Uncertainty Inequality Research Articles

Generalized uncertainty relations and coherent and squeezed states.

Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.

An uncertainty inequality involving $L^1$ norms

An uncertainty inequality involving $L^1$ norms

Uncertainty Inequalities for Fourier Series of Pairs of Reciprocal Positive Functions

In the early 1930s, Wiener proved that if f ( x ) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g ( x )=1/ f ( x ) is also absolutely convergent [ 8 , pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [ 4 , pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g ( x )= F ( f ( x )), where F is holomorphic in an open subset of [Copf ] that contains the range of f ( x ) (for x ∈ℝ). In this context, Wiener's original problem corresponds to the choice F ( z )=1/ z . In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ ( n ) and ĝ ( n ) as n →∞. We shall focus on g =1/ f , but we shall also study the general case g = F ( f ). In either case, there are obviously no constraints when f is a constant function. Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ ( n ) and ĝ ( n ) cannot vanish too quickly at the same time as n →∞, unless f ( x ) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ( x ) and its Fourier Transform ϕcirc;( u ) cannot vanish too quickly at the same time as x →∞ and u →∞, unless ϕ( x ) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [ 5 ]; for a thorough and recent introduction to this vast subject, see [ 3 ].

Uncertainty inequalities for the continuous wavelet transform

Uncertainty inequalities of Heisenberg-type are established for the continuous wavelet transform resting on classical Heisenberg and Heisenberg-von Keumann principles.

The uncertainty principle: A mathematical survey

We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

Generalized Heisenberg commutation relations and uncertainty inequality for relativistic harmonic oscillators

Generalized Heisenberg commutation relations and a Hamiltonian are proposed for relativistic harmonic oscillators. Using a Lie algebra and Schwinger's dual-bosonic representation of x, p, and the Hamiltonian, expressions for the quantized energy are derived and a revised uncertainty inequality for ( Δx) 2( Δp) 2 is given.

Generalized uncertainty relations and characteristic invariants for the multimode states.

The close relationship between the zero-point energy, the uncertainty relation, coherent states, squeezed states, and correlated states for one mode is investigated. This group theoretic perspective of the problem enables the parametrization and identification of their multimode generalization. A simple and efficient method of determining the canonical structure of the generalized correlated states is presented. Implication of canonical commutation relations for correlations are not exhausted by the Heisenberg uncertainty relation, not even by the Schr\"odinger-Robertson uncertainty inequality, but there are relations in the multimode case that are the generalization of the Schr\"odinger-Robertson relation.

Uncertainty inequalities for hopf algebras

The uncertainty principle was recently formulated for locally compact Abelian groups and for finite groups. In this paper similar inequalities are found for certain Hopf algebras. Some cases of equ...

An uncertainty inequality for groups of order pq

We are concerned with some relations between the support of a function defined on a finite group and the support of its Fourier transform. Let D p,q be the non-commutative group of order pq , where p, q are primes, and p | q −1. We prove an uncertainty-type inequality for D N p,q = {( x 1 , …, x N ): x i ∈ D p,q }, which has the following application: If H is a subgroup of D N p,q and x 1 … x N = 1 for all ( x 1 , …, x N ) ∈ H , then ( D N p,q : H ) ⩾ q ½ p ( N −1)/2 .

Uncertainty principle, quantum fluctuations, and broken symmetries

We discuss a natural generalization of the uncertainty inequality holding in the case of non-hermitian operators. The inequality is employed to derive useful constraints on the behaviour of quantum fluctuations in problems with continuous group symmetries. Applications to Bose superfluids, antiferromagnets and crystals at zero temperature are discussed. We provide, in particular, a simple and direct proof of the absence of long range order at zero temperature in the1D case. A new inequality involving the spectral function at finite temperature is finally derived.

An uncertainty inequality and zero subsums

Let G be a finite abelian group, and let m be the maximal order of elements in G. It is shown that if s>m 1 + log |G| m , then any sequence a 1, …, a s of elements in G, has a on-empty subsequence which sums to zero. The result is a consequence of an inequality for the finite Fourier transform.

Some uncertainty inequalities

We prove an uncertainty inequality for the Fourier transform on the Heisenberg group analogous to the classical uncertainty inequality for the Euclidean Fourier transform. Inequalities of similar form are obtained for the Hermite and Laguerre expansions.

Local Uncertainty Inequalities for Locally Compact Groups

Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left [ {0,\tfrac {1} {2}} \right .)$ there exists a constant ${K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, \[ {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f)) d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}\] where ${\phi _\theta }$ is a certain weight function on $G$ (for which an explicit formula is given). When $G = {{\mathbf {R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = |x{|^{k\theta }}$.

Local Uncertainty Inequalities for Compact Groups

Local Uncertainty Inequalities for Compact Groups

Principal squeezing of vacuum fluctuations

With regard to the squeezing detection process a more substantial definition of squeezing is proposed based on the sharpened Heisenberg uncertainty inequality. This notion involves the quadrature correlation and is rotationally invariant.

Local uncertainty inequalities for compact groups

Conditions are established on α , β ∈ R \alpha ,\beta \in {\mathbf {R}} for there to exist a constant K = K ( α , β ) K = K(\alpha ,\beta ) such that \[ ∑ γ ∈ E d ( γ ) tr ⁡ ( f ^ ( γ ) ∗ f ^ ( γ ) ) ≤ K ( ∑ γ ∈ E d ( γ ) 2 ) α ‖ w β f ‖ 2 \sum \limits _{\gamma \in E} {d(\gamma )\operatorname {tr} (\hat f{{(\gamma )}^ * }\hat f(\gamma )) \leq K{{\left ( {\sum \limits _{\gamma \in E} {d{{(\gamma )}^2}} } \right )}^\alpha }{{\left \| {{w^\beta }f} \right \|}_2}} \] for all f ∈ L 1 ( G ) f \in {L^1}(G) and E ⊆ G ^ E \subseteq \hat G where G G is a compact metric group, G ^ \hat G is its dual, f ^ \hat f is the Fourier transform of f f and w : G → R + w:G \to {{\mathbf {R}}^ + } is the function taking x ∈ G x \in G to the area of the ball in G G with centre e e and x x on its boundary. This is followed by a partial analogy for compact riemannian manifolds.

Local uncertainty inequalities for locally compact groups

Let G G be a locally compact unimodular group equipped with Haar measure m m , G ^ \hat G its unitary dual and μ \mu the Plancherel measure (or something closely akin to it) on G ^ \hat G . When G G is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given θ ∈ [ 0 , 1 2 ) \theta \in \left [ {0,\tfrac {1} {2}} \right .) there exists a constant K θ {K_\theta } such that for all f f in a certain class of functions on G G and all measurable E ⊆ G ^ E \subseteq \hat G , \[ ( ∫ E Tr ⁡ ( π ( f ) ∗ π ( f ) ) d μ ( π ) ) 1 / 2 ⩽ K θ μ ( E ) θ | | ϕ θ f | | 2 {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f))\,d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2} \] where ϕ θ {\phi _\theta } is a certain weight function on G G (for which an explicit formula is given). When G = R k G = {{\mathbf {R}}^k} the inequality has been established with ϕ θ ( x ) = | x | k θ {\phi _\theta }(x) = |x{|^{k\theta }} .

Extensions of the Heisenberg‐Weyl inequality

In this paper a number of generalizations of the classical Heisenberg‐Weyl uncertainty inequality are given. We prove the n‐dimensional Hirschman entropy inequality (Theorem 2.1) from the optimal form of the Hausdorff‐Young theorem and deduce a higher dimensional uncertainty inequality (Theorem 2.2). From a general weighted form of the Hausdorff‐Young theorem, a one‐dimensional weighted entropy inequality is proved and some weighted forms of the Heisenberg‐Weyl inequalities are given.

Local Uncertainty Inequalities for Fourier Series

Local Uncertainty Inequalities for Fourier Series

Local uncertainty inequalities for Fourier series

Necessary and sufficient conditions are given on α \alpha , β \beta and t t for there to exist a constant K K such that \[ ( ∑ n ∈ E | f ^ ( n ) | 2 ) 1 / 2 ⩽ K | E | α ‖ f | x | β ‖ t {\left ( {{{\sum \limits _{n \in E} {\left | {\hat f(n)} \right |} }^2}} \right )^{1/2}} \leqslant K{\left | E \right |^\alpha }{\left \| {f{{\left | x \right |}^\beta }} \right \|_t} \] for all f ∈ L 1 ( T d ) f \in {L^1}({T^d}) and finite E ⊂ Z d E \subset {Z^d} .