Resource Theory of Quantum Coherence

• Resource theory of quantum coherence defines superposition as an operational resource, quantifying and manipulating coherence through unitary extraction methods.
• It establishes that the maximal coherence identical to a state’s purity limits the generation of entanglement and discord, unifying different quantum resources.
• The framework provides experimentally accessible measures, such as the swap test, to directly assess purity and its impact on coherence and other quantum correlations.

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 (Streltsov et al., 2016).

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 (Streltsov et al., 2016).

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 $D$ (Streltsov et al., 2016). 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 (Streltsov et al., 2016).

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 (Streltsov et al., 2016).

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" (Streltsov et al., 2016)
• For additional context on operational and dynamical extensions see also (Yang et al., 2017, Saxena et al., 2019, Dana et al., 2017).