Contextuality Can be Verified with Noncontextual Experiments

We uncover features of generalized contextuality by connecting it to the Kirkwood-Dirac (KD) quasiprobability distribution. Quantum states can be represented by KD distributions, which take values in the complex unit disk. Only for “KD-positive” states are the KD distributions joint probability distributions. A KD distribution can be measured by a series of weak and projective measurements. We design such an experiment and show that it is contextual iff the underlying state is not KD positive. We analyze this connection with respect to mixed KD-positive states that cannot be decomposed as convex combinations of pure KD-positive states. Our result is the construction of a noncontextual experiment that enables an experimenter to verify contextuality.

Kirkwood-Dirac classical pure states

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

Majorana representation (MR) of symmetric N-qubit pure states has been used successfully in entanglement classification. Generalization of this has been a long standing open problem due to the difficulties faced in the construction of a Majorana like geometric representation for symmetric mixed state. We have overcome this problem by developing a method of classifying local unitary (LU) equivalent classes of symmetric N-qubit mixed states based on the geometrical multiaxial representation (MAR) of the density matrix. In addition to the two parameters defined for the entanglement classification of the symmetric pure states based on MR, namely, diversity degree and degeneracy configuration, we show that another parameter called rank needs to be introduced for symmetric mixed state classification. Our scheme of classification is more general as it can be applied to both pure and mixed states. To bring out the similarities/differences between the MR and MAR, N-qubit GHZ state is taken up for a detailed study. We conclude that pure state classification based on MR is not a special case of our classification scheme based on MAR. We also give a recipe to identify the most general symmetric N-qubit pure separable states. The power of our method is demonstrated using several well-known examples of symmetric two-qubit pure and mixed states as well as three-qubit pure states. Classification of uniaxial, biaxial and triaxial symmetric two-qubit mixed states which can be produced in the laboratory is studied in detail.

Selective weak measurement reveals super-ergotropy

In this paper we establish a unique factorization theorem for pure quantum states expressed in computational basis. We show that there always exists unique factorization for any given N-qubit pure quantum state in terms of the tensor product of non-factorable or ``prime'' pure quantum states. This result is based on a simple criterion: Given N-qubit pure quantum state in computational basis can be factorized as the tensor product of an m-qubit pure quantum state and an n-qubit pure quantum state, where (m + n) = N, if and only if the rank of the certain associated matrix is equal to one. This simple criterion leads to a factorization algorithm which when applied to an N-qubit pure quantum state factorizes that state into the tensor product of non-factorable or ``prime'' pure quantum states. This paper shows that for any given N-qubit pure quantum state the said factorization always ``exists'' and is ``unique''. We demonstrated our work here on a computational basis. PACS Number: 03.67.Mn, 03.65.Ca, 03.65.Ud

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, i.e., that its integral is one and that the marginal properties are satisfied. However, this is generally not true. We introduced a class of quantum states for which this property is satisfied; these states are dubbed “Feichtinger states” because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980s by H. Feichtinger. The properties of these states were studied, giving us the opportunity to prove an extension to the general case of a result due to Jaynes on the non-uniqueness of the statistical ensemble, generating a density operator.

Classicality and anticlassicality measures of pure and mixed quantum states

Representation of quantum states by statistical ensembles on the quantum phase space in the Hamiltonian form of quantum mechanics is analyzed. Various mathematical properties and some physical interpretations of the equivalence classes of ensembles representing a mixed quantum state in the Hamiltonian formulation are examined. In particular, non-uniqueness of the quantum phase space probability density associated with the quantum mixed state, Liouville dynamics of the probability densities and the possibility to represent the reduced states of bipartite systems by marginal distributions are discussed in detail. These considerations are used to study ensembles of hybrid quantum-classical systems. In particular, nonlinear evolution of a single hybrid system in a pure state and unequal evolutions of initially equivalent ensembles are discussed in the context of coupled hybrid systems.

In the Aharonov–Albert–Vaidman (AAV) weak measurement, it is assumed that the measuring device or the pointer is in a quantum mechanical pure state. In reality, however, it is often not the case. In this paper, we generalize the AAV weak measurement scheme to include more generalized situations in which the measuring device is in a mixed state. We also report an optical implementation of the weak value measurement in which the incoherent pointer is realized with the pseudo-thermal light. The theoretical and experimental results show that the measuring device under the influence of partial decoherence could still be used for amplified detection of minute physical changes and is applicable for implementing the weak value measurement for massive particles.

In this paper, we propose a protocol to deterministically teleportan unknown mixed state of qubit by utilizing a maximally bipartiteentangled state of qubits as quantum channel. If a non-maximallyentangled bipartite pure state is employed as quantum channel, theunknown mixed quantum state of qubit can be teleported with1−(1−C2)1/2 probability, where C is the concurrence of thequantum channel. The protocol can also be generalized to teleport amixed state of qudit or a multipartite mixed state. More importantpurpose is that, on the basis of the protocol, the teleportation ofan arbitrary multipartite (pure or mixed) quantum state can bedecomposed into the teleportation of each subsystem by employingseparate entangled states as quantum channels. In the case ofdeterministic teleportation, Bob only needs to perform unitarytransformations on his single particles in order to recover theinitial teleported multipartite quantum state.

Quantum states are the key mathematical objects in quantum theory. It is therefore surprising that physicists have been unable to agree on what a quantum state truly represents. One possibility is that a pure quantum state corresponds directly to reality. However, there is a long history of suggestions that a quantum state (even a pure state) represents only knowledge or information about some aspect of reality. Here we show that any model in which a quantum state represents mere information about an underlying physical state of the system, and in which systems that are prepared independently have independent physical states, must make predictions which contradict those of quantum theory.

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function (representing a pure state) but by a density matrix (representing a mixed state). I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research.

Investigating positivity of polynomials over the complex unit disc is a relevant problem in electrical and computer engineering. This paper provides two sufficient and necessary conditions for solving this problem via linear matrix inequalities (LMIs). These conditions are obtained by exploiting trigonometric transformations, a key tool for the representation of polynomials, and results from the theory of positive polynomials. Some numerical examples illustrate the proposed conditions.

The restricted numerical range WR(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A—a normalized probability distribution on the complex plane supported in WR(A). Its value at point is equal to the probability that the expectation value 〈ψ|A|ψ〉 is equal to z, where |ψ〉 represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size D = NANB we analyse the geometry of sets of separable and maximally entangled states of the NA × NB composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow, we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 23-dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.