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arXiv:1502.05390 · International Journal of Theoretical Physics

The Essence of Nonclassicality: Non-Vanishing Signal Deficit

S. Aravinda, R. Srikanth

Published Jul 21, 2015• 3 citations
arXivHTMLPDF
DOI
10.1007/s10773-015-2735-7

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We argue that the essence of nonclassicality of a bipartite correlation is a positive signal deficit-- the communication cost excess over the available signaling. By this criterion, while violations of Bell-type and contextuality inequalities are necessarily non-classical, some violations of the Leggett-Garg inequality are classical. Further, signaling tends to diminish nonclassical properties, such as intrinsic randomness, no-cloning, complementarity, etc. Signal deficit is shown to have its ultimate origin in intrinsic randomness. A possible analogy of nonclassicality to the metamathematical concept of Gödel incompleteness is noted.
This preprint introduces an exact Hilbert-space decomposition of the Robertson--Schrödinger uncertainty relation into a canonical quantum minimum, a covariance contribution, and a non-negative orthogonal remainder. The remainder, denoted R_DN[ψ], is interpreted as a Dirichlet--Neumann-type compatibility deficit between the position and momentum fluctuation vectors. The work does not introduce a new uncertainty principle. Rather, it gives a geometric interpretation of the Robertson--Schrödinger saturation condition: the deficit vanishes precisely when the momentum fluctuation vector is a complex scalar multiple of the position fluctuation vector. This characterizes Robertson--Schrödinger-saturating Gaussian states, possibly with a quadratic phase. The approach is motivated by the Hamilton--Jacobi relation between coordinate-type and momentum-type boundary data. Examples involving an uncorrelated Gaussian state, a quadratic-phase Gaussian state, and a Fock state illustrate the role of the deficit.
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 of 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.