Editing Uncertainty principle (section)

===Mixed states===

The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components <math>\varrho_k</math> in any decomposition of the [[density matrix]] given as
<math display="block">
\varrho=\sum_k p_k \varrho_k.
</math>
Here, for the probabilities <math>p_k\ge0</math> and <math>\sum_k p_k=1</math> hold. Then, using the relation
<math display="block">
\sum_k a_k \sum_k b_k \ge \left(\sum_k \sqrt{a_k b_k}\right)^2
</math>
for <math> a_k,b_k\ge 0</math>,
it follows that<ref name="PhysRevResearch21">{{cite journal |last1=Tóth |first1=Géza |last2=Fröwis |first2=Florian |title=Uncertainty relations with the variance and the quantum Fisher information based on convex decompositions of density matrices |journal=Physical Review Research |date=31 January 2022 |volume=4 |issue=1 |pages=013075 |doi=10.1103/PhysRevResearch.4.013075|arxiv=2109.06893 |bibcode=2022PhRvR...4a3075T |s2cid=237513549 }}</ref>
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\sum_k p_k L(\varrho_k)\right]^2,
</math>
where the function in the bound is defined
<math display="block">
L(\varrho) = \sqrt{\left | \frac{1}{2}\operatorname{tr}(\rho\{A,B\}) - \operatorname{tr}(\rho A)\operatorname{tr}(\rho B)\right |^2 +\left | \frac{1}{2i} \operatorname{tr}(\rho[A,B])\right | ^2}.
</math>
The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation
<math display="block">
\sigma_A^2 \sigma_B^2 \geq \left[\max_{p_k,\varrho_k} \sum_k p_k L(\varrho_k)\right]^2,
</math>
where on the right-hand side there is a concave roof over the decompositions of the density matrix.
The improved relation above is saturated by all single-qubit quantum states.<ref name="PhysRevResearch21" />

With similar arguments, one can derive a relation with a convex roof on the right-hand side<ref name="PhysRevResearch21" />
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq 4 \left[\min_{p_k,\Psi_k} \sum_k p_k L(\vert \Psi_k\rangle\langle \Psi_k\vert)\right]^2
</math>
where <math>F_Q[\varrho,B]</math> denotes the [[quantum Fisher information]] and the density matrix is decomposed to pure states as
<math display="block">
\varrho=\sum_k p_k \vert \Psi_k\rangle \langle \Psi_k\vert.
 </math>
The derivation takes advantage of the fact that the [[quantum Fisher information]] is the convex roof of the variance times four.<ref>{{cite journal |last1=Tóth |first1=Géza |last2=Petz |first2=Dénes |title=Extremal properties of the variance and the quantum Fisher information |journal=Physical Review A |date=20 March 2013 |volume=87 |issue=3 |pages=032324 |doi=10.1103/PhysRevA.87.032324|bibcode=2013PhRvA..87c2324T |arxiv=1109.2831 |s2cid=55088553 }}</ref><ref>{{cite arXiv |last1=Yu |first1=Sixia |title=Quantum Fisher Information as the Convex Roof of Variance |date=2013 |eprint=1302.5311|class=quant-ph }}</ref>

A simpler inequality follows without a convex roof<ref>{{cite journal |last1=Fröwis |first1=Florian |last2=Schmied |first2=Roman |last3=Gisin |first3=Nicolas |title=Tighter quantum uncertainty relations following from a general probabilistic bound |journal=Physical Review A |date=2 July 2015 |volume=92 |issue=1 |pages=012102 |doi=10.1103/PhysRevA.92.012102|arxiv=1409.4440 |bibcode=2015PhRvA..92a2102F |s2cid=58912643 }}</ref>
<math display="block">
\sigma_A^2 F_Q[\varrho,B] \geq \vert \langle i[A,B]\rangle\vert^2,
</math>
which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have
<math display="block">
F_Q[\varrho,B]\le 4 \sigma_B,
</math>
while for pure states the equality holds.