, both weak one-way deficit and super-quantum discord converge to the same value, which is either the one-way deficit or the quantum discord both quantifying quantum correlation. In this sense, weak measurement does not necessarily capture more quantumness of correlations. We also give the geometry of super-quantum discord of the Bell-diagonal states with explicit geometrical figures. As an application, the dynamic behavior of super-quantum correlation including super-quantum discord and weak one-way deficit under decoherence is investigated. We find that the order relation of the super-quantum correlation and the quantum correlation remained unchanged under the phase flipping channel for the Bell-diagonal states and the Werner states.     

Analytic expressions of quantum correlations in qutrit Werner states

Quantum correlations in qutrit Werner states are extensively investigated with five popular methods, namely, original quantum discord (OQD) (Ollivier and Zurek in Phys Rev Lett 88:017901, 2001), measurement-induced disturbance (MID) (Luo in Phys Rev A 77:022301, 2008), ameliorated MID (AMID) (Girolami et al. in J Phys A Math Theor 44:352002, 2011), relative entropy (RE) (Modi et al. in Phys Rev Lett 104:080501, 2010) and geometric discord (GD) (Daki? et al. in Phys Rev Lett 105:190502, 2010). Two different analytic expressions of quantum correlations are derived. Quantum correlations captured by the former four methods are same and bigger than those obtained via the GD method. Nonetheless, they all qualitatively characterize quantum correlations in the concerned states. Moreover, as same as the qubit case, there exist quantum correlations in separable qutrit Werner states, too.     

A comparative study of protocols for secure quantum communication under noisy environment: single-qubit-based protocols versus entangled-state-based protocols

The effect of noise on various protocols of secure quantum communication has been studied. Specifically, we have investigated the effect of amplitude damping, phase damping, squeezed generalized amplitude damping, Pauli type as well as various collective noise models on the protocols of quantum key distribution, quantum key agreement, quantum secure direct quantum communication and quantum dialogue. From each type of protocol of secure quantum communication, we have chosen two protocols for our comparative study: one based on single-qubit states and the other one on entangled states. The comparative study reported here has revealed that single-qubit-based schemes are generally found to perform better in the presence of amplitude damping, phase damping, squeezed generalized amplitude damping noises, while entanglement-based protocols turn out to be preferable in the presence of collective noises. It is also observed that the effect of noise depends upon the number of rounds of quantum communication involved in a scheme of quantum communication. Further, it is observed that squeezing, a completely quantum mechanical resource present in the squeezed generalized amplitude channel, can be used in a beneficial way as it may yield higher fidelity compared to the corresponding zero squeezing case.     

Decoherence dynamics of geometric measure of quantum discord and measurement induced nonlocality for noninertial observers at finite temperature

Quantum discord quantifies the total non-classical correlations in mixed states. It is the difference between total correlation, measured by quantum mutual information, and the classical correlation. Another step forward towards the quantification of quantum discord was by Daki? et al. (Phys Rev Lett 105:190502, 2010) who introduced the geometric measure of quantum discord (GMQD) and derived an explicit formula for a two-qubit state. Recently, Luo and Fu (Phys Rev Lett 106:120401, 2011) introduced measurement-induced nonlocality (MIN) as a measure of nonlocality for a bipartite quantum system. The dynamics of GMQD is recently considered by Song et al. (arXiv: quant/ph.1203.3356) and Zhang et al. (Eur Phys J D 66:34, 2012) for inertial observers. However, the topic requires due attention in noninertial frames, particularly, from the perspective of MIN. Here I consider $X$ -structured bipartite quantum system in noninertial frames and analyze the decoherence dynamics of GMQD and MIN at finite temperature. The dynamics under the influence of amplitude damping, depolarizing and phase flip channels is discussed. It is worth-noting that initial state entanglement plays an important role in bipartite states. It is possible to distinguish the Bell, Werner and general type initial quantum states using GMQD. Sudden transition in the behaviour of GMQD and MIN occurs depending upon the mean photon number of the local environment. The transition behaviour disappears for larger values of $\bar{n},$ i.e. $\bar{n}>0.3.$ It becomes more prominent, when environmental noise is introduced in the system. In the presence of environmental noise, as we increase the value of acceleration $r$ , GMQD and MIN decay due to Unruh effect. The effect is prominent for the phase flip and amplitude damping channels. However, in case of depolarizing channel, no sudden change in the behaviour of GMQD and MIN is observed. The environmental noise has stronger affect on the dynamics of GMQD and MIN as compared to the Unruh effect. Furthermore, Werner like states are more robust than General type initial states at finite temperature.     

Fault tolerant authenticated quantum direct communication immune to collective noises

This study proposes two new coding functions for GHZ states and GHZ-like states, respectively. Based on these coding functions, two fault tolerant authenticated quantum direct communication (AQDC) protocols are proposed. Each of which is robust under one kind of collective noises: collective-dephasing noise and collective-rotation noise, respectively. Moreover, the proposed AQDC protocols enable a sender to send a secure as well as authenticated message to a receiver within only one step quantum transmission without using the classical channels.     

Protection of quantum correlations against decoherence

The protection of different quantum correlations, such as Bell nonlocality, quantum discord, and geometric quantum discord as trace distance against noise, is explored. By weak measurement and quantum measurement reversal, we show that the mentioned quantum correlations can be effectively preserved probabilistically from the decoherence due to amplitude damping. The results will play an important role in the experiments using the quantum correlations as resource.     

Robustness measure of hybrid intra-particle entanglement,discord, and classical correlation with initial Werner state

Quantum information processing is largely dependent on the robustness of non-classical correlations, such as entanglement and quantum discord. However, all the realistic quantum systems are thermodynamically open and lose their coherence with time through environmental interaction. The time evolution of quantum entanglement, discord, and the respective classical correlation for a single, spin-1/2 particle under spin and energy degrees of freedom, with an initial Werner state, has been investigated in the present study. The present intra-particle system is considered to be easier to produce than its inter-particle counterpart. Experimentally, this type of system may be realized in the well-known Penning trap. The most stable correlation was identified through maximization of a system-specific global objective function. Quantum discord was found to be the most stable, followed by the classical correlation. Moreover, all the correlations were observed to attain highest robustness under initial Bell state, with minimum possible dephasing and decoherence parameters.     

Genuine correlations of tripartite system

We define genuine total, classical and quantum correlations in tripartite systems. The genuine tripartite quantum discord can be interpreted as ‘quantum advantage’ in tripartite superdense coding. We find in a symmetrical tripartite state, for total correlation and classical correlation, the genuine tripartite correlations are no less than the pair-wise correlations. However, the genuine quantum tripartite correlation can be surpassed by the pair-wise quantum correlations. Analytical expressions for genuine tripartite correlations are obtained for pure states and rank-2 symmetrical states. The genuine correlations in both entangled and separable states are calculated.     

Quantum discord protection of a two-qutrit V-type atomic system from decoherence by partially collapsing measurements

In this paper, by exploiting the weak measurement and quantum measurement reversal procedure, we propose a scheme to show how one can protect the geometric quantum discord (GQD) of a two-qutrit V-type atomic system each of which interacts with a dissipative reservoir independently. We examine the scheme for the GQD of the initial two-qutrit Werner and Horodecki states for different classes of weak measurement strengths. It is found out that the presented protocol enables us to suppress decoherence due to the amplitude damping channel and preserve the quantum discord of the two-qutrit system successfully.     

Dynamical behavior of quantum correlations between two qubits coupled to an external environment

We investigate the dynamics of quantum correlations of a two-qubit system coupled to an external environment. We have considered both cases: a spin environment and a bosonic environment. In all cases, we have chosen the Bell-diagonal state as the initial state and computed the evolution of quantum correlations in terms of entanglement, quantum discord and trace distance geometric quantum discord. Special attention is paid to the singular quantum phenomena, such as entanglement sudden death, sudden transition and double sudden transitions from classical to quantum decoherence, which all depend on the initial state and the parameters related to the system and the environment. We find the trace distance geometric quantum discord has a good robustness in resisting the spin and bosonic environmental noise.     

Quantum correlations in Gaussian states via Gaussian channels: steering,entanglement, and discord

Here we study the quantum steering, quantum entanglement, and quantum discord for Gaussian Einstein–Podolsky–Rosen states via Gaussian channels. And the sudden death phenomena for Gaussian steering and Gaussian entanglement are theoretically observed. We find that some Gaussian states have only one-way steering, which confirms the asymmetry of quantum steering. Also we investigate that the entangled Gaussian states without Gaussian steering and correlated Gaussian states own no Gaussian entanglement. Meanwhile, our results support the assumption that quantum entanglement is intermediate between quantum discord and quantum steering. Furthermore, we give experimental recipes for preparing quantum states with desired types of quantum correlations.     

Decoherence dynamics of measurement-induced nonlocality and comparison with geometric discord for two qubit systems

We check the decoherence dynamics of Measurement-induced Nonlocality (in short, MIN) and compare it with geometric discord for two qubit systems. There are quantum states, on which the action of dephasing channel cannot destroy MIN in finite or infinite time. We check the additive dynamics of MIN on a qubit state under two independent noise. Geometric discord also follows such additive dynamics like quantum discord. We have further compared non-Markovian evolution of MIN and geometric discord under dephasing and amplitude damping noise for pure state and it shows distinct differences between their dynamics.     

Quantifying quantum correlations in fermionic systems using witness operators

We present a method to quantify quantum correlations in arbitrary systems of indistinguishable fermions using witness operators. The method associates the problem of finding the optimal entanglement witness of a state with a class of problems known as semidefinite programs, which can be solved efficiently with arbitrary accuracy. Based on these optimal witnesses, we introduce a measure of quantum correlations which has an interpretation analogous to the Generalized Robustness of entanglement. We also extend the notion of quantum discord to the case of indistinguishable fermions, and propose a geometric quantifier, which is compared to our entanglement measure. Our numerical results show a remarkable equivalence between the proposed Generalized Robustness and the Schliemann concurrence, which are equal for pure states. For mixed states, the Schliemann concurrence presents itself as an upper bound for the Generalized Robustness. The quantum discord is also found to be an upper bound for the entanglement.     

Multipartite quantum and classical correlations in symmetric <Emphasis Type="Italic">n</Emphasis>-qubit mixed states

We discuss how to calculate genuine multipartite quantum and classical correlations in symmetric, spatially invariant, mixed n-qubit density matrices. We show that the existence of symmetries greatly reduces the amount of free parameters to be optimized in order to find the optimal measurement that minimizes the conditional entropy in the discord calculation. We apply this approach to the states exhibited dynamically during a thermodynamic protocol to extract maximum work. We also apply the symmetry criterion to a wide class of physically relevant cases of spatially homogeneous noise over multipartite entangled states. Exploiting symmetries we are able to calculate the non-local and genuine quantum features of these states and note some interesting properties.     

Geometric measure of quantum discord for a two-parameter class of states in a qubit–qutrit system under various dissipative channels

Quantum discord, as a measure of all quantum correlations, has been proposed as the key resource in certain quantum communication tasks and quantum computational models without containing much entanglement. Daki? et al. (Phys Rev Lett 105:190502, 2010) introduced a geometric measure of quantum discord (GMQD) and derived an explicit formula for any two-qubit state. Luo and Fu (Phys Rev A 82:034302, 2010) introduced another form of GMQD and derived an explicit formula for arbitrary state in a bipartite quantum system. However, the explicit analytical expression for any bipartite system was not given. In this work, we give out the explicit analytical expressions of the GMQD for a two-parameter class of states in a qubit–qutrit system and study its dynamics for the states under various dissipative channels in the first time. Our results show that all these dynamic evolutions do not lead to a sudden vanishing of GMQD. Quantum correlations vanish at an asymptotic time for local or multi-local dephasing, phase-flip, and depolarizing noise channels. However, it does not disappear even though t → ∞ for local trit-flip and local trit-phase-flip channels. Our results maybe provide some important information for the application of GMQD in hybrid qubit–qutrit systems in quantum information.     

Relationship between the field local quadrature and the quantum discord of a photon-added correlated channel under the influence of scattering and phase fluctuation noise

We study quantum correlations and discord in a bipartite continuous variable hybrid system formed by linear combinations of coherent states \(\mathinner {|{\alpha }\rangle }\) and single photon-added coherent states of the form \(\mathinner {|{\psi }\rangle }_{\text {dp(pa)}}= \mathcal {N}/\sqrt{2} (\hat{a}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b \pm \hat{b}^\dagger \mathinner {|{\alpha }\rangle }_a\mathinner {|{\alpha }\rangle }_b)\). We stablish a relationship between the quantum discord with a local observable (the quadrature variance for one subsystem) under the influence of scattering and phase fluctuation noise. For the pure states the quantum correlations are characterized by means of measurement induced disturbance (MID) with simultaneous quadrature measurements. In a scenario where homodyne conditional measurements are available we show that the MID provides an easy way to select optimal phases to obtain information of the maximal correlations in the channels. The quantum correlations of these entangled states with channel losses are quantitatively characterized with the quantum discord (QD) with a displaced qubit projector. We observe that as scattering increases, QD decreases monotonically. At the same time for the state \(\mathinner {|{\psi }\rangle }_{\text {dp}}\), QD is more resistant to high phase fluctuations when the average photon number \(n_0\) is bigger than zero, but if phase fluctuations are low, QD is more resistant if \(n_0=0\). For the dp model with scattering, we obtain an analytical expression of the QD as a function of the observable quadrature variance in a local subsystem. This relation allows us to have a way to obtain the degree of QD in the channel by just measuring a local property observable such as the quadrature variance. For the other model this relation still exists but is explored numerically. This relation is an important result that allows to identify quantum processing capabilities in terms of just local observables.     

Analytical expression of quantum discord for rank-2 two-qubit states

Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary \(d\otimes 2\) quantum states and von Neumann entropy-based classical correlation for arbitrary \(2\otimes 2\) rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.     

Trojan horse attack free fault-tolerant quantum key distribution protocols

This work proposes two quantum key distribution (QKD) protocols—each of which is robust under one kind of collective noises—collective-dephasing noise and collective-rotation noise. Due to the use of a new coding function which produces error-robust codewords allowing one-time transmission of quanta, the proposed QKD schemes are fault-tolerant and congenitally free from Trojan horse attacks without having to use any extra hardware. Moreover, by adopting two Bell state measurements instead of a 4-GHZ state joint measurement for decoding, the proposed protocols are practical in combating collective noises.