---
title: Resource Theory of Quantum Coherence
url: https://www.emergentmind.com/topics/resource-theory-of-quantum-coherence
type: topic
---

# Resource Theory of Quantum Coherence

The resource theory of quantum coherence formalizes superposition as a physical resource, establishing a rigorous framework for its quantification, manipulation, and operational application. In this context, the resource theory of purity considers deviations from the maximally mixed state as a basis-independent resource. Recent developments have demonstrated a deep unification of these theories: maximal coherence extractable from a quantum state via unitary operations is identified with the state's purity. This perspective situates purity as a fundamental, basis-independent resource underpinning all quantum processing capabilities arising from coherence, entanglement, and discord.

## 1. Resource Theory of Quantum Coherence and Purity

In the standard resource theory of quantum coherence, free states—called incoherent states—are diagonal in a fixed reference basis $\{ |i\rangle \}$:
\[
\sigma \in \mathbb{I} \iff \sigma = \sum_i p_i |i\rangle\langle i|.
\]
The main classes of free operations include:

- **Maximally Incoherent Operations (MIO):** CPTP maps $\Lambda$ such that $\Lambda[\sigma] \in \mathbb{I}$ for all $\sigma \in \mathbb{I}$.
- **Incoherent Operations (IO):** MIO maps admitting a Kraus decomposition in which each $K_k$ maps basis vectors to other basis vectors (up to phases).

Various coherence quantifiers have been proposed:
- **$\ell_1$-norm of coherence:** $C_{\ell_1}(\rho) = \sum_{i \neq j} |\rho_{ij}|$, which is an IO monotone.
- **Relative entropy of coherence:** $C_r(\rho) = S(\Delta[\rho]) - S(\rho)$, where $\Delta[\rho]$ projects $\rho$ onto its diagonal and $S(\cdot)$ denotes von Neumann entropy.
- **Distance-based coherence measures:** $C_D(\rho) = \min_{\sigma \in \mathbb{I}} D(\rho, \sigma)$, for a contractive metric $D$.

The **resource theory of purity** is a basis-independent framework where the only free state is the maximally mixed state $I/d$, and all deviations from this state are viewed as a resource. Notably, the purity measured by a contractive distance $D( \rho, I/d )$ is operationally equivalent to the maximal coherence extractable from $\rho$ via unitaries [1612.07570].

## 2. Maximal Coherence via Unitary Operations and Unification with Purity Theory

Given a state $\rho$, the maximal coherence achievable by unitaries is defined, for a given coherence monotone $C$, as:
\[
C_{\max}(\rho) := \max_{U} C( U \rho U^\dagger )
\]
where $U$ runs over all unitary operators.

**Theorem 1 (Universal Maximally Coherent Mixed States):**
For any $\rho$ with eigenvalues $\{ p_n \}$, choose a basis $\{ |n_+\rangle \}$ mutually unbiased with respect to the incoherent basis. Then,
\[
\rho_{\max} = \sum_{n=1}^d p_n | n_+ \rangle \langle n_+ |
\]
simultaneously maximizes $C(U \rho U^\dagger)$ for every MIO-monotone $C$. Thus, the maximal coherence extractable from $\rho$ is always equal to the coherence of $\rho_{\max}$, and the maximizing basis is universal for all contractive distance measures [1612.07570].

## 3. Quantitative Equivalence: Maximal Coherence and Purity

Specializing to distance-based measures, for any contractive distance $D$, it holds that:
\[
C_{\max}(\rho) = D( \rho, I/d )
\]
where $I/d$ is the maximally mixed state. This establishes the operational equivalence
between maximal coherence and purity:
- **Relative-entropy purity:** $P_r(\rho) = S(\rho \| I/d ) = \log_2 d - S(\rho)$
- **Hilbert-Schmidt purity:** $P_2(\rho) = \sqrt{ \operatorname{Tr} \rho^2 - 1/d }$
- **Trace-norm purity:** $P_1(\rho) = \| \rho - I/d \|_1$

Explicitly, for $\rho_{\max}$,
\[
C_D(\rho_{\max}) = D( \rho_{\max}, \Delta[\rho_{\max}] ) = D( \rho, I/d )
\]
for any distance $D$ [1612.07570]. The maximal extractable relative-entropy coherence is $C_r^\mathrm{max}(\rho) = \log_2 d - S(\rho)$.

This reveals that, operationally, the coherence resource theory with optimal unitaries is subsumed into the purity resource theory: purity is the ultimate, basis-free resource for generating coherence.

## 4. Purity Bounds on Generation of Entanglement and Discord

Let $D$ be any contractive distance and consider distance-based quantifiers of discord and entanglement,
\[
D(\rho) = \min_{\sigma \in Z} D(\rho, \sigma), \quad E(\rho) = \min_{\sigma \in S} D(\rho, \sigma)
\]
with $Z$ the set of zero-discord (classical-quantum) states and $S$ the set of separable states.

For any $U$, the following hierarchy obtains:
\[
P_D(\rho) = D(\rho, I/d) \geq C_D( U \rho U^\dagger ) \geq D( U \rho U^\dagger ) \geq E( U \rho U^\dagger )
\]
Thus,
\[
P_D(\rho) \geq \max_U D( U \rho U^\dagger ) \geq \max_U E( U \rho U^\dagger )
\]
For relative-entropy, this implies:
\[
E_{\max}(\rho) \leq D_{\max}(\rho) \leq C_{\max}(\rho) = \log_2 d - S(\rho)
\]
That is, the purity of $\rho$ strictly upper-bounds the maximal entanglement or discord that can be generated from $\rho$ via any unitary transformation [1612.07570].

## 5. Single-Shot Regime and Operational Purity Measures

Single-copy versions are given by Rényi-$\alpha$ purities:
\[
P_\alpha(\rho) = \log_2 d - S_\alpha(\rho)
\]
yielding
- **Single-shot distillable purity:** $P_d^{(1)}(\rho) = \lfloor \lim_{\alpha \to 0} P_\alpha(\rho) \rfloor = \lfloor \log_2 ( d / r ) \rfloor$, with $r$ the rank of $\rho$.
- **Single-shot purity cost:** $P_c^{(1)}(\rho) = \lceil \lim_{\alpha \to \infty} P_\alpha(\rho) \rceil = \lceil \log_2 ( d \lambda_{\max} ) \rceil$, with $\lambda_{\max}$ the largest eigenvalue.

This framework clarifies the conversion between different manifestations of quantumness, with purity as the elementary currency.

## 6. Experimental Considerations and Protocols

Operationally, purity is directly measurable in experiments via two-copy interference protocols (such as the swap test), which provide access to the Hilbert-Schmidt purity $\operatorname{Tr} \rho^2$. Thus, the maximal achievable coherence—and hence the maximal entanglement or discord attainable by unitary transformations—can be tightly bounded with experimentally accessible observables.

Further, protocols such as incoherent-operation activation demonstrate that rotating $\rho$ to its $\rho_{\max}$ basis, followed by an optimal IO, achieves entanglement production $P_r(\rho) = \log_2 d - S(\rho)$, again equating purity to a meaningful operational resource. For two-qubit states, e.g., applying a CNOT to $\rho^A \otimes |0\rangle \langle 0|^B$, the produced negativity is $N = C_{\ell_1}(\rho^A)/2$, which is always upper-bounded by the geometric purity $P_g(\rho^A)$.

## 7. Synopsis and Unified Framework

Purity—as measured by contractive distance from the maximally mixed state—emerges as the fundamental, basis-independent resource governing coherence generation, entanglement activation, and discord creation. Every quantum informational resource generable from a state $\rho$ via basis changes or interaction is strictly limited by the state's purity. This unifies the resource theories of purity, coherence, discord, and entanglement, with purity as the elementary quantifier [1612.07570].

This unification has profound implications:
- Operational resource interconversion is fundamentally purity-limited.
- All typical coherence and correlation activation processes can be reframed in terms of maximal extractable purity.
- Experimental strategies can, in many scenarios, bypass coherence quantification in favor of purity measurement.
- The result clarifies resource-theoretic connections between protocols in quantum information, quantum thermodynamics, and foundational quantum theory.

**References:**
- "Maximal Coherence and the Resource Theory of Purity" [1612.07570]
- For additional context on operational and dynamical extensions see also [1704.04868], [1910.00708], [1704.03710].

Source: https://www.emergentmind.com/topics/resource-theory-of-quantum-coherence