---
title: lp-Norm Coherence in Quantum Resource Theory
url: https://www.emergentmind.com/topics/lp-norm-coherence
type: topic
---

# lp-Norm Coherence in Quantum Resource Theory

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 study 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 $1 \leq p \leq 2$ [2008.04362].

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 [2008.04362] 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 $C_{\log}(\rho) = \log_2[1 + C_{\ell_1}(\rho)]$ [1612.09234]:
\[
C_r(\rho) \leq C_{\log}(\rho),
\]
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 [2511.05937]. 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 [2002.00165]. 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" [2511.05937].

## 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 [1312.4280].
- **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 [2104.03303].
- **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 [2008.04362] 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.

Source: https://www.emergentmind.com/topics/lp-norm-coherence