lp-Norm Coherence in Quantum Resource Theory

• lp-Norm Coherence is a quantitative measure that evaluates quantum state coherence by assessing off-diagonality via specific lp-norms (with q=1 and 1 ≤ p ≤ 2).
• It unifies existing coherence metrics through a closed-form formulation based on the diagonal approximation, ensuring computational efficiency at O(n²) complexity.
• The framework bridges quantum information, radar waveform design, and sparse recovery by offering actionable insights into trade-offs and resource quantification.

The lp-norm coherence formalism generalizes the quantification of quantum coherence—one of the central resources in quantum information theory—by associating coherence with matrix norm distances from the set of incoherent states. For a state (density matrix) $\rho$ in a fixed reference basis, lp-norm coherence measures assess the "off-diagonality" of $\rho$ via suitable matrix norms. This framework unifies existing approaches and clarifies under what circumstances norm-induced functionals yield bona fide resource-theoretic coherence measures. The paper of lp-norm coherence also connects quantum resource theories, radar waveform design, and linear inverse problems, making it a flexible analytic and algorithmic tool.

1. Resource-Theoretic Definitions and lp-Norm Coherence Measures

A coherence measure $C$ on quantum states $\rho \in D_n$ (the set of $n \times n$ density matrices) is required to satisfy four axioms:

1. Nonnegativity and Faithfulness: $C(\rho) \geq 0$ for all $\rho$, and $C(\rho) = 0$ if and only if $\rho$ is diagonal (i.e., incoherent).
2. Monotonicity under Incoherent Operations: $C(\rho) \geq C(\Lambda(\rho))$ for every incoherent CPTP map $\Lambda$ (all Kraus operators permute the incoherent set $\mathcal{I}_n$ into itself).
3. Strong Monotonicity: Decompositions via incoherent Kraus maps cannot increase average coherence: $C(\rho) \geq \sum_j p_j C(\rho_j)$ with $p_j = \operatorname{Tr}[K_j \rho K_j^\dagger]$ and $\rho_j = K_j \rho K_j^\dagger/p_j$.
4. Convexity: $\sum_{i} p_i C(\rho_i) \geq C(\sum_{i} p_i \rho_i)$ for all probabilistic mixtures.

Letting $\|A\|_{q,p}$ denote the matrix $\ell_{q,p}$-norm: $$\|A\|_{q,p} = \left( \sum_{j=1}^n \|A_j\|_q^p \right)^{1/p},$$ where $A_j$ is the $j$-th column, the norm-induced coherence measure is defined as: $$C_{q,p}(\rho) = \min_{\delta \in \mathcal{I}_n} \|\rho - \delta\|_{q,p}.$$ A central result is that $C_{q,p}$ is a valid coherence measure (satisfying all four axioms) if and only if $q=1$ and $1 \leq p \leq 2$ (Jing et al., 2020).

For this admissible regime, the minimizer is always $\mathrm{diag}(\rho)$ (the diagonal part), yielding the explicit closed form: $$C_{1,p}(\rho) = \left( \sum_{j=1}^n \left( \sum_{i \neq j} |\rho_{ij}| \right)^p \right)^{1/p}$$ In particular, $p=1$ yields the familiar $\ell_1$-coherence; $p=2$ corresponds to an $\ell_2$-type norm on column-wise off-diagonal sums.

2. Characterization Theorems and Computational Properties

The key theorem established in (Jing et al., 2020) asserts that:

• No unitary similarity invariant (USI) norm induces a proper coherence measure.
• Among all $\ell_{q,p}$ norms, only those with $q=1$ and $1 \leq p \leq 2$ produce legitimate coherence monotones.

The proof leverages counterexamples based on mixtures of maximally coherent states (for $q \neq 1$) and explicit Kraus maps (for $p > 2$) to demonstrate violation of strong monotonicity. Conversely, for $q=1$ and $p \leq 2$, operator inequalities together with a contractivity argument for incoherent maps establish satisfaction of all resource-theoretic criteria.

These coherence measures admit highly efficient computation: no eigenvalue decomposition is required, and the closed formula depends only on off-diagonal elements with computational complexity $O(n^2)$. The smoothness increases with $p$, with $p=2$ marking the sharp boundary where the measure remains valid.

3. Special Cases, Relationships, and Operational Significance

Standard $\ell_1$-Coherence and Its Role

The $\ell_1$-norm coherence, $C_{1,1}(\rho) = \sum_{i \neq j} |\rho_{ij}|$, is the prototypical example and enjoys the strongest operational status. It directly upper bounds the distillable coherence (equal to relative entropy of coherence), via the logarithmic coherence $C_{\log}(\rho) = \log_2[1 + C_{\ell_1}(\rho)]$ (Rana et al., 2016): $C_r(\rho) \leq C_{\log}(\rho),$ with the upper bound being tight for special classes such as pure states and qubits. The robustness of coherence $C_R$ also admits a similar bound, demonstrating that $C_{\ell_1}$, despite its algebraic simplicity, captures resource-theoretic constraints with surprising fidelity.

Generalization: $\ell_{1,p}$-Coherence

For $1 < p < 2$, $C_{1,p}$ interpolates between entrywise sums and columnwise $\ell_2$-aggregated asymmetries. This family enables, for example, trade-offs in robustness and smoothing: increasing $p$ reduces sensitivity to large individual off-diagonal elements but maintains strong monotonicity. The family $\{C_{1,p}: 1 \leq p \leq 2\}$ thus forms the maximal family of entrywise norms yielding genuine coherence measures.

lp-norm and l2-norm Connections

Quantitative relationships have been established between $C_{l_p}$ and $C_{l_2}$ (with $$C_{l_p}(\rho) = (\sum_{i \neq j} |\rho_{ij}|^p)^{1/p}$$), most notably in the context of wave–particle–mixedness trialities (Li et al., 8 Nov 2025). For $1 \leq p < 2$: $C_{l_p}(\rho) \leq [d(d-1)]^{(2-p)/2p} C_2(\rho),$

$$C_{l_p}(\rho) \geq \sqrt{2} \, [d(d-1)]^{-(p-1)/p} C_2(\rho),$$

where $d$ is the Hilbert space dimension. This provides tight upper and lower constraints on lp-norm coherence in terms of the more tractable $l_2$-norm.

4. Trade-Off Relations and Multipartite Extensions

The structure of lp-norm coherence enables fundamental trade-off inequalities for multipartite systems, establishing that total coherence must distribute among subsystems and their entanglements.

For the $\ell_1$-case: $C_{1...n} \geq \sum_{\text{marginals}} C_\alpha$ where the sum is over all $(n-1)$-partite marginals. For three qubits, a sharper statement holds: $C_{123} \geq C_{12} + C_{13} + C_{23} + \tau_3$ where $\tau_3$ is the three-tangle measuring genuine tripartite entanglement (Jiang et al., 2020). For $p > 1$, the proof becomes obstructed by the nonlinear nature of the $p$-norm power, suggesting that convexity and combinatorics play more intricate roles as $p$ increases. The extension to general $p$ remains an open problem, though qualitative patterns are expected.

Furthermore, triality relations for wave, particle, and mixedness properties—originating in studies of quantum complementarity—arise naturally in terms of $l_p$-norm coherence. Specifically, exact equalities of the form: $$\frac{d}{d-1} C_2^2(\rho) + M_l(\rho) + P^2(\rho) = 1$$ and related $p$-dependent generalizations, provide exact trade-off surfaces for quantum systems, interpolating between $l_1$ and $l_2$ measures and clarifying the allocation of quantum "resources" (Li et al., 8 Nov 2025).

5. Algorithmic and Practical Applications

lp-norms have been deployed as objective functions in various algorithmic settings, notably:

• Sparse Signal Recovery: The $\ell_p$-induced operator norm and associated mutual coherence concepts provide uniqueness conditions for sparse solutions in underdetermined linear systems, with the $\ell_p$-norm for $0 < p < 1$ approximating hard $\ell_0$ minimization (Xu et al., 2013).
• Radar Waveform Design: lp-norm coherence criteria have been used as design objectives on sets of unimodular sequences to optimize autocorrelation and cross-correlation sidelobe levels for MIMO radars. Here, minimizing a weighted lp-norm of sidelobes enables flexible trade-offs between sparsity ($p \to 0$), integrated power ($p = 2$), and peak level ($p \to \infty$), leveraging algorithms such as block successive upper bound minimization (BSUM) to efficiently tackle non-convex optimization landscapes (Raei et al., 2021).
• Quantum Information: lp-norm coherence measures with $1 \leq p \leq 2$ offer closed-form functionals for benchmarking algorithms and certifying bounds on distillable coherence, single-shot protocols, and robustness costs, without the need for diagonalization or eigenvalue decomposition.

6. Limitations, Generalizations, and Open Directions

The structure of the characterization theorem (Jing et al., 2020) reveals significant limitations: unitarily invariant norms, including Schatten-$p$ norms, are incompatible with the axioms of quantum resource theory for coherence. Only absolute-entrywise norms with $q=1$ and $1 \leq p \leq 2$ are valid. An open question remains as to whether further norm generalizations might yield bona fide coherence measures by relaxing or modifying the strong monotonicity requirements, or by considering non-norm-based distance functionals.

Attempts to generalize trade-off or distribution relations for $p > 1$ encounter obstacles due to the failure of simple power manipulations; new combinatorial or analytic tools will be needed for tight bounds in this regime. In algorithmic contexts, while $\ell_p$-norm objectives are powerful, computational complexity generally grows as $p$ departs from tractable norms (e.g., $p=2$), particularly as NP-hardness manifests for certain non-convex code designs.

A plausible implication is that lp-norm coherence, with its direct connection to matrix entries and clear resource-theoretic status for $1 \leq p \leq 2$, will remain central both in the analytic structure of quantum resource theories and as a bridge to practical optimization problems in signal processing and quantum information.