We now try to obtain the uncertainty relation which directly relates the two conditional standard deviations of the LPP, σ t | ω 2 and σ ω | t 2. Following the procedure in [8], we get σ t | ω 2 σ ω | t 2 ≥ ∫ d dτ | η t | 2 dτ ∫ d dw | μ ω ( w ) | 2 dw, (38).
Recently, technical terms related to the uncertainty relation, such as variance of distributions, mean square measurement errors, and disturbances due to measurements, have been clearly understood [4, 5], and a universally valid uncertainty relation has been obtained [6, 7, 8].
Here we omit the σ-component of FIM because it is not related to the uncertainty relation.
The Kennard-Robertson type of uncertainty relation arises for two physical quantities related to one another by Fourier transformation.
Equation (13) is another type of Heisenberg uncertainty relation; generally, if a measurement with duration τ is made of the energy in a system, the measurement disturbs the system, causing the energy to be uncertain by an amount ΔE, the magnitude of which is given by the above equation.
Indeed, Heisenberg says: "the uncertainty relation does not hold for the past".
But, remarkably, this proposal does not use the uncertainty relation as one of its fundamental principles.
Here, we investigate this matter in a quantitative way using the new uncertainty relation.
The necessary conditions for violation (reduction) of the Heisenberg-type uncertainty relation are discussed in detail.
A function h is called extremal, if equality holds for it in the uncertainty relation.
Suppose now that the uncertainty relation is σ p σ r ≥ fu.
