Hierarchic Superposition Revisited

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory

Concurrence of superposition

The bounds on concurrence of the superposition state in terms of those of the states being superposed are studied in this paper. The bounds on concurrence are quite different from those on the entanglement measure based on von Neumann entropy (Phys. Rev. Lett. 97, 100502 (2006)). In particular, a nonzero lower bound can be provided if the states being superposed are properly constrained.Comment: 4 page

Hierarchic Superposition Revisited

Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory

Measurement- and comparison-based sizes of Schrödinger cat states of light

We extend several measurement-based definitions of effective superposition size to coherent state superpositions with branches composed of either single coherent states or tensor products of coherent states. These measures of superposition size depend on determining the maximal quantum distinguishability of certain states associated with the superposition state: e.g., in one measure, the maximal distinguishability of the branches of the superposition is considered as in quantum binary decision theory; in another measure, the maximal distinguishability of the initial superposition and its image after a one-parameter evolution generated by a local Hermitian operator is of interest. The scaling of the size of the superposition with the number of modes and mode intensity (i.e., photon number) is compared to the scaling of certain geometric properties of the Wigner function of the superposition and also to the superposition size estimated experimentally from decoherence. We also apply earlier comparison-based methods for determining macroscopic superposition size that require a reference Greenberger-Horne-Zeilinger (GHZ) state. The case of a hierarchical Schrödinger cat state with branches composed of smaller superpositions is also analyzed from a measurement-based perspective. © 2014 American Physical Society

Quantum resource studied from the perspective of quantum state superposition

Quantum resources,such as discord and entanglement, are crucial in quantum information processing. In this paper, quantum resources are studied from the aspect of quantum state superposition. We define the local superposition (LS) as the superposition between basis of single part, and nonlocal superposition (NLS) as the superposition between product basis of multiple parts. For quantum resource with nonzero LS, quantum operation must be introduced to prepare it, and for quantum resource with nonzero NLS, nonlocal quantum operation must be introduced to prepare it. We prove that LS vanishes if and only if the state is classical and NLS vanishes if and only if the state is separable. From this superposition aspect, quantum resources are categorized as superpositions existing in different parts. These results are helpful to study quantum resources from a unified frame.Comment: 9 pages, 4 figure

A Comparison of Superposition Coding Schemes

There are two variants of superposition coding schemes. Cover's original superposition coding scheme has code clouds of the identical shape, while Bergmans's superposition coding scheme has code clouds of independently generated shapes. These two schemes yield identical achievable rate regions in several scenarios, such as the capacity region for degraded broadcast channels. This paper shows that under the optimal maximum likelihood decoding, these two superposition coding schemes can result in different rate regions. In particular, it is shown that for the two-receiver broadcast channel, Cover's superposition coding scheme can achieve rates strictly larger than Bergmans's scheme.Comment: 5 pages, 3 figures, 1 table, submitted to IEEE International Symposium on Information Theory (ISIT 2013

Fast macroscopic-superposition-state generation by coherent driving

We propose a scheme to generate macroscopic superposition states in spin ensembles, where a coherent driving field is applied to accelerate the generation of macroscopic superposition states. The numerical calculation demonstrates that this approach allows us to generate a superposition of two classically distinct states of the spin ensemble with a high fidelity above 0.96 for 300 spins. For the larger spin ensemble, though the fidelity slightly decline, it maintains above 0.85 for an ensemble of 500 spins. The time to generate a macroscopic superposition state is also numerically calculated, which shows that the significantly shortened generation time allows us to achieve such macroscopic superposition states within a typical coherence time of the system.Comment: 17 pages, 15 figure

Solution of the problem of definite outcomes of quantum measurements

Theory and experiment both demonstrate that an entangled quantum state of two subsystems is neither a superposition of states of its subsystems nor a superposition of composite states but rather a coherent superposition of nonlocal correlations between incoherently mixed local states of the two subsystems. Thus, even if one subsystem happens to be macroscopic as in the entangled "Schrodinger's cat" state resulting from an ideal measurement, this state is not the paradoxical macroscopic superposition it is generally presumed to be. It is, instead, a "macroscopic correlation," a coherent quantum correlation in which one of the two correlated sub-systems happens to be macroscopic. This clarifies the physical meaning of entanglement: When a superposed quantum system A is unitarily entangled with a second quantum system B, the coherence of the original superposition of different states of A is transferred to different correlations between states of A and B, so the entangled state becomes a superposition of correlations rather than a superposition of states. This transfer preserves unitary evolution while permitting B to be macroscopic without entailing a macroscopic superposition. This resolves the "problem of outcomes" but is not a complete resolution of the measurement problem because the entangled state is still reversible.Comment: 21 pages, 3 figures, 1 tabl