Background Independent Quantum Mechanics, Metric of Quantum States, and Gravity: A Comprehensive Perspective

This paper presents a comprehensive perspective of the metric of quantum states with a focus on the background independent metric structures. We also explore the possibilities of geometrical formulations of quantum mechanics beyond the quantum state space and Kahler manifold. The metric of quantum states in the classical configuration space with the pseudo-Riemannian signature and its possible applications are explored. On contrary to the common perception that a metric for quantum state can yield a natural metric in the configuration space with the limit when Planck constant vanishes, we obtain the metric of quantum states in the configuration space without imposing this limiting condition. Here, Planck constant is absorbed in the quantity like Bohr radii. While exploring the metric structure associated with Hydrogen like atom, we witness another interesting finding that the invariant lengths appear in the multiple of Bohr radii.Comment: 25 Pages;journal reference added:Published in- Int. J. Theor. Phys. 46 (2007) 3216-3229. References revise

Inseparability of Quantum Parameters

In this work, we show that 'splitting of quantum information' [6] is an impossible task from three different but consistent principles of unitarity of Quantum Mechanics, no-signalling condition and non increase of entanglement under Local Operation and Classical Communication.Comment: 9 pages, Presented in Quantum Computing Back Action in IIT Kanpur (2006). Accepted in International Journal of Theoretical Physic

Self Replication and Signalling

It is known that if one could clone an arbitrary quantum state one could send signal faster than the speed of light. However it remains interesting to see that if one can perfectly self replicate an arbitrary quantum state, does it violate the no signalling principle? Here we see that perfect self replication would also lead to superluminal signalling.Comment: Modified version of quant-ph/0510221, Accepted in International Journal of Theoretical Physic

Fluctuation, time-correlation function and geometric Phase

We establish a fluctuation-correlation theorem by relating the quantum fluctuations in the generator of the parameter change to the time integral of the quantum correlation function between the projection operator and force operator of the ``fast'' system. By taking a cue from linear response theory we relate the quantum fluctuation in the generator to the generalised susceptibility. Relation between the open-path geometric phase, diagonal elements of the quantum metric tensor and the force-force correlation function is provided and the classical limit of the fluctuation-correlation theorem is also discussed.Comment: Latex, 12 pages, no figures, submitted to J. Phys. A: Math & Ge

General impossible operations in quantum information

We prove a general limitation in quantum information that unifies the impossibility principles such as no-cloning and no-anticloning. Further, we show that for an unknown qubit one cannot design a universal Hadamard gate for creating equal superposition of the original and its complement state. Surprisingly, we find that Hadamard transformations exist for an unknown qubit chosen either from the polar or equatorial great circles. Also, we show that for an unknown qubit one cannot design a universal unitary gate for creating unequal superpositions of the original and its complement state. We discuss why it is impossible to design a controlled-NOT gate for two unknown qubits and discuss the implications of these limitations.Comment: 15 pages, no figures, Discussion about personal quantum computer remove

Sum Uncertainty Relation in Quantum Theory

We prove a new sum uncertainty relation in quantum theory which states that the uncertainty in the sum of two or more observables is always less than or equal to the sum of the uncertainties in corresponding observables. This shows that the quantum mechanical uncertainty in any observable is a convex function. We prove that if we have a finite number $N$ of identically prepared quantum systems, then a joint measurement of any observable gives an error $\sqrt N$ less than that of the individual measurements. This has application in quantum metrology that aims to give better precision in the parameter estimation. Furthermore, this proves that a quantum system evolves slowly under the action of a sum Hamiltonian than the sum of individuals, even if they are non-commuting.Comment: LaTeX file, no figure, 4 page

Minimum cbits for remote preperation and measurement of a qubit

We show that a qubit chosen from equatorial or polar great circles on a Bloch spehere can be remotely prepared with one cbit from Alice to Bob if they share one ebit of entanglement. Also we show that any single particle measurement on an arbitrary qubit can be remotely simulated with one ebit of shared entanglement and communication of one cbit.Comment: Latex, 7 pages, minor changes, references adde

Universal quantum Controlled-NOT gate

An investigation of an optimal universal unitary Controlled-NOT gate that performs a specific operation on two unknown states of qubits taken from a great circle of the Bloch sphere is presented. The deep analogy between the optimal universal C-NOT gate and the `equatorial' quantum cloning machine (QCM) is shown. In addition, possible applications of the universal C-NOT gate are briefly discussed.Comment: 18 reference

Quantum Dissension: Generalizing Quantum Discord for Three-Qubit States

We introduce the notion of quantum dissension for a three-qubit system as a measure of quantum correlations. We use three equivalent expressions of three-variable mutual information. Their differences can be zero classically but not so in quantum domain. It generalizes the notion of quantum discord to a multipartite system. There can be multiple definitions of the dissension depending on the nature of projective measurements done on the subsystems. As an illustration, we explore the consequences of these multiple definitions and compare them for three-qubit pure and mixed GHZ and W states. We find that unlike discord, dissension can be negative. This is because measurement on a subsystem may enhance the correlations in the rest of the system. This approach can pave a way to generalize the notion of quantum correlations in the multiparticle setting.Comment: 9 pages 6 figures typo fixed and some arguments adde

Necessity of integral formalism

To describe the physical reality, there are two ways of constructing the dynamical equation of field, differential formalism and integral formalism. The importance of this fact is firstly emphasized by Yang in case of gauge field [Phys. Rev. Lett. 33 (1974) 445], where the fact has given rise to a deeper understanding for Aharonov-Bohm phase and magnetic monopole [Phys. Rev. D. 12 (1975) 3845]. In this paper we shall point out that such a fact also holds in general wave function of matter, it may give rise to a deeper understanding for Berry phase. Most importantly, we shall prove a point that, for general wave function of matter, in the adiabatic limit, there is an intrinsic difference between its integral formalism and differential formalism. It is neglect of this difference that leads to an inconsistency of quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 (2004) 160408]. It has been widely accepted that there is no physical difference of using differential operator or integral operator to construct the dynamical equation of field. Nevertheless, our study shows that the Schrodinger differential equation (i.e., differential formalism for wave function) shall lead to vanishing Berry phase and that the Schrodinger integral equation (i.e., integral formalism for wave function), in the adiabatic limit, can satisfactorily give the Berry phase. Therefore, we reach a conclusion: There are two ways of describing physical reality, differential formalism and integral formalism; but the integral formalism is a unique way of complete description.Comment: 13Page; Schrodinger differential equation shall lead to vanishing Berry phas