Editing Uncertainty principle (section)

===The Maccone–Pati uncertainty relations===
The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Lorenzo Maccone and [[Arun K. Pati]] give non-trivial bounds on the sum of the variances for two incompatible observables.<ref>{{cite journal|last1=Maccone|first1=Lorenzo|last2=Pati|first2=Arun K.|title=Stronger Uncertainty Relations for All Incompatible Observables|journal=Physical Review Letters|date=31 December 2014|volume=113| issue=26|page=260401|doi=10.1103/PhysRevLett.113.260401|pmid=25615288|arxiv=1407.0338|bibcode=2014PhRvL.113z0401M|s2cid=21334130 }}</ref> (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref.<ref>{{cite journal |last1=Huang |first1=Yichen |title=Variance-based uncertainty relations |journal=Physical Review A |date=10 August 2012 |volume=86 |issue=2 |page=024101 |doi=10.1103/PhysRevA.86.024101|arxiv=1012.3105 |bibcode=2012PhRvA..86b4101H |s2cid=118507388 }}</ref> due to Yichen Huang.) For two non-commuting observables <math>A</math> and <math>B</math> the first stronger uncertainty relation is given by
<math display="block"> \sigma_{A}^2 + \sigma_{ B}^2 \ge \pm i \langle \Psi\mid [A, B]|\Psi \rangle + \mid \langle \Psi\mid(A \pm i B)\mid{\bar \Psi} \rangle|^2, </math>
where <math> \sigma_{A}^2 = \langle \Psi |A^2 |\Psi \rangle - \langle \Psi \mid A \mid \Psi \rangle^2 </math>, <math> \sigma_{B}^2 = \langle \Psi |B^2 |\Psi \rangle - \langle \Psi \mid B \mid\Psi \rangle^2 </math>, <math>|{\bar \Psi} \rangle </math> is a normalized vector that is orthogonal to the state of the system <math>|\Psi \rangle </math> and one should choose the sign of <math>\pm i \langle \Psi\mid[A, B]\mid\Psi \rangle </math> to make this real quantity a positive number.

The second stronger uncertainty relation is given by
<math display="block"> \sigma_A^2 + \sigma_B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} \mid(A + B)\mid \Psi \rangle|^2 </math>
where <math>| {\bar \Psi}_{A+B} \rangle </math> is a state orthogonal to <math> |\Psi \rangle </math>.
The form of <math>| {\bar \Psi}_{A+B} \rangle </math> implies that the right-hand side of the new uncertainty relation is nonzero unless <math>| \Psi\rangle </math> is an eigenstate of <math>(A + B)</math>. One may note that <math>|\Psi \rangle </math> can be an eigenstate of <math>( A+ B)</math> without being an eigenstate of either <math> A</math> or <math> B </math>. However, when <math> |\Psi \rangle </math> is an eigenstate of one of the two observables the Heisenberg–Schrödinger uncertainty relation becomes trivial. But the lower bound in the new relation is nonzero unless <math> |\Psi \rangle </math> is an eigenstate of both.