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arXiv:quant-ph/0005086 · Journal of Physics A Mathematical and General

State extended uncertainty relations

D. A. Trifonov

Published Aug 3, 2000• 35 citations
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Abstract

A scheme for construction of uncertainty relations for n observables and m states is presented. Several lowest-order inequalities are displayed and briefly discussed. For two states |ψ⟩ and |ϕ⟩ and canonical observables the (entangled) extension of the Heisenberg relation reads [Δ p (ψ)] 2 [Δ q (ϕ)] 2 + [Δ p (ϕ)] 2 [Δ q (ψ)] 2 ⩾½.

DOI
10.1088/0305-4470/33/32/102

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A scheme for construction of uncertainty relations for n observables and m states is presented. Several lowest-order inequalities are displayed and briefly discussed. For two states |ψ⟩ and |ϕ⟩ and canonical observables the (entangled) extension of the Heisenberg relation reads [Δ p (ψ)] 2 [Δ q (ϕ)] 2 + [Δ p (ϕ)] 2 [Δ q (ψ)] 2 ⩾½.
The Heisenberg inequality ΔX ΔP \geq \hbar/2 can be replaced by an exact equality, for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. The significance of this "exact" uncertainty relation is discussed, and results are generalised to angular momentum and phase, photon number and phase, time and frequency, and to states described by density operators. Connections to optimal estimation of an observable from the measurement of a second observable, Wigner functions, energy bounds, and entanglement are also given.
The proof of the Heisenberg uncertainty relation is modified to produce two improvements: (a) The resulting inequality is stronger because it includes the covariance between the two observables, and (b) the proof lifts certain restrictions on the state to which the relation is applied, increasing its generality. The restrictions necessary for the standard inequality to apply are not widely known, and they are discussed in detail. The classical analog of the Heisenberg relation is also derived, and the two are compared. Finally, the modified relation is used to address the apparent paradox that eigenfunctions of the z component of angular momentum Lz do not satisfy the φ–Lz Heisenberg relation; the resolution is that the restrictions mentioned above make the usual inequality inapplicable to these states. The modified relation does apply, however, and it is shown to be consistent with explicit calculations.
Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, rather little is known about entropic uncertainty relations with more than two measurement settings. In this note we review known results and open questions. The uncertainty principle is one of the fundamental ideas of quantum mechanics. Since Heisenberg’s uncertainty relations for canonically conjugate variables, they have been one of the most prominent examples of how quantum mechanics differs from the classical world (Heisenberg, 1927). Uncertainty relations today are probably best known in the form given by (Robertson, 1929), who extended Heisenberg’s result to two arbitrary observables A and B. Robertson’s relation states that if we prepare many copies of the state |ψ〉, and measure each copy individually using either A or B, we have
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.
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 α 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 the present article, we discuss two types of uncertainty relations in Quantum Mechanics–multiplicative and additive inequalities for two canonical observables. The multiplicative uncertainty relation was discovered by Heisenberg. Few years later (1930) Erwin Schrödinger has generalized and made it more precise than the original. The additive uncertainty relation is based on the three independent statistical moments in Quantum Mechanics–Cov(q,p), Var(q) and Var(p). We discuss the existing symmetry of both types of relations and applicability of the additive form for the estimation of the total error.