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

In this Letter, we point out that the widely used quantitative conditions in the adiabatic theorem are insufficient in that they do not guarantee the validity of the adiabatic approximation. We also reexamine the inconsistency issue raised by Marzlin and Sanders [Phys. Rev. Lett. 93, 160408 (2004)] and elucidate the underlying cause.

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 ...

A kinematic approach to the geometric phase for mixed quantal states in nonunitary evolution is proposed. This phase is manifestly gauge invariant and can be experimentally tested in interferometry. It leads to well-known results when the evolution is unitary.

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The novel experimental realization of three-level optical quantum systems is presented. We use the polarization state of biphotons to generate a specific sequence of states that are used in the extended version of four-state QKD protocol quantum key distribution protocol. We experimentally verify the orthogonality of the basic states and demonstrate the ability to easily switch between them. The tomography procedure is employed to reconstruct the density matrices of generated states.

We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it, along with additional conditions, is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all N-dimensional quantum systems. Moreover, we illustrate the use of the general quantitative criterion in some physical models.