Symmetric orthogonalization and probabilistic weights in resource quantification

TL;DR

The paper tackles the challenge of quantifying quantum resources when the natural basis is non-orthogonal. It advocates Löwdin symmetric orthogonalization (LSO) to map between non-orthogonal and orthonormal frames, enabling a consistent probabilistic description through Löwdin weights. Key contributions include proving that LSO preserves state symmetry and minimizes distortion relative to the original basis, defining non-negative Löwdin weights from the overlap matrix via $\rho_L = \frac{O_d^{1/2} \rho O_d^{1/2}}{\text{Tr}(O_d^{1/2} \rho O_d^{1/2})}$, and decomposing total coherence into a geometry dependent floor and a genuine quantum resource. The framework then defines a relative superposition measure and associates an entropic localization metric with the Löwdin weights, providing a geometry aware method to quantify and compare coherence and superposition. Together, these results offer a robust, basis democratic approach to resource theory in non-orthogonal settings with potential impact on quantum metrology and information processing where non-orthogonal bases are natural.

Abstract

Transforming non-orthogonal bases into orthogonal ones often compromises essential properties or physical meaning in quantum systems. Here, we demonstrate that Löwdin symmetric orthogonalization (LSO) outperforms the widely used Gram-Schmidt orthogonalization (GSO) in characterizing and quantifying quantum resources, with particular emphasis on coherence and superposition. We employ LSO both to construct an orthogonal basis from a non-orthogonal one and to obtain a non-orthogonal basis from an orthogonal set, thereby mitigating ambiguity related to the basis choice in defining quantum coherence. Unlike GSO, which depends on the ordering of input states, LSO applies a symmetric transformation that treats all vectors equally and minimizes deviation from the original basis. This procedure yields basis sets with enhanced stability, preserving the closest possible correspondence to the original physical states while satisfying orthogonality. Building on LSO, we also introduce Löwdin weights -- probabilistic weights for non-orthogonal representations that provide a consistent measure of resource content. We explicitly contrast these with Chirgwin-Coulson weights, demonstrating that Löwdin weights ensure non-negativity, a prerequisite for information-theoretic measures. These weights further enable the quantification of coherence and the characterization of superposition, providing a degree of superposition as a distinct measure, as well as facilitating the assessment of state delocalization through entropy and participation ratios. Our theoretical and numerical analyses confirm LSO's superior preservation of quantum state symmetry and resource characteristics, underscoring the critical role of orthogonalization methods and Löwdin weights in resource theory frameworks involving non-orthogonal bases.

Figures (4)

• Figure 1: Order dependence of the GSO process. (a) Two non-orthogonal vectors $\left\{\ket{c_1} (=c_1), \ket{c_2} (=c_2)\right\}$ with relative angle $\theta \in (0, \pi/2)$. (b) Orthogonalization results differ based on which vector is processed first: Starting with $\ket{c_1}$ yields $\left\{\ket{{e}_{1}^{\text{GS}}} (={c_1}), \ket{{e}_{2}^{\text{GS}}} (={\bar{c}_1})\right\}$; Starting with $\ket{c_2}$ gives $\left\{\ket{{e}_{1}^{\text{GS}}} (={c_2}), \ket{{e}_{2}^{\text{GS}}} (={\bar{c}_2})\right\}$. This illustrates the intrinsic order dependence of Gram-Schmidt, the two bases are distinct. The same applies in higher dimensions. For clarity, all vectors are assumed unit-norm.
• Figure 2: Geometric illustration of Löwdin SO for a two-dimensional system. (a) The original non-orthogonal vectors $\ket{c_1} (=c_1)$ and $\ket{c_2} (=c_2)$ exhibit a nonzero overlap (with $\theta \in (0, \pi/2)$). (b) After symmetrically rotating both vectors by an angle $\beta$, they are transformed into the orthonormal Löwdin basis $\ket{{e}_{1}^{\text{L}}}_{\mathrm{sym}} (=\bar{c}_1)$ and $\ket{{e}_{2}^{\text{L}}}_{\mathrm{sym}} (=\bar{c}_2)$, where $\theta +2\beta = \pi/2$ ensures mutual orthogonality. The same applies in higher dimensions. All vectors are assumed unit-norm.
• Figure 3: Total coherence as a function of the basis overlap parameter $s$ in the Löwdin representation. The shaded region delimits the accessible coherence space, encompassing the full continuum of valid quantum states --- both superposition and superposition-free regimes. The lower bound (dashed blue curve) defines the superposition-free limit (classical mixtures), quantifying the irreducible "geometric coherence" imposed solely by the non-orthogonality of the basis. The upper bound (solid red line) marks the saturation limit corresponding to maximal superposition states, where the coherence reaches its theoretical maximum of unity Torun23Low. Note that the specific form of the maximal state transitions from the symmetric combination $\ket{c_1} + \ket{c_2}$ for $s < 0$ to the antisymmetric form $\ket{c_1} - \ket{c_2}$ for $s > 0$.
• Figure 4: Density plot quantifying the superposition degree $\mathcal{S}_{\text{off}}$ (Eq. \ref{['ASupDegreeMeasure']}) within the accessible parameter space of the Löwdin representation. The color gradient maps the normalized proximity to the maximal superposition limit, scaling from superposition-free classical mixtures (blue, $\mathcal{S}_{\text{off}}=0$) to maximal superposition states (red, $\mathcal{S}_{\text{off}}=1$). The accessible region is delimited by the irreducible geometric coherence $|s|$ (lower bound) and the saturation limit of unity (upper bound). Thus, the gradient visualizes the emergence of genuine quantum resources (i.e., superposition) exceeding the baseline coherence inherent to the non-orthogonal basis.