Generalized conditional entropy optimization for qudit-qubit states

We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local projective measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the post-measurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest non-trivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\times 3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.Comment: 11 pages, 6 figures. Final published versio

Generalized conditional entropy in bipartite quantum systems

We analyze, for a general concave entropic form, the associated conditional entropy of a quantum system A+B, obtained as a result of a local measurement on one of the systems (B). This quantity is a measure of the average mixedness of A after such measurement, and its minimum over all local measurements is shown to be the associated entanglement of formation between A and a purifying third system C. In the case of the von Neumann entropy, this minimum determines also the quantum discord. For classically correlated states and mixtures of a pure state with the maximally mixed state, we show that the minimizing measurement can be determined analytically and is universal, i.e., the same for all concave forms. While these properties no longer hold for general states, we also show that in the special case of the linear entropy, an explicit expression for the associated conditional entropy can be obtained, whose minimum among projective measurements in a general qudit-qubit state can be determined analytically, in terms of the largest eigenvalue of a simple 3x3 correlation matrix. Such minimum determines the maximum conditional purity of A, and the associated minimizing measurement is shown to be also universal in the vicinity of maximal mixedness. Results for X states, including typical reduced states of spin pairs in XY chains at weak and strong transverse fields, are also provided and indicate that the measurements minimizing the von Neumann and linear conditional entropies are typically coincident in these states, being determined essentially by the main correlation. They can differ, however, substantially from that minimizing the geometric discord.Comment: 11 pages, 2 figures; References adde

Quantum discord and information deficit in spin chains

We examine the behavior of quantum correlations of spin pairs in a finite anisotropic $XY$ spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one way-information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.Comment: 14 pages, 5 figure

History state formalism for Dirac's theory

We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Dirac's equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Dirac's norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $\tau$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of an hypothetical superposition of different masses. Analytical expressions for both space-time density and electron-time entanglement are provided for two particular families of electron's states, the former including Pryce localized particles.Comment: 9 pages, 2 figures, final versio

Measurements, quantum discord and parity in spin 1 systems

We consider the evaluation of the quantum discord and other related measures of quantum correlations in a system formed by a spin 1 and a complementary spin system. A characterization of general projective measurements in such system in terms of spin averages is thereby introduced, which allows to easily visualize their deviation from standard spin measurements. It is shown that the measurement optimizing these measures corresponds in general to a non-spin measurement. The important case of states that commute with the total $S_z$ spin parity is discussed in detail, and the general stationary measurements for such states (parity preserving measurements) are identified. Numerical and analytical results for the quantum discord, the geometric discord and the one way information deficit in the relevant case of a mixture of two aligned spin 1 states are also presented.Comment: 6 pages, 2 figures, References adde