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Register a new userSemiclassical Double-Inequality on Heisenberg Uncertainty Relation in 1D
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Authors
Klaus Bering
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We prove a double-inequality for the product of uncertainties for position and momentum of bound states for 1D quantum mechanical systems in the semiclassical limit.
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We prove the uncertainty relation $sigma_T , sigma_E geq hbar/2$ between the time $T$ of detection of a quantum particle on the surface $partial Omega$ of a region $Omegasubset mathbb{R}^3$ containing the particles initial wave function, using the absorbing boundary rule for detection time, and the energy $E$ of the initial wave function. Here, $sigma$ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since $T$ is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrodinger. We also prove that if there is nonzero probability that the particle never reaches $partial Omega$ (in which case we write $T=infty$), and if $sigma_T$ denotes the standard deviation conditional on the event $T<infty$, then $sigma_T , sigma_E geq (hbar/2) sqrt{mathrm{Prob}(T<infty)}$.
In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of hypothetical future measurements, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. We study what happens under such circumstances with an atomic ensemble containing $10^{11}$ $^{87}text{Rb}$ atoms, initiated nearly in the ground state in presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $hat{x}_A$ and $hat{p}_A$ that satisfy $[hat{x}_A,hat{p}_A]=ihbar$. Quantum non-demolition measurements of $hat{p}_A$ before and of $hat{x}_A$ after time $t$ allow precise estimates of both observables at time $t$. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.
We consider the uncertainty bound on the sum of variances of two incompatible observables in order to derive a corresponding steering inequality. Our steering criterion when applied to discrete variables yields the optimum steering range for two qubit Werner states in the two measurement and two outcome scenario. We further employ the derived steering relation for several classes of continuous variable systems. We show that non-Gaussian entangled states such as the photon subtracted squeezed vacuum state and the two-dimensional harmonic oscillator state furnish greater violation of the sum steering relation compared to the Reid criterion as well as the entropic steering criterion. The sum steering inequality provides a tighter steering condition to reveal the steerability of continuous variable states.
Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order $alpha$ rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.
In this paper, Wielandts inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bound, dichotomy theorems for the zero--error capacity of quantum channels and for the Matrix Product State (MPS) dimension of ground states of frustration-free Hamiltonians are derived. The obtained inequalities also imply new bounds on the required interaction-range of Hamiltonians with unique MPS ground state.
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