Abstract. Among various definitions of quantum correlations, quantum discord has attracted considerable attention. The connection between the quantum discord and the entanglement of formation is described by Koashi-Winter relation. We investigate this relation from the viewpoint of the quantum channel that is isomorphic to the given state. It is shown that in the case of two-qubit states the channel, on the one hand, determines the form of the quantum steering ellipsoid of the given state, and on the other hand, is closely related to the concurrence of the complement state of the given state. We also point out that, for two-qubit rank-two state, von Neumann measurement is the optimal choice to achieve the quantum discord. However, for some two-qubit states with the rank larger than two, the three-element POVM measurement is optimal.

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 analytical 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.

We present an efficient method to solve the quantum discord of two-qubit X states exactly. A geometric picture is used to clarify whether and when the general POVM measurement is superior to von Neumann measurement. We show that either the von Neumann measurement or the threeelement POVM measurement is optimal, and more interestingly, in the latter case the components of the postmeasurement ensemble are invariant for a class of states.The notation of quantum discord was proposed in 2001 [1]. It is regarded as a measure of quantumness of correlation, even in the absence of quantum entanglement. For ten years, many works have been devoted to the significance and application of quantum discord (see, for example, [2]). The analytical expressions for quantum discord have been obtained only in a few cases including two-qubit Bell-diagonal states [3], rank-two states [4] and Gaussian state [5]. However there is no exact result so far for two-qubit X states (i.e., the states such that the non-zero elements of the density matrix only lie along the diagonal or skew diagonal). In this paper we present an efficient method to solve this problem.As we have known, the major difficulty in the calculation quantum discord is how to acquire the maximal information about one particle by measuring the other particle. Given a bipartite state ρ AB , perform on particle A a generalized measurement, denoted by POVMQuantum discord Q is given by Q = I − C, where I is the total correlation quantified by the mutual information,. It is a formidable task to find the optimal measurement among all M to achieve the minimal value of the conditional entropy S(ρ B |M ). Much effort [6-9], analytical or numerical, has been made in studying the optimization for two-qubit states. However there is no definite answer as to whether and how the quantum discord is determined by the general POVM measurements.The measurement M on A induces the decomposition of ρ B into the ensemble {p i , ρ B|Mi }. For two-qubit states, we have known that all ρ B|Mi are distributed, in terms of the Bloch vectors, in an ellipsoidal region in three-dimensional real space. This region is called quantum steering ellipsoid [10], which we denote by E. It has been shown that this geometric picture is very useful in the discussion of the quantum discord of two-qubit states [4,11]: We need only consider the decomposition with the form of ρ B = i p i ρ B i , where all ρ B i are distributed on the surface of E. The problem is then to find the minimal value of the average entropy S B = i p i S(ρ B i ), which we denoted by S B min . As we show later, we benefit greatly from this geometric picture in the case of two-qubit X states: The optimal measurement on A, or the optimal decomposition of ρ B , can be determined unambiguously, and thus the exact result of quantum discord is attained.Note that there are infinitely many states corresponding to a given E. Denote by [ρ AB ] E the set of all X states having the identical E. We show that all steering ellipsoids associated with X states are c...

Time-resolved magnetic sensing is of great importance from fundamental studies to applications in physical and biological sciences. Recently the nitrogen-vacancy (NV) defect center in diamond has been developed as a promising sensor of magnetic field under ambient conditions. However the methods to reconstruct time-resolved magnetic field with high sensitivity are not yet fully developed. Here, we propose and demonstrate a novel sensing method based on spin echo, and Haar wavelet transform. Our method is exponentially faster in reconstructing time-resolved magnetic field with comparable sensitivity over existing methods. Further, the wavelet's unique features enable our method to extract information from the whole signal with only part of the measuring sequences. We then explore this feature for a fast detection of simulated nerve impulses. These results will be useful to time-resolved magnetic sensing with quantum probes at nano-scales. Sensing of weak signals with high spatial resolution is of great importance in diverse areas ranging from fundamental physics and material science to biological sciences. NV center in diamond has recently emerged as one multifunctional sensor with high sensitivity and nano-scale spatial resolution under ambient conditions. It is applied for sensing magnetic fields [1][2][3][4][5][6][7][8], electric fields[9], temperature [10][11][12] and magnetic resonance imaging [13][14][15][16][17].For NV based magnetic sensing applications, the basic idea is measuring magnetic field, for example, via Ramsey interferometry [18]: the electron spin is first prepared in an equal superposition of eigenstates |0 and |1 and obtains a relative phase in an external field b(t). After an acquisition time T the accumulated phase T 0 γb(t)dt results in a frequency shift that is measured optically where γ is gyromagnetic ratio. This scheme is often called DC sensing scheme which could in principle extract the complete dynamics of the field required by general applications. By a successive monitoring of the resonance energy, either through increasing the acquisition period or sequential small acquisition steps, we can reconstruct the temporal magnetic field and extract its dynamics. However these reconstruction schemes are inefficient in sampling rate and also suffer from the electron spin's short coherence time (i.e., T * 2 ) which limits the measurement sensitivity. Dynamical decoupling sequences [19][20][21][22][23], acting as a controllable frequency band-pass filter, could be used to enhance the coherence time by filtering out a large part of the noise [24][25][26]. High sensitivity of constant oscillating magnetic fields is achieved based on this method. But the temporal field's reconstruction is not straightforward.To deal with this problem, recently Cooper et.al designed a group of decoupling sequences associated with the Wash functions [27]. While protecting against dephasing noise, each Walsh decoupling sequence is measuring a Wash transform coefficient of temporal magnetic field on the correspon...

A symmetric measure of quantum correlation based on the Hilbert—Schmidt distance is presented in this paper. For two-qubit states, we considerably simplify the optimization procedure so that numerical evaluation can be performed efficiently. Analytical expressions for the quantum correlation are attained for some special states. We further investigate the dynamics of quantum correlation of the system qubits in the presence of independent dissipative environments. Several nontrivial aspects are demonstrated. We find that the quantum correlation can increase even if the system state is suffering from dissipative noise. Sudden changes occur, even twice, in the time evolution of quantum correlation. There exists a certain correspondence between the evolution of quantum correlation in the systems and that in the environments, and the quantum correlation in the systems will be transferred into the environments completely and asymptotically.

We suggest an experimental scheme that a single nitrogen-vacancy (NV) center coupled to a nearest neighbor 13 C nucleus as a sensor in diamond can be used to detect a static vector magnetic field. By means of optical detection magnetic resonance (ODMR) technique, both the strength and the direction of the vector field could be determined by relevant resonance frequencies of continuous wave (CW) and Ramsey spectrums. In addition, we give a method that determines the unique one of eight possible hyperfine tensors for an (NV-13 C) system. Finally, we propose an unambiguous method to exclude the symmetrical solution from eight possible vector fields, which correspond to nearly identical resonance frequencies due to their mirror symmetry about 14 N-Vacancy-13 C ( 14 N-V-13 C) plane.

We perform a proof-of-principle experiment that uses a single negatively charged nitrogen-vacancy (NV) color center with a nearest neighbor 13 C nuclear spin in diamond to detect the strength and direction (including both polar and azimuth angles) of a static vector magnetic field by optical detection magnetic resonance (ODMR) technique. With the known hyperfine coupling tensor between an NV center and a nearest neighbor 13 C nuclear spin, we show that the information of static vector magnetic field could be extracted by observing the pulsed continuous wave (CW) spectrum.

We carried out a proof-of-principle demonstration of the reconstruction of a static vector magnetic field involving adjacent three nitrogen-vacancy (NV) sensors with corresponding different NV symmetry axes in a bulk diamond. By means of optical detection of the magnetic resonance (ODMR) techniques, our experiment employs the continuous wave (CW) to monitor resonance frequencies and it extracts the information of the detected field strength and polar angles with respect to each NV frame of reference. Finally, the detected magnetic field relative to a fixed laboratory reference frame was reconstructed from the information acquired by the multi-NV sensor.