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Weak measurement is a new way to manipulate and control quantum systems. Different from projection measurement, weak measurement only makes a small change in status. Applying weak measurement to quantum discord, Singh and Pati proposed a new kind of quantum correlations called super quantum discord (SQD) [Annals of Physics textbf{343},141(2014)]. Unfortunately, the super quantum discord is also difficult to calculate. There are only few explicit formulae about SQD. We derive an analytical formulae of SQD for general X-type two-qubit states, which surpass the conclusion for Werner states and Bell diagonal states. Furthermore, our results reveal more knowledge about the new insight of quantum correlation and give a new way to compare SQD with normal quantum discord. Finally, we analyze its dynamics under nondissipative channels.
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Quantum discord Q is a function of density matrix elements. The domain of such a function in the case of two-qubit system with X density matrix may consist of three subdomains at most: two ones where the quantum discord is expressed in closed analytical forms (Q_{pi/2} and Q_0) and an intermediate subdomain for which, to extract the quantum discord Q_theta, it is required to solve in general numerically a one-dimensional minimization problem to find the optimal measurement angle thetain(0,pi/2). Hence the quantum discord is given by a piecewise-analytic-numerical formula Q=min{Q_{pi/2}, Q_theta, Q_0}. Equations for determining the boundaries between these subdomains are obtained. The boundaries consist of bifurcation points. The Q_{theta} subdomains are discovered in the generalized Horodecki states, in the dynamical phase flip channel model, in the anisotropic spin systems at thermal equilibrium, in the heteronuclear dimers in an external magnetic field. We found that transitions between Q_{theta} subdomain and Q_{pi/2} and Q_0 ones occur suddenly but continuously and smoothly, i.e., nonanalyticity is hidden and can be observed in higher derivatives of discord function.
The study of classical and quantum correlations in bipartite and multipartite systems is crucial for the development of quantum information theory. Among the quantifiers adopted in tripartite systems, the genuine tripartite quantum discord (GTQD), estimating the amount of quantum correlations shared among all the subsystems, plays a key role since it represents the natural extension of quantum discord used in bipartite systems. In this paper, we derive an analytical expression of GTQD for three-qubit systems characterized by a subclass of symmetrical X-states. Our approach has been tested on both GHZ and maximally mixed states reproducing the expected results. Furthermore, we believe that the procedure here developed constitutes a valid guideline to investigate quantum correlations in form of discord in more general multipartite systems.
Quantum discord is a function of density-matrix elements (and through them, e.~g., of temperature, applied fields, time, and so forth). The domain of such a function in the case of two-qubit system with X or centrosymmetric (CS) density matrix can consist at most of three subdomains: two ones, where the quantum discord is expressed in closed analytical forms (Q_0 and Q_{pi/2}), and an intermediate subdomain in which for determining the quantum discord Q_theta it is required to solve numerically a one-dimensional minimization problem to find the optimal measurement angle thetain(0,pi/2). Exact equations for determining the boundaries between these subdomains are obtained and solved for a number of models. The Q_theta subdomains are discovered in the anisotropic spin dimers in external field. On the other hand, coinciding boundaries and therefore sudden transitions between optimal measurement angles theta=pi/2 and theta=0 are observed in dynamics of spin currying particles in closed nanopore and also in phase flip channels. In latter cases the solutions are entirely analytical.
Super-quantum discord is an extension of the quantum discord concept where weak measurements are made instead of projective measurements. We study the temperature behavior of the super-quantum discord for two real magnetic materials. We extract information about super-quantum discord from the magnetic susceptibility data for iron nitrosyl complexes $mathrm{ Fe_2(SC_3 H_5N_2)_2(NO)_4}$ and binuclear Cu(II) acetate complex $mathrm{[Cu_2 L(OAc)]cdot 6 H_2O}$, where $mathrm{ L}$ is a ligand that allows us to compare the super-quantum discord with the standard one. The dependence of the super-quantum discord on the parameter describing weak measurements is studied. Obtained difference between super-quantum and quantum discords confirms the detection of additional quantum correlations that are usually destroyed during projective measurements. The use of the approach allows to predict quantitatively in advance the advantages of use of weak measurements versus projective one for quantum technologies in real settings where quantum discord is used as resource. This can be relevant particularly in macroscopic quantum systems where weak measurements can be used to extract information about the system.
Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find analytical expression of quantum discord is an intractable task. Exact results are known only for very special states, namely, two-qubit X-shaped states. We present in this paper a geometric viewpoint, from which two-qubit quantum discord can be described clearly. The known results about X state discord are restated in the directly perceivable geometric language. As a consequence, the dynamics of classical correlations and quantum discord for an X state in the presence of decoherence is endowed with geometric interpretation. More importantly, we extend the geometric method to the case of more general states, for which numerical as well as analytica results about quantum discord have not been found yet. Based on the support of numerical computations, some conjectures are proposed to help us establish geometric picture. We find that the geometric picture for these states has intimate relationship with that for X states. Thereby in some cases analytical expressions of classical correlations and quantum discord can be obtained.
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