Stronger uncertainty relations for the sum of variances (1407.0338v3)

Abstract: Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.

• The paper introduces two new uncertainty inequalities that yield reliable nonzero lower bounds for the sum of variances even when traditional bounds vanish.
• It refines the Heisenberg-Robertson relation by incorporating orthogonal states to effectively capture the incompatibility of quantum observables.
• The proposed bounds have significant implications for quantum cryptography, entanglement detection, and precision measurements in quantum systems.

Stronger Uncertainty Relations for the Sum of Variances

The paper by Lorenzo Maccone and Arun K. Pati presents an advancement in the field of quantum mechanics by proposing refined uncertainty relations for the sum of variances. Traditional uncertainty principles, like the Heisenberg-Robertson relation, establish a lower bound on the product of the variances of two observables based on their commutator. However, this classic approach sometimes yields a null lower bound, which fails to highlight the incompatibility of observables for certain quantum states. This paper addresses this limitation by introducing two new uncertainty relations that are always nontrivial for incompatible observables.

Background and Motivation

The Heisenberg-Robertson uncertainty relation, represented by the inequality $\Delta A^2 \Delta B^2 \geqslant \left|\tfrac{1}{2}\< [A,B] \>\right|^2$, serves as a cornerstone in quantum mechanics, capturing the essence of measurement uncertainty between pairs of non-commuting operators $A$ and $B$. However, this relation is not sufficiently robust, as the bound can be null even for non-compatible observables that don't share common eigenstates. The paper identifies this problem and extends existing work by proposing uncertainty relations based on the sum $\Delta A^2 + \Delta B^2$, which provide more stringent lower bounds.

Main Results

The authors present two principal inequalities:

1. First Inequality: $\Delta A^2 + \Delta B^2 \geqslant \pm i\<[A,B]\> + \left|\<\psi |A \pm iB |\psi^\perp\>\right|^2$. This relation involves an arbitrary state $|\psi^\perp\>$ orthogonal to the system's state $|\psi\>$. The inequality captures incompatibility effectively because the lower bound remains non-zero unless the system’s state is a joint eigenstate of $A$ and $B$.
2. Second Inequality: $\Delta A^2 + \Delta B^2 \geqslant \tfrac{1}{2} |\<\psi_{A+B}^\perp |A+B |\psi\>|^2$. This formulation provides a nontrivial lower bound unless the state of the system is an eigenstate of the sum $A+B$.

The combined uncertainty relation $$\Delta A^2+\Delta B^2\geqslant\max({\cal L}_{(3)},{\cal L}_{(4)})$$, uniting these bounds, highlights their collective robustness against common inaccuracies of previous models.

Implications and Speculation on Future Developments

These results have profound implications for quantum theory and applications like quantum cryptography and entanglement detection, where precise characterizations of observable incompatibilities are essential. Given this enhanced capability to capture quantum uncertainty, future research could explore the adaptation of these relations to broader classes of observables or develop corresponding experimental validations.

Speculatively, these stronger uncertainty relations may influence the development of quantum technologies by providing comprehensive descriptions of state preparations critical for quantum computing and communication protocols. Moreover, aligning such refined theoretical constructs with advancements in quantum measurement precision could yield novel insights into both fundamental physics and applied quantum mechanics.

Conclusion

In sum, the presented research refines the understanding of quantum uncertainty by implementing a more stringent approach based on the sum of variances. This address a longstanding gap in existing models, having the potential to significantly influence theoretical explorations and practical applications in quantum science. The paper successfully introduces robust tools that enable a more accurate depiction of quantum uncertainty, potentially guiding further advancements in the domain.