Uncertainty Principle for the Anti-commutator


Recently I’ve sent a paper to PRL which is now under appeal. It claims that anti-commutators generate further constraints to observable dispersion that have to be appended to those well known from the standard uncertainty principle for the commutator. Here’s the traditional statement in its general form:

\Delta_{\left|\psi\right\rangle }A\,\Delta_{\left|\psi\right\rangle }B\geq\frac{1}{2}\left|\left\langle \psi\left|i\left[A,B\right]\right|\psi\right\rangle \right|

Now, any pair of Hermitian operators obviously generate both a commutator and an anti-commutator. If one generalises the proof by Robertson by constructing,

x\left(\lambda\right)\overset{{\scriptstyle \textrm{def}}}{=}\left\Vert \left(A'+\lambda B'\right)\left|\psi\right\rangle \right\Vert ^{2}\geq0

with \lambda real, one gets,

\triangle_{\left|\psi\right\rangle }A\:\triangle_{\left|\psi\right\rangle }B\geq\frac{1}{2}\left|\left\langle AB+BA\right\rangle _{\left|\psi\right\rangle }-2\left\langle A\right\rangle _{\left|\psi\right\rangle }\left\langle B\right\rangle _{\left|\psi\right\rangle }\right|

The consequences are almost always immaterial, as the new constraint does not make uncertainty worse than that already implied in the commutator. But in my proposed paper I apply this to the spin singlet state, which is one of the interesting cases where the anti-commutator makes things significantly worse for determining the variables (more uncertainty). It is interesting to apply it to this case: Imagine you have a particle with a certain wave function in the localisation variable x and the corresponding probability density \left|\psi\left(x\right)\right|^2. Now consider the following pair of discrete localisation observables: E_1; E_2; E_1E_2=0; E_1+E_2=I. They trivially anticommute (in fact they also commute, they do not “connect”, as their product is identically zero!). Because they commute, the standard uncertainty principle leads to \Delta_{\left|\psi\right\rangle }E_1\,\Delta_{\left|\psi\right\rangle }E_2\geq 0, which is devoid of any content. But because they also anti-commute, the anti-commutator gives something very different,

\triangle_{\left|\psi\right\rangle }E_{1}\:\triangle_{\left|\psi\right\rangle }E_{2}\geq\left|\left\langle E_{1}\right\rangle _{\left|\psi\right\rangle }\left\langle E_{2}\right\rangle _{\left|\psi\right\rangle }\right|

which means that the dispersions of localising a particle inside and outside of a region can never improve the constraint given by the product of the “inside” and “ouside” probability. The worst-case scenario is when p\left(E_1\right)=p\left(E_2\right)=1/2, which is the maximum of \epsilon\,\left(1-\epsilon\right).

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