Entropic Coherence in Quantum Resources

• Entropic coherence is defined as the entropy gap between a quantum state and its dephased counterpart, capturing basis-dependent superposition.
• It extends the resource theory by incorporating quantum-addition channels, Jensen–Shannon constructions, and energy constraints to unify uncertainty, interferometry, and thermodynamics.
• Applications include quantifying coherence distribution in multipartite systems, establishing coherence-based uncertainty relations, and assessing thermodynamic irreversibility.

Entropic coherence is the family of quantum-coherence quantifiers defined through entropy differences or relative-entropic distances to an incoherent reference set. In the resource theory of coherence, the canonical instance is the relative entropy of coherence, $$C_r(\rho)=\min_{\sigma\in\mathcal I}S(\rho\|\sigma)=S(\Delta(\rho))-S(\rho)$$, where $\mathcal I$ is the set of basis-diagonal states and $\Delta$ is the dephasing map in a fixed reference basis. Subsequent work generalized this template to quantum-addition channels, Jensen–Shannon-divergence constructions, incoherent-quantum reference sets, and Hamiltonian-resolved notions based on energy twirling, thereby turning entropic coherence into a unifying language across resource theory, uncertainty relations, interferometry, thermodynamics, and symmetry-constrained control (Winter et al., 2015, Mukhopadhyay et al., 2018, Castellano et al., 1 Sep 2025).

1. Resource-theoretic definition and formal scope

Entropic coherence is basis dependent. Given a preferred orthonormal basis $\{|i\rangle\}$, the incoherent states are exactly the diagonal density operators $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$. The relative entropy of coherence measures the informational gap between a state and its dephased counterpart:

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$

This quantity is nonnegative, vanishes exactly on incoherent states, and admits a direct asymptotic operational interpretation: the distillable coherence equals $C_r(\rho)$, while the coherence cost equals the coherence of formation $C_f(\rho)$, so mixed-state coherence theory is generally irreversible but has no bound coherence (Winter et al., 2015).

The same entropic pattern reappears in several variants. In unilateral settings, dephasing is applied only on one subsystem; in Hamiltonian-constrained settings, incoherence is defined relative to energy eigenspaces rather than a computational basis; and in multipartite settings the reference set can be enlarged from fully incoherent states to separable or incoherent-quantum states. The common structure is that coherence is quantified as an entropy-theoretic discrepancy between a full state and a constrained reference object.

Measure	Definition	Setting
Relative entropy of coherence	$C_r(\rho)=S(\Delta(\rho))-S(\rho)$	Fixed basis (Winter et al., 2015)
Quantum-addition coherence	$$C_\alpha(\rho)=\min_{\sigma\in\mathcal I}S(\rho\boxplus_\alpha \sigma\|\alpha\rho+(1-\alpha)\sigma)$$	Quantum addition vs classical mixture (Mukhopadhyay et al., 2018)
QJSD coherence	$\mathcal I$0	Metric-like entropic distance (Chandrashekar et al., 2016)
Unilateral relative entropy coherence	$\mathcal I$1	Bipartite memory-assisted tasks (Dolatkhah et al., 2018)
Hamiltonian entropic coherence	$\mathcal I$2	Energy-superselection setting (Castellano et al., 1 Sep 2025)

A recurrent misconception is that entropic coherence should be basis independent. The literature instead treats the basis, or more generally the Hamiltonian and its eigenspace decomposition, as part of the definition of the resource. A second misconception is that entropy-based coherence is merely an alternative notation for measurement entropy. In fact, several works explicitly separate state mixedness from basis-dependent quantumness, so that the coherence term isolates the nonclassical contribution rather than raw unpredictability (Yuan et al., 2016).

2. Principal entropic constructions

The standard form $\mathcal I$3 is not the only entropic realization of coherence. A distinct construction arises from quantum addition channels. For qudit states $\mathcal I$4 and weight $\mathcal I$5,

$\mathcal I$6

The corresponding classical mixture is $\mathcal I$7. The reverse entropy power equality

$\mathcal I$8

implies $\mathcal I$9, with strict inequality when $\Delta$0 and $\Delta$1 do not commute. This motivates the coherence of quantum addition,

$\Delta$2

which interprets mixed-state coherence as the irreducible discrepancy between quantum mixing and classical mixing (Mukhopadhyay et al., 2018).

The same paper shows that $\Delta$3 satisfies the standard requirements of a coherence monotone. It is faithful, monotone under incoherent channels, and obeys the direct-sum relation

$\Delta$4

It is also upper bounded in terms of the relative entropy of coherence:

$\Delta$5

where $\Delta$6 is the binary entropy. This relation ties the newer quantum-addition measure back to the standard entropic monotone (Mukhopadhyay et al., 2018).

A second line of generalization uses the quantum Jensen–Shannon divergence,

$\Delta$7

with induced distance

$\Delta$8

The square root of the QJSD is metric for pure states and numerically supports the triangle inequality more broadly. This metric-like property enables decompositions into local and intrinsic coherence, because one can compare the state not only to the nearest incoherent state but also to the nearest separable state (Chandrashekar et al., 2016).

More recent work pushes the entropic program further through majorization. Using the Schur–Horn theorem, the diagonal of a density operator is majorized by its spectrum, which guarantees positivity of entropy-based coherence measures and motivates additional quantities such as the relative cross-entropy of coherence

$\Delta$9

together with partial variants defined on the first $\{|i\rangle\}$0 ordered components. This line of work also embeds coherence into a von Neumann–Tsallis entropy geometry and introduces degenerate coherence distillation for spectra with symmetry-induced degeneracies (Aziz et al., 12 Mar 2025).

3. Measurement, uncertainty, and interferometric complementarity

In measurement theory, entropic coherence functions as the genuinely quantum part of uncertainty. For a basis $\{|i\rangle\}$1, the relative entropy of coherence can be written as

$\{|i\rangle\}$2

and similarly for another basis $\{|i\rangle\}$3. The subtraction of $\{|i\rangle\}$4 removes basis-independent mixedness, so the remaining term quantifies basis-dependent quantumness rather than generic randomness. This viewpoint leads to coherence-based uncertainty relations between incompatible bases and yields explicit qubit bounds for the relative entropy of coherence, coherence of formation, and the $\{|i\rangle\}$5 norm (Yuan et al., 2016).

In bipartite settings with quantum memory, unilateral coherence supplies a direct bridge to memory-assisted entropic uncertainty relations. For a measurement $\{|i\rangle\}$6 on subsystem $\{|i\rangle\}$7,

$\{|i\rangle\}$8

so lower bounds on conditional entropies immediately become lower bounds on sums of unilateral coherences. This identity underlies strengthened two-basis and multi-basis coherence uncertainty relations with correction terms involving Holevo quantities and mutual information. In the multi-measurement setting, the conversion rule

$\{|i\rangle\}$9

systematically upgrades no-memory entropic uncertainty bounds into memory-assisted ones, and the resulting coherence bounds are tighter when the extra $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$0 term is positive (Dolatkhah et al., 2018, Dolatkhah et al., 2019).

These connections have been tested experimentally. In an all-optical platform using Bell-like and Bell-like diagonal states, measurements over the three qubit mutually unbiased bases verified both entropic uncertainty relations and coherence-based uncertainty relations. The experiment found that the lower bounds can be tightened by mutual-information and Holevo corrections, and that entropic uncertainty is inversely correlated with coherence (Ding et al., 2019).

Interferometry provides another canonical arena. For an $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$1-path interferometer, the path state $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$2 has coherence

$\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$3

which equals $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$4 because the joint particle-detector state is pure. When path information is characterized by mutual information between detector labels and measurement outcomes, the complementarity relation becomes

$\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$5

or equivalently with accessible information. When which-way information is obtained by zero-error identification, the normalized coherence $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$6 and distinguishability $\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$7 obey the duality relation

$\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$8

Allowing both errors and failures yields the mixed-strategy generalization

$\mathcal I=\{\rho=\sum_i p_i |i\rangle\langle i|\}$9

which reduces to the zero-error bound when $C_r(\rho)=S(\Delta(\rho))-S(\rho).$0 (Bagan et al., 2015, Bagan et al., 2020).

4. Multipartite distribution, localization, and monogamy

Entropic coherence is not only a single-system quantity; it also diagnoses how coherence is distributed across composite systems. For bipartite states, distributed coherence is defined by the gap between global coherence and the sum of local coherences,

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$1

and can be rewritten as

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$2

Thus distributed coherence is the mutual information destroyed by dephasing. Genuine distributed coherence is then obtained by minimizing $C_r(\rho)=S(\Delta(\rho))-S(\rho).$3 over incoherent unitaries, which preserve total coherence while possibly localizing it on subsystems (Kraft et al., 2018).

This framework separates coherence from entanglement. A maximally entangled state can have vanishing genuine distributed coherence if an incoherent unitary concentrates all global coherence locally. In the two-qubit pure-state case, every maximally entangled state has $C_r(\rho)=S(\Delta(\rho))-S(\rho).$4, whereas the state

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$5

maximizes genuine distributed coherence among two-qubit pure states, despite having neither maximal global coherence nor maximal coherence rank (Kraft et al., 2018).

The QJSD-based entropic distance supplies a complementary multipartite decomposition. One defines intrinsic coherence by minimizing over separable states and local coherence by the residual distance to the nearest incoherent state. The triangle inequality then yields

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$6

and in multipartite systems one can further resolve coherence into site-local and genuinely collective parts. This leads to trade-off relations and a monogamy indicator

$C_r(\rho)=S(\Delta(\rho))-S(\rho).$7

where $C_r(\rho)=S(\Delta(\rho))-S(\rho).$8 indicates monogamy and $C_r(\rho)=S(\Delta(\rho))-S(\rho).$9 indicates polygamy. GHZ states are strictly monogamous in this sense, while W states are strictly polygamous (Chandrashekar et al., 2016).

A more refined entropic treatment uses incoherent-quantum reference sets. For bipartite $C_r(\rho)$0, the relative entropy of IQ coherence,

$C_r(\rho)$1

together with its smooth max- and min-relative entropy analogues, yields distribution laws of the form

$C_r(\rho)$2

Here the total coherence is lower bounded by the sum of local coherence and genuine multipartite entanglement. The same framework produces monogamy relations such as

$C_r(\rho)$3

capturing the non-sharability of coherence relative to a common quantum memory (Bu et al., 2017).

5. Thermodynamic and dynamical interpretations

In nonequilibrium thermodynamics, entropic coherence appears as a directly quantifiable contribution to irreversibility. For a closed system initially in equilibrium and driven unitarily to $C_r(\rho)$4, the irreversible entropy production obeys

$C_r(\rho)$5

where $C_r(\rho)$6 is the relative entropy of coherence in the instantaneous energy basis and $C_r(\rho)$7 is the final equilibrium Gibbs state at the initial inverse temperature. The first term measures irreversibility due to created superpositions between energy levels; the second measures population mismatch after dephasing. The same decomposition holds at intermediate times, coherence becomes the leading contribution to non-adiabaticity in slow driving, and the generated coherence satisfies an integral fluctuation theorem (Francica et al., 2017).

This decomposition has been verified experimentally in a driven two-qubit NMR quantum processor. In that setting, the irreversible entropy production was resolved into a coherence contribution

$C_r(\rho)$8

and a population-mismatch term. The experiment confirmed the generalized Clausius inequality

$C_r(\rho)$9

and observed that faster driving produces more entropy and more coherence, while slower driving suppresses coherence so that population mismatch dominates the residual irreversibility (Shende et al., 2024).

Open-system formulations lead to a related decomposition. Writing $C_f(\rho)$0, one has

$C_f(\rho)$1

and consequently

$C_f(\rho)$2

For the spin systems studied in phase space via Husimi $C_f(\rho)$3 functions and Wehrl entropy, larger initial coherence generally produces a larger entropy production rate under both dephasing and amplitude damping. The phase-space formalism is especially useful at zero temperature or for pure states, where the von Neumann-relative-entropy formulation can become singular (Zicari et al., 2024).

Entropic coherence also characterizes how quantum evolutions generate or destroy superposition. The cohering power of a unitary $C_f(\rho)$4 is

$C_f(\rho)$5

which reduces to a maximization over incoherent basis states. For a one-qubit unitary $C_f(\rho)$6, the cohering power is

$C_f(\rho)$7

The same work defines the decohering power of channels, proves that controlled-unitary cohering power reduces to that of the target unitary, and derives the chain

$C_f(\rho)$8

linking local coherence, global coherence, quantum correlations, and entanglement (Xi et al., 2015).

6. Symmetry-constrained resources and recent extensions

In Hamiltonian-constrained resource theories, incoherence is defined relative to energy eigenspaces. Let

$C_f(\rho)$9

be the energy twirling map. The entropic coherence relative to $C_r(\rho)=S(\Delta(\rho))-S(\rho)$0 is

$C_r(\rho)=S(\Delta(\rho))-S(\rho)$1

where $C_r(\rho)=S(\Delta(\rho))-S(\rho)$2 is the set of states with no coherence between different energy eigenspaces. In the task of implementing a non-energy preserving gate $C_r(\rho)=S(\Delta(\rho))-S(\rho)$3 using an energy-preserving interaction with a battery, this quantity is a necessary resource. If the gate is approximated with worst-case infidelity $C_r(\rho)=S(\Delta(\rho))-S(\rho)$4, then the battery coherence must satisfy logarithmic lower bounds such as

$C_r(\rho)=S(\Delta(\rho))-S(\rho)$5

and, for a proportionate battery family,

$C_r(\rho)=S(\Delta(\rho))-S(\rho)$6

An immediate corollary is that any finite-dimensional battery has a strictly positive minimum gate-implementation error (Castellano et al., 1 Sep 2025).

Gravitational and relativistic settings have recently supplied further applications. For GHZ and W states of a fermionic field in Einstein–Gauss–Bonnet gravity, coherence is quantified by both the $C_r(\rho)=S(\Delta(\rho))-S(\rho)$7 norm and the relative entropy of coherence. Hawking radiation increases uncertainty and degrades coherence; in $C_r(\rho)=S(\Delta(\rho))-S(\rho)$8 dimensions increasing horizon radius decreases uncertainty and increases coherence, whereas in $C_r(\rho)=S(\Delta(\rho))-S(\rho)$9 both quantities become non-monotonic. The W state is more robust in preserving coherence, while the GHZ state is more resistant to Hawking-induced growth of entropic uncertainty (Li et al., 15 Oct 2025).

Many-body dynamics has introduced a distinct coherence-controlled transition. In random permutation dynamics, the reduced density matrix of a subsystem always thermalizes to the maximally mixed state, but the projected ensemble of post-measurement conditional pure states undergoes a deep-thermalization transition between a Haar-random phase and a classical bit-string phase. The relevant order parameter is the ensemble-averaged relative entropy of coherence,

$$C_\alpha(\rho)=\min_{\sigma\in\mathcal I}S(\rho\boxplus_\alpha \sigma\|\alpha\rho+(1-\alpha)\sigma)$$0

not the coherence of the averaged reduced state. In the mixed-basis model, the transition occurs at

$$C_\alpha(\rho)=\min_{\sigma\in\mathcal I}S(\rho\boxplus_\alpha \sigma\|\alpha\rho+(1-\alpha)\sigma)$$1

demonstrating that injected coherence can control ergodicity breaking at the level of conditional-state ensembles even when ordinary thermalization remains intact (Liu et al., 21 Oct 2025).

Taken together, these developments show that entropic coherence is not a single formula but a coherent research program. Its core invariant is the entropy-theoretic discrepancy between a quantum state and a constrained incoherent reference; its main structural features are basis or symmetry dependence, monotonicity under the relevant free operations, and direct convertibility into statements about distinguishability, uncertainty, distributed correlations, irreversibility, and symmetry-limited control. The field’s current trajectory suggests further integration of majorization geometry, one-shot resource theory, and Hamiltonian-resolved operational tasks within a common entropic framework (Aziz et al., 12 Mar 2025).