A note on Hardy's theorem

Hardy's theorem for the Riemann zeta-function $\zeta(s)$ says that it admits infinitely many complex zeros on the line $\Re({s}) = \frac{1}{2}$. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.Comment: 9 pages; To appear in Hardy Ramanujan Journa

From quantum stochastic differential equations to Gisin-Percival state diffusion

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy (Comm. Math. Phys. 93, 301 (1984)) and exploiting the Wiener-Ito-Segal isomorphism between the Boson Fock reservoir space $\Gamma(L^2(\mathbb{R}_+)\otimes (\mathbb{C}^{n}\oplus \mathbb{C}^{n}))$ and the Hilbert space $L^2(\mu)$, where $\mu$ is the Wiener probability measure of a complex $n$-dimensional vector-valued standard Brownian motion $\{\mathbf{B}(t), t\geq 0\}$, we derive a non-linear stochastic Schrodinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion $\mathbf{B}$. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation (J. Phys. A, 167, 315 (1992)). This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a radomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.Comment: 28 pages, one pdf figure. An error in the multiplying factor in Eq. (102) corrected. To appear in Journal of Mathematical Physic

Separability bounds on multiqubit moments due to positivity under partial transpose

Positivity of the density operator reflects itself in terms of sequences of inequalities on observable moments. Uncertainty relations for non-commuting observables form a subset of these inequalities. In addition, criterion of positivity under partial transposition (PPT) imposes distinct bounds on moments, violations of which signal entanglement. We present bounds on some novel sets of composite moments, consequent to positive partial transposition of the density operator and report their violation by entangled multiqubit states. In particular, we derive separability bounds on a multiqubit moment matrix (based on PPT constraints on bipartite divisions of the density matrix) and show that three qubit pure states with non-zero tangle violate these PPT moment constraints. Further, we recover necessary and sufficient condition of separability in a multiqubit Werner state through PPT bounds on moments.Comment: 16 pages, no figures, minor revisions, references added; To appear in Phys. Rev.

Quantumness of correlations and entanglement

Generalized measurement schemes on one part of bipartite states, which would leave the set of all separable states insensitive are explored here to understand quantumness of correlations in a more general perspecitve. This is done by employing linear maps associated with generalized projective measurements. A generalized measurement corresponds to a quantum operation mapping a density matrix to another density matrix, preserving its positivity, hermiticity and traceclass. The Positive Operator Valued Measure (POVM) -- employed earlier in the literature to optimize the measures of classical/quatnum correlations -- correspond to completely positive (CP) maps. The other class, the not completely positive (NCP) maps, are investigated here, in the context of measurements, for the first time. It is shown that that such NCP projective maps provide a new clue to the understanding the quantumness of correlations in a general setting. Especially, the separability-classicality dichotomy gets resolved only when both the classes of projective maps (CP and NCP) are incorporated as optimizing measurements. An explicit example of a separable state -- exhibiting non-zero quantumn discord when possible optimizing measurements are restricted to POVMs -- is re-examined with this extended scheme incorporating NCP projective maps to elucidate the power of this approach.Comment: 14 pages, no figures, revision version, Accepted for publication in the Special Issue of the International Journal of Quantum Information devoted to "Quantum Correlations: entanglement and beyond

A scheme for amplification and discrimination of photons

A scheme for exploring photon number amplification and discrimination is presented based on the interaction of a large number of two-level atoms with a single mode radiation field. The fact that the total number of photons and atoms in the excited states is a constant under time evolution in Dicke model is exploited to rearrange the atom-photon numbers. Three significant predictions emerge from our study: Threshold time for initial exposure to photons, time of perception (time of maximum detection probability), and discrimination of first few photon states.Comment: 8 pages, 3 figures, RevteX, Minor revision, References adde