Geometric approach to extend Landau-Pollak uncertainty relations for positive operator-valued measures

We provide a twofold extension of Landau-Pollak uncertainty relations for mixed quantum states and for positive operator-valued measures, by recourse to geometric considerations. The generalization is based on metrics between pure states, having the form of a function of the square of the inner product between the states. The triangle inequality satisfied by such metrics plays a crucial role in our derivation. The usual Landau-Pollak inequality is thus a particular case (derived from Wootters metric) of the family of inequalities obtained, and, moreover, we show that it is the most restrictive relation within the family.

Figure 1

Illustration of the uncertainty relations given in Theorem 1 in the case of nondegenerate observables and $N=3$. Snapshots of the uncertainty sum ${\mathcal{U}}_f({\mathcal{A}}_k;{\rho }_l)+{\mathcal{U}}_f({\mathcal{B}}_k;{\rho }_l)$ vs the corresponding generalized overlap $c_{{\mathcal{A}}_k,{\mathcal{B}}_k}$ (points), and comparison to the bound $f\left(c_{\mathcal{A},\mathcal{B}}^2\right)$ (solid line) for (a) $f\left(x\right)=arccos\sqrt{x}$ (Wootters); (b) $f\left(x\right)=\sqrt{2(1-\sqrt{x})}$ (Bures); (c) $f\left(x\right)=\sqrt{1-x}$ (root infidelity). In the simulation, $k=1,...,{10}^4$ and $l=1,...,25$. (d) Domain ${\mathbb{D}}_{\mathrm{LP}}(1,1,0.75)$ that corresponds to Wootters metric (solid line), and functions $h_{c_{\mathcal{A},\mathcal{B}}}^f$ for the Bures (dashed-dotted line) and root-infidelity (dashed line) metrics; the points represent the pairs $(P_{\mathcal{A},{\rho }_l},P_{\mathcal{B},{\rho }_l})$ with $(\mathcal{A},\mathcal{B})$ fixed and $l=1,...,5\times {10}^4$.

Figure 2

Illustration of Corollary 3: domain ${\mathbb{D}}_{\mathrm{LP}}(c_{\mathcal{A}},c_{\mathcal{B}},c_{\mathcal{A},\mathcal{B}})$ and snapshots of pairs $(P_{\mathcal{A};{\rho }_l},P_{\mathcal{B};{\rho }_l})$ (points). In each figure, $N=3,N_A=4,N_B=5$, and the dotted lines represent, respectively, $P_{\mathcal{A};\rho }=\frac{1}{N_A},P_{\mathcal{B};\rho }=\frac{1}{N_B},P_{\mathcal{A};\rho }=c_{\mathcal{A}}^2,P_{\mathcal{B};\rho }=c_{\mathcal{B}}^2$, and $P_{\mathcal{B};\rho }=h_{c_{\mathcal{A},\mathcal{B}}}\left(P_{\mathcal{A};\rho }\right)$. The domain ${\mathbb{D}}_{\mathrm{LP}}(c_{\mathcal{A}},c_{\mathcal{B}},c_{\mathcal{A},\mathcal{B}})$ is delimited by the solid line. In (a) and (b), $(c_{\mathcal{A}},c_{\mathcal{B}},c_{\mathcal{A},\mathcal{B}})=(0.92,0.95,0.60)$; in (c) and (d), $(c_{\mathcal{A}},c_{\mathcal{B}},c_{\mathcal{A},\mathcal{B}})=(0.84,0.86,0.84)$. The number of snapshots is of the order of ${10}^4$.