linguistic:voice_change
Improving the Robertson–Schrödinger uncertainty relation based on decompositions of the density matrix

The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components ϱ k {\displaystyle \varrho _{k}} in any decomposition of the density matrix given as ϱ = ∑ k p k ϱ k. {\displaystyle \varrho =\sum _{k}p_{k}\varrho _{k}.} Here, for the probabilities p k ≥ 0 {\displaystyle p_{k}\geq 0} and ∑ k p k = 1 {\displaystyle \sum _{k}p_{k}=1} hold. Then, using the relation ∑ k a k ∑ k b k ≥ (∑ k a k b k) 2 {\displaystyle \sum _{k}a_{k}\sum _{k}b_{k}\geq \left(\sum _{k}{\sqrt {a_{k}b_{k}}}\right)^{2}} for a k, b k ≥ 0 {\displaystyle a_{k},b_{k}\geq 0},it follows that σ A 2 σ B 2 ≥ 2, {\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq \left^{2},} where the function in the bound is defined L (ϱ) = | 1 2 tr (ρ { A, B }) − tr (ρ A) tr (ρ B) | 2 + | 1 2 i tr (ρ) | 2. {\displaystyle 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)\right|^{2}}}.} 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 σ A 2 σ B 2 ≥ 2, {\displaystyle \sigma _{A}^{2}\sigma _{B}^{2}\geq \left^{2},} 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.

With similar arguments, one can derive a relation with a convex roof on the right-hand side σ A 2 F Q ≥ 4 2 {\displaystyle \sigma _{A}^{2}F_{Q}\geq 4\left^{2}} where F Q {\displaystyle F_{Q}} denotes the quantum Fisher information and the density matrix is decomposed to pure states as ϱ = ∑ k p k | Ψ k ⟩ ⟨ Ψ k |. {\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert.} The derivation takes advantage of the fact that the quantum Fisher information is the convex roof of the variance times four.

A simpler inequality follows without a convex roof σ A 2 F Q ≥ | ⟨ i ⟩ | 2, {\displaystyle \sigma _{A}^{2}F_{Q}\geq \vert \langle i\rangle \vert ^{2},} which is stronger than the Heisenberg uncertainty relation, since for the quantum Fisher information we have F Q ≤ 4 σ B, {\displaystyle F_{Q}\leq 4\sigma _{B},} while for pure states the equality holds.