arXiv subjects

Gebhard Gruebl

Publications and source records attributed to Gebhard Gruebl.

7 recordsLinked to original sources

arXiv2012

Reaching Fleming's dicrimination bound

Any rule for identifying a quantum system's state within a set of two non-orthogonal pure states by a single measurement is flawed. It has a non-zero probability of either yielding the wrong result or leaving the query undecided. This also holds if the measurement of an observable $A$ is repeated on a finite sample of $n$ state copies. We formulate a state identification rule for such a sample. This rule's probability of giving the wrong result turns out to be bounded from above by $1/nδ_{A}^{2}$ with $δ_{A}=| _{1}- _{2}|/(Δ_{1}A+Δ_{2}A).$ A larger $δ_{A}$ results in a smaller upper bound. Yet, according to Fleming, $δ_{A}$ cannot exceed $\tanθ$ with $θ\in(0,π/2) $ being the angle between the pure states under consideration. We demonstrate that there exist observables $A$ which reach the bound $\tanθ$ and we determine all of them.

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arXiv2010

Non-differentiable Bohmian trajectories

A solution $ψ$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown ψ/mψ\right) .$ In the case of one specific non-differentiable weak solution $Ψ$ we show how Bohmian trajectories can be obtained for $Ψ$ from the trajectories of a sequence $Ψ_{n}\rightarrow Ψ.$ (For any real $t$ the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $Ψ$ in the Hilbert space on which the Schrödinger evolution $% Ψ\mapsto e^{-iht}Ψ$ acts.

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arXiv2010

Lower bound for the mean square distance between classical and quantum spin correlations

Bell's theorem prevents local Kolmogorov-simulations of the singlet state of two spin-1/2 particles. We derive a positive lower bound for the $L^{2}% $-distance between the quantum mechanical spin singlet anticorrelation function $\cos$ and any of its classical approximants $C$ formed by the stationary autocorrelation functions of mean-square-continuous, $2π$-periodic, $\pm1$-valued, stochastic processes. This bound is given by $\Vert C-\cos\Vert \geq(1-\frac{8}{π^{2}}) /\sqrt{2}\approx0.133\,95.$

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arXiv2003

Bohmian arrival time without trajectories

The computation of detection probabilities and arrival time distributions within Bohmian mechanics in general needs the explicit knowledge of a relevant sample of trajectories. Here it is shown how for one-dimensional systems and rigid inertial detectors these quantities can be computed without calculating any trajectories. An expression in terms of the wave function and its spatial derivative, both restricted to the boundary of the detector's spacetime volume, is derived for the general case, where the probability current at the detector's boundary may vary its sign.

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arXiv2002

Bohmian trajectories and Klein's paradox

We compute the Bohmian trajectories of the incoming scattering plane waves for Klein's potential step in explicit form. For finite norm incoming scattering solutions we derive their asymptotic space-time localization and we compute some Bohmian trajectories numerically. The paradox, which appears in the traditional treatments of the problem based on the outgoing scattering asymptotics, is absent.

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arXiv2002

Time of Arrival from Bohmian Flow

We develop a new conception for the quantum mechanical arrival time distribution from the perspective of Bohmian mechanics. A detection probability for detectors sensitive to quite arbitrary spacetime domains is formulated. Basic positivity and monotonicity properties are established. We show that our detection probability improves and generalises earlier proposals by Leavens and McKinnon. The difference between the two notions is illustrated through application to a free wave packet.

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arXiv2002

The quantum measurement problem enhanced

The quantum measurement problem as formalised by Bassi and Ghirardi [Phys. Lett. A 275 (2000)373] without taking recourse to sharp apparatus observables is extended to cover impure initial states.

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