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 on various physical systems. 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 non-contextual realistic models while being violated by all qutrit states. Thus our inequality will practically facilitate a state-independent test of the quantum contextuality for an indivisible quantum system. We provide also a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a three by three magic cube.It is believed, almost religiously, that every effect has its own cause and the same cause shall lead to the same effect. The predictions of quantum mechanics (QM) are however probabilistic and the effect that different outcomes appear in different runs of a measurement seem to have no definite cause, at least unexplainable using QM alone. Einstein, Podolsky, and Rosen [1] initiated a longlasting quest for a quantum reality by questioning the completeness of quantum mechanics. Hidden variable (HV) models are introduced in order to explain why a certain outcome appears in each run of a measurement, attempting to make QM complete. Years later Kochen, Specker [2], and Bell [3] discovered that quantum mechanics can be completed only by a hidden variable model that is contextual: the outcome of a measurement depends on which compatible observable might be measured alongside. Simply put, Kochen-Specker (KS) theorem states that non-contextual HV models cannot reproduce all the predictions of QM or quantum mechanics is contextual.In any non-contextual HV model all observables have definite values determined only by some HVs λ that are distributed according to a given probability distribution ̺ λ with normalization dλ̺ λ = 1. Two observables are compatible if they can be measured in a single experimental setup and a maximal set of mutually compatible observables defines a context. Non-contextuality is a typical classical property: the value of an observable revealed by a measurement is predetermined by HVs λ only regardless of which compatible observable might be measured alongside. Local realism is a form of non-contextuality enforced by the locality and thus Bell's inequalities [4] are a special form of KS inequalities [5][6][7][8][9], experimentally testable inequalities that are satisfied by all noncontextual HV models, some of which have been tested in recent measurements [10][11][12][13][14][15][16][17][18]. In general KS inequalities reveal the nonclassical nature of single systems demanding neither space-like separation nor entanglement, i.e....

Quantum discord provides a measure for quantifying quantum correlations beyond entanglement and is very hard to compute even for two-qubit states because of the minimization over all possible measurements. Recently a simple algorithm to evaluate the quantum discord for two-qubit X-states is proposed by Ali, Rau and Alber [Phys. Rev. A 81, 042105 (2010)] with minimization taken over only a few cases. Here we shall at first identify a class of X-states, whose quantum discord can be evaluated analytically without any minimization, for which their algorithm is valid, and also identify a family of X-states for which their algorithm fails. And then we demonstrate that this special family of X-states provides furthermore an explicit example for the inequivalence between the minimization over positive operator-valued measures and that over von Neumann measurements. For an important family of two-qubit states, the so called X-states [25], an algorithm has been proposed to calculate their quantum discord with minimization taken over only a few simple cases [26], which is unfortunately impeded by a counter example [27]. In this paper we shall at first identify a vast class of X-states, whose quantum discord can be evaluated analytically without any minimization at all, for which their algorithm is valid, and also identify a family of X-states X m , the so-called maximally discordant mixed states [24], for which the above mentioned algorithm fails. And then for this family of Xstates X m we construct a POVM showing that the quantum discord obtained by minimization over all POVMs is strictly smaller than that over all possible von Neumann measurements.For a given quantum state ̺ of a composite system AB the total amount of correlations, including classical and quantum correlations, is quantified by the quantum mutual information I(ρ) = S(̺ A ) + S(̺ B ) − S(̺) where S(̺) = −Tr(̺ log 2 ̺) denotes the von Neumann entropy and ̺ A , ̺ B are reduced density matrices for subsystem A, B respectively. An alternative version of the mutual information can be defined aswhere the minimum is taken over all possible POVMs {E defines the quantum discord that quantifies the quantum correlation. Also the minimum in Eq.(1) can be taken over all von Neumann measurements [3] and we

We present a single inequality as the necessary and sufficient condition for two unsharp observables of a two-level system to be jointly measurable in a single apparatus and construct explicitly the joint observables. A complementarity inequality arising from the condition of joint measurement, which generalizes Englert's duality inequality, is derived as the trade-off between the unsharpnesses of two jointly measurable observables.PACS numbers: 03.65. Ta, Built in the standard formalism of quantum mechanics, there are mutually exclusive but equally real aspects of quantum systems, as summarized by the complementarity principle of Bohr [1]. Mutually exclusive aspects are often exhibited via noncommuting observables, for which the complementarity is quantitatively characterized by two kinds of uncertainty relationships, namely, the preparation uncertainty relationships (PURs) and the measurement uncertainty relationships (MURs).The PURs stem from the semi positive definiteness of the density matrix describing the quantum state and characterize the predictability of two noncommuting observables in a given quantum state. To test PURs two different projective measurements will be performed on two identically prepared ensembles of the quantum system and these measurements cannot be performed within one experimental setup on a single ensemble.On the other hand MURs characterize the trade-off between the precisions of unsharp measurements of two noncommuting observables in a single experimental setup. The very first effort of Heisenberg [2] in deriving the uncertainty relationships was based on a simultaneous measurement of the position and momentum, with the rigorous form of MUR established recently by Werner [3]. In the interferometry the wave-particle duality between the path-information and the fringe visibility of interference pattern is characterized quantitatively by Englert's duality inequality [4], which turns out to be originated from the joint measurability of two special unsharp observables encoding the path information and the fringe visibility [5]. To establish a general MUR the condition for joint measurement has to be explored, which can be turned into some kinds of MURs when equipped with proper measure of the precisions (e.g., distinguishability).In this Letter we shall consider the joint measurability of two general unsharp observables of a qubit and derive a simple necessary and sufficient condition with joint observables explicitly constructed. We also present a MUR arising from the condition of joint measurement that generalizes Englert's duality inequality.Joint measurability -Generally an observable is described by a positive-operator valued measure (POVM), a set of positive operators {O k } K k=1 summed up to the identity (O k ≥ 0 and k O k = I) with K being the number of outcomes. By definition, a joint measurement of two observables {O k } and {O ′ l } is described by a joint observable {M kl } whose outcomes can be so grouped thatHere we shall consider the qubits, any two-level systems such ...

We derive new inequalities for the probabilities of projective measurements in mutually unbiased bases of a qudit system. These inequalities lead to wider ranges of validity and tighter bounds on entropic uncertainty inequalities previously derived in the literature.

We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of a 3×3 bound entangled state [P. Horodecki, Phys. Lett. A 232, 333 (1997)] is explicitly constructed and a family of d × d bound entangled states is introduced, whose entanglement can be detected by permuting local orthogonal observables. The proposed criterion of separability can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables. There have been many approaches to the problem such as the partial transposition criterion [2,3], the realignment criterion [4], the symmetric extension criterion [5,6], and the equation-solving method [7], to name a few. Many criteria such as the partial transposition criterion and the reduction criterion arise from positive maps that are not completely positive (non-CP). A state is separable if and only if the state keeps its positivity under all non-CP maps [3]. The states with positive partial transposition (PPT) belong to bound entangled states [8] while the states violating the reduction criterion can be distilled, or free entangled [9]. The non-CP maps are not very easy to find and they are not physically realizable. There are also some physical approaches including Bell inequalities [10,11,12], local uncertainty relationships [13,14,15], and entanglement witnesses [9,11]. A 3-setting Bell like inequality is found to be a sufficient and necessary condition for the 2 × 2 system [13]. A local uncertainty relation is found to be violated by bound entangled states [14,15].In this Letter we shall at first construct a family of entanglement witnesses, from which a generalization of the reduction criterion can be derived, based on local orthogonal observables. Then we apply our criterion of separability to several bound entangled states, including a family of bound entangled states where the criterion is sufficient and necessary. Finally we reformulate the criterion in terms of physically measurable quantities, namely Hermitian correlation matrices.Local orthogonal observables.-We consider a d×d system, a bipartite system with two d-level subsystems labelled by A and B, whose Hilbert space is spanned by |m, n = |m ⊗ |n , (m, n = 1, 2, . . . , d). For each system a complete set of local orthogonal observables (LOOs) is a

We show that a single Bell's inequality with two dichotomic observables for each observer, which originates from Hardy's nonlocality proof without inequalities, is violated by all entangled pure states of a given number of particles, each of which may have a different number of energy levels. Thus Gisin's theorem is proved in its most general form from which it follows that for pure states Bell's nonlocality and quantum entanglement are equivalent.

A 3-setting Bell-type inequality enforced by the indeterminacy relation of complementary local observables is proposed as an experimental test of the 2-qubit entanglement. The proposed inequality has an advantage of being a sufficient and necessary criterion of the separability. Therefore any entangled 2-qubit state cannot escape the detection by this kind of tests. It turns out that the orientation of the local testing observables plays a crucial role in our perfect detection of the entanglement.PACS numbers: 03.67. Mn, 03.65.Ud, 03.65.Ta The entanglement or the quantum correlation has become a key concept in the nowadays quantum mechanics. From a fundamental point of view the entangled states of two spacelike separated quantum systems give rise to the question of the completeness of quantum mechanics starting with Einstein-Podolsky-Rosen paper [1] and culminating in Bell's theorem [2]. Form a practical point of view the entanglement has found numerous applications in the quantum information such as quantum computation and quantum teleportation [3,4]. A practical question arises as to how we can detect the entanglement experimentally.As is well known, the entangled states of a bipartite system are states that cannot be prepared locally. More precisely, entangled states are not classically correlated states, i.e, separable states which are convex combinations of product states. The entanglement, though simply defined, is notoriously difficult to detect from both the mathematical and physical point of view. There are plenty separability criteria for the separability of bipartite systems [5,6,7,8], among which the Peres-Horodecki (PH) partial transpose criterion [5,6] is an operationalfriendly criterion and the Bell inequality distinguishes itself as an experimentally doable test for the entanglement.Initially, Bell's inequalities and its generalizations aimed at ruling out various kinds of local realistic theories quantitatively, providing a sufficient and necessary condition for the existence of local hidden variable model in the case of two settings [9,10,11]. Since any separable state admits a local hidden variable model it obeys the Bell inequality. For all separable states of two qubits the Bell-Clauser-Horne-Shimony-Holt inequality [12]holds true. Here A i = a i · σ and B i = b i · τ (i = 1, 2) are two arbitrary sets of local testing observables with σ and τ being the Pauli matrices for two qubits respectively; the norms of the real vectors a i , b i are less than or equal to 1; AB ρ = Tr(ρAB) denotes the average of the observable AB in the state ρ as usual. Since the Bell inequality can be viewed as a property of separable states, it provides a sufficient criterion for the entanglement. One needs only choose properly the testing observables such that the above inequality is violated in order to ensure an entangled states. For multi particles the generalization of the Bell inequalities can be employed to detect the totally separable states [13,14] and fully entangled states [15,16,17], and to classify th...

Quantum Fisher information places the fundamental limit to the accuracy of estimating an unknown parameter. Here we shall provide the quantum Fisher information an operational meaning: a mixed state can be so prepared that a given observable has the minimal averaged variance, which equals exactly to the quantum Fisher information for estimating an unknown parameter generated by the unitary dynamics with the given observable as Hamiltonian. In particular we shall prove that the quantum Fisher information is the convex roof of the variance, as conjectured by Tóth and Petz based on numerical and analytical evidences, by constructing explicitly a pure-state ensemble of the given mixed state in which the averaged variance of a given observable equals to the quantum Fisher information.