Multi-observable Uncertainty Relations in Product Form of Variances

Abstract

We investigate the product form uncertainty relations of variances for n (n ≥ 3) quantum observables. In particular, tight uncertainty relations satisfied by three observables has been derived, which is shown to be better than the ones derived from the strengthened Heisenberg and the generalized Schrödinger uncertainty relations, and some existing uncertainty relation for three spin-half operators. Uncertainty relation of arbitrary number of observables is also derived. As an example, the uncertainty relation satisfied by the eight Gell-Mann matrices is presented.

Variance uncertainty relations without covariances for three and four observables

“…Unfortunately, this inequality is rather complicated for N > 2 observables, because it contains, in addition to N variances X kk and N (N − 1)/2 mean values of commutators, numerous sums and products of various combinations of N (N − 1)/2 covariances X jk with j = k. For example, if N = 3, then (3) can be written in the form (see, e.g., [25,26])…”

A Note on the Strong Unitary Uncertainty Relations

“…It would be interesting to compare the n-observable relations from the approach used in [14] to the ones from the approach used in [15]. By a bijection map between the unitary operators and Hermitian operators, we show here that the strong unitary uncertainty relations given in [14] are equivalent to the Hermitian uncertainty relations presented in [15], although these two kinds of uncertainty relations are derived from quite different approaches. In proving this equivalence the general representations of n-observable uncertainty relations for both approaches are provided.…”

“…In terms of the covariance matrices of the mean values of Hermitian operators, in [15] uncertainty relations for n observables have been also derived. It would be interesting to compare the n-observable relations from the approach used in [14] to the ones from the approach used in [15].…”

“…In terms of the covariance matrices of the mean values of Hermitian operators, in [15] uncertainty relations for n observables have been also derived. It would be interesting to compare the n-observable relations from the approach used in [14] to the ones from the approach used in [15]. By a bijection map between the unitary operators and Hermitian operators, we show here that the strong unitary uncertainty relations given in [14] are equivalent to the Hermitian uncertainty relations presented in [15], although these two kinds of uncertainty relations are derived from quite different approaches.…”

Uncertainty relations for two observables coupled with the third one

“…Recently, it was re-derived in Ref. [17]. Inequality (14) has the form X 11 X 22 X 33 ≥ aX 11 + bX 22 + c, where coefficients a, b and c do not contain variances X 11 and X 22 .…”