Uncertainty Relations for Angular Momentum (1505.00049v2)

Abstract: In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the "uncertainty regions'' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s goes to 1 as s tends to infinity.

Uncertainty Relations for Angular Momentum: An Analytical Perspective

The paper "Uncertainty Relations for Angular Momentum" by Lars Dammeier, René Schwonnek, and Reinhard F. Werner provides an in-depth analysis of uncertainty in the context of angular momentum for quantum systems represented in terms of the Lie group $SU(2)$. The authors focus on characterizing uncertainty regions articulated through the variances of angular momentum operators, delivering comprehensive insights into state-dependent and independent uncertainty bounds. The results encapsulate both traditional variance-based approaches and extensions to entropic uncertainty measures, enriching our understanding of quantum uncertainty mechanisms.

Key Insights and Methodology

1. Variance-Based Uncertainty: The paper begins by generalizing variance uncertainty, traditionally expressed by Robertson's inequality, to include all three components of angular momentum. A critical component is the determination of "uncertainty regions" for variance triples $(\var_{\rho}(L_1), \var_{\rho}(L_2), \var_{\rho}(L_3))$. Here, the paper derives lower bounds for sums of variances, identifying optimal bounds for specific cases (like $s=1/2, 1, 3/2$), and extends these to higher values of angular momentum quantum number $s$. For larger dimensions, the authors effectively employ numerical techniques to support analytical explorations.
2. General Bounds and Asymptotic Behavior: A significant contribution is the development of bounds using a generalized Robertson method which involves combining the typical pairwise treatment into a holistic consideration of triples, resulting in inequalities like $4\var_{\rho}(L_1)(\var_{\rho}(L_2)+\var_{\rho}(L_3))\geq s(s+1)-\sum \var_{\rho}(L_i)$. Additionally, large-$s$ behavior is studied using Holstein-Primakoff approximations, leading to asymptotic scaling insights, specifically that $c_2(s) \sim s^{2/3}$, describing the minimal sum of variances for two-component scenarios.
3. Entropic Uncertainty: Delving into entropic measures, the paper compares variance and entropy as tools for gauging uncertainty. It critiques the applicability of Maassen-Uffink bounds, revealing situations where these bounds become sharp, especially examining the large-$s$ limit where traditional variance-based bounds lack precision. Here, entropic assessments provide an alternative, potentially more versatile measure for jointly appraising different quantum states.
4. Measurement Uncertainty: Beyond preparation uncertainty, measurement-based uncertainty relations are examined, addressing joint measurement scenarios of angular momentum components through positive operator-valued measures (POVMs). An explicit covariant observable minimizing measurement uncertainty is determined, extending Busch, Lahti, and Werner's theoretical framework from position and momentum in infinite-dimensional systems to finite angular momentum systems. This approach yields practical methods to decipher the worst-case measurement errors.

Implications and Future Directions

The research strengthens foundational insights in quantum mechanics regarding the indeterminacy of angular momentum observables, providing analytic and numerical techniques for bounding uncertainties both in preparation and measurement contexts. The exploration of entropic uncertainty offers a promising direction for extending classical uncertainty frameworks, aligning more closely with the information-theoretic approach prevalent in modern quantum information science.

The work opens avenues for deeper investigations into higher-dimensional and more complex systems, potentially unifying uncertainty across diverse quantum observables within varied algebraic structures. Moreover, utilizing the results as a benchmark, experimental techniques dealing with quantum measurements can be fine-tuned, particularly in areas such as quantum optics, magnetic resonance, and quantum metrology.

Finally, this paper accentuates the significant role of state-dependent analyses in understanding quantum systems, suggesting broader applications in fields that require explicit handling of quantum state preparations and measurements, pushing toward increasingly precise control and manipulation of quantum information.