Relative Entropy of Coherence (REC)

• Relative Entropy of Coherence (REC) is a basis-dependent measure defined as the entropy increase from dephasing a quantum state.
• REC applies to finite- and infinite-dimensional systems and underpins operational tasks like coherence distillation and metrology.
• REC satisfies standard coherence axioms, links to entanglement measures, and enables experimental assessment without full state tomography.

Relative Entropy of Coherence (REC), also denoted $C_r$, $C_{RE}$, or $C_{\mathrm{rel}}$ in different parts of the literature, is a basis-dependent coherence monotone in the resource theory of coherence. For a density operator $\rho$ and a fixed incoherent basis $\{|i\rangle\}$, REC is defined as the minimum quantum relative entropy from $\rho$ to the set of incoherent states, and admits the closed form

$$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$

where $\Delta$ is the dephasing map that deletes all off-diagonal entries in the chosen basis. This entropy-increment characterization under dephasing is the starting point for its axiomatic analysis, its operational identification with distillable coherence, and its extensions to infinite-dimensional systems, generalized measurements, Bayesian metrology, and quantum channels (Winter et al., 2015, Bai et al., 2015, Zhang et al., 2015, Lecamwasam et al., 2024, Kamin et al., 2020).

1. Definition and entropy-difference form

In the standard finite-dimensional formulation, one fixes once and for all an orthonormal incoherent basis $\{|i\rangle\}$, declares as free states the density operators diagonal in that basis, and defines the dephasing map

$$\Delta(\rho)=\sum_i |i\rangle\langle i|\,\rho\,|i\rangle\langle i|.$$

REC is then the minimum relative entropy distance to the incoherent set: $C_{RE}$0 The minimizer is $C_{RE}$1, so REC reduces exactly to the entropy increase induced by dephasing (Winter et al., 2015, Bai et al., 2015).

This formulation makes explicit that REC is basis-dependent. It measures how far $C_{RE}$2 is from the set of diagonal states in the chosen basis, or equivalently how much entropy is generated by erasing its off-diagonal terms. In the high-energy application to top–antitop spin states, for example, the relevant basis is the helicity basis, and a larger $C_{RE}$3 indicates stronger quantum superpositions in that basis (Chen et al., 3 Apr 2026).

For pure states $C_{RE}$4, REC reduces to the Shannon or von Neumann entropy of the decohered population distribution: $C_{RE}$5 Thus, for pure states, coherence is completely determined by the basis populations of the state vector (Zhu et al., 2017).

2. Structural properties and extremal states

REC satisfies the standard Baumgratz–Cramer–Plenio coherence-measure axioms. It is faithful, in the sense that $C_{RE}$6 if and only if $C_{RE}$7 is incoherent; monotone under incoherent CPTP maps; strongly monotone under selective incoherent measurements; and convex under mixing. It is also additive on tensor products,

$C_{RE}$8

and super-additive on bipartite states,

$C_{RE}$9

An asymptotic continuity bound is also available (Winter et al., 2015, Bai et al., 2015, Zhu et al., 2017).

The maximal value of REC on a $C_{\mathrm{rel}}$0-dimensional Hilbert space is $C_{\mathrm{rel}}$1. States achieving this value are the maximally coherent states (MCSs). A complete characterization is known: a state is maximally coherent if and only if it is pure and has the form

$C_{\mathrm{rel}}$2

with $C_{\mathrm{rel}}$3 any diagonal unitary. In particular, no mixed state can attain maximal REC (Bai et al., 2015).

The bipartite structure of maximally coherent states is constrained. For a bipartite MCS with equal local dimensions $C_{\mathrm{rel}}$4, equality in super-additivity,

$C_{\mathrm{rel}}$5

holds if and only if $C_{\mathrm{rel}}$6. At the same time, there exist states that are simultaneously maximally coherent on the total space and maximally entangled between the subsystems, showing that maximal coherence and maximal entanglement are compatible in the equal-dimension setting (Bai et al., 2015).

REC is also related to robustness-type coherence measures. One has

$C_{\mathrm{rel}}$7

and the equality condition is fully characterized: $C_{\mathrm{rel}}$8 if and only if $C_{\mathrm{rel}}$9 commutes with $\rho$0 and $\rho$1, where $\rho$2 is the support projector (Zhu et al., 2017).

3. Operational significance

REC has a precise operational meaning in asymptotic resource conversion. In the operational resource theory of coherence, the distillable coherence $\rho$3 is defined as the optimal asymptotic rate at which copies of the maximally coherent qubit state $\rho$4 can be extracted from many copies of $\rho$5 using incoherent operations. The central single-letter result is

$\rho$6

This identifies REC as the asymptotic distillation rate of coherent bits (Winter et al., 2015).

The corresponding formation task is quantified by the coherence cost $\rho$7, which equals the coherence of formation $\rho$8. Since generally $\rho$9, coherence theory is generically irreversible; the reversible states are special block-diagonal states in which each block is pure. An immediate corollary is that there is no bound coherence in the asymptotic incoherent-operations setting: any state with nonzero REC has distillable coherence (Winter et al., 2015).

A complementary operational connection links coherence to entanglement generation. The relative entropy of coherence equals the maximal relative-entropy entanglement that incoherent operations can create from $\{|i\rangle\}$0, and for maximally correlated states the relative entropy of coherence coincides with the corresponding relative-entropy entanglement measure. This places REC at the interface between single-system superposition and bipartite entanglement theory (Zhu et al., 2017).

In Bayesian metrology, REC acquires a further operational interpretation through ensembles. For an ensemble $\{|i\rangle\}$1 with average state $\{|i\rangle\}$2, the ensemble coherence is defined as

$\{|i\rangle\}$3

For a projective measurement $\{|i\rangle\}$4 in the incoherent basis, one has the CXI equality

$\{|i\rangle\}$5

where $\{|i\rangle\}$6 is the Holevo information and $\{|i\rangle\}$7 is the mutual information between the encoded parameter and the measurement outcome. The same equality extends to arbitrary POVMs via Naimark dilation, yielding a POVM-coherence $\{|i\rangle\}$8. In this formulation, REC quantifies exactly the information gap between a chosen measurement and the Holevo-optimal performance, including unitary, dissipative, and discrete encoding settings (Lecamwasam et al., 2024).

4. Extensions beyond the standard finite-dimensional setting

REC extends to infinite-dimensional bosonic systems in Fock space provided an energy constraint is imposed. For a single-mode state $\{|i\rangle\}$9 with mean photon number bounded by

$\rho$0

one defines

$\rho$1

where $\rho$2 dephases in the number basis. Under this constraint, the diagonal entropy is finite and bounded by the thermal photon-number distribution, implying

$\rho$3

Thus REC remains a bona fide coherence measure in infinite dimensions, with the usual monotonicity and convexity properties supplemented by a finiteness-under-finite-energy condition (Zhang et al., 2015).

The same construction generalizes to multi-mode Fock space. At fixed total mean photon number, the maximal relative entropy of coherence increases strictly with the number of modes. The paper further reports the same ordering numerically for physically relevant states such as two-mode coherent states, two-mode squeezed vacuum, and beam-splitter outputs (Zhang et al., 2015).

A major line of generalization replaces the Umegaki relative entropy with Rényi, Tsallis, or sandwiched Rényi divergences. In the Rényi case, the distance-based coherence measure

$\rho$4

recovers the ordinary REC in the limit $\rho$5, is additive, and supports analytic formulas and bounds. For maximally correlated states, each Rényi relative entropy of coherence equals the corresponding Rényi relative entropy of entanglement (Zhu et al., 2017).

The generalized entropy-increment picture is not uniform across families. For Rényi coherence, the closest incoherent state is a weighted diagonal state $\rho$6, and the distance-based and entropy-increment formulations coincide. By contrast, the Tsallis entropy-difference form does not define a genuine coherence monotone under incoherent operations, except under the restrictive class of genuinely incoherent operations; the distance-based Tsallis coherence is monotone under incoherent operations for $\rho$7 but fails strong monotonicity for $\rho$8 (Vershynina, 2022). Sandwiched Rényi coherence measures provide another interpolation family satisfying the BCP axioms for $\rho$9, and they reduce to standard REC as $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$0 (Xu, 2018). A common misconception is therefore that any entropy-difference analogue of REC automatically inherits the same resource-theoretic status; the Tsallis case shows that this is not generally so.

5. Channel coherence, discord, and non-Markovianity

REC admits a bipartite quantum–incoherent extension in which coherence is treated as a resource only on one subsystem. For a bipartite state $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$1, the quantum–incoherent relative entropy of coherence is

$$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$2

where $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$3 dephases only subsystem $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$4. This quantity upper-bounds the rate at which incoherent ancilla states can be distilled into maximally coherent qubit states on $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$5 via LQICC operations (Wu et al., 2019).

Its dynamical behavior is sharply linked to CP-divisibility. Under CP-divisible evolution acting on one subsystem, $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$6 decreases monotonically for all bases; any temporal revival signals non-CP-divisibility and information backflow. By contrast, the ordinary single-system REC can be non-monotonic even for CP-divisible processes when the dynamics is not incoherent in the chosen basis. This is why QI REC, rather than local REC, was proposed as a basis-robust witness of non-Markovianity (Wu et al., 2019).

The channel-level resource theory formulates coherence directly for quantum channels by combining the Choi–Jamiołkowski isomorphism with coherence-breaking channels (CBCs) as free operations. For a channel $$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$7 with Choi state

$$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$8

the channel quantum–incoherent REC is

$$C_r(\rho)=\min_{\delta\in\mathcal I} S(\rho\|\delta)=S(\Delta(\rho))-S(\rho),$$9

where $\Delta$0 is the set of Choi states of CBCs and $\Delta$1 dephases only the system register. For qubit channels, this channel quantity coincides with the ordinary REC of the Choi state,

$\Delta$2

The same framework yields a decomposition

$\Delta$3

so the basis-dependent asymmetric quantum discord of the Choi state never exceeds channel REC. The paper also states that for qubit channels the basis-dependent quantum symmetric discord can never exceed the coherence, and proves monotonic decrease of $\Delta$4 for divisible quantum incoherent evolutions (Kamin et al., 2020).

6. Measurement protocols and representative applications

REC can be measured directly without full state tomography. The key observation is that the von Neumann entropy can be obtained as the minimum Shannon entropy over projective measurement bases,

$\Delta$5

A direct protocol therefore consists of scanning a family of measurements to identify the minimum outcome entropy, which yields $\Delta$6, and then measuring in the incoherent basis to obtain $\Delta$7. Their difference gives $\Delta$8. Explicit quantum-optical implementations were proposed for two-path spatial coherence and for polarization coherence, both avoiding full tomography (Bernardo, 2018).

In open-system dynamics, REC and entanglement can exhibit markedly different decay patterns. For a two-qubit Werner state under local bit-flip, phase-damping, and amplitude-damping channels, the relative-entropy coherence is more robust than concurrence. The bit-flip channel does not produce a constant nonzero frozen REC; under phase damping and amplitude damping, REC decays smoothly and vanishes only at $\Delta$9, whereas concurrence can undergo temporary disappearance or complete attenuation to zero earlier in the evolution (Sun et al., 2024).

The high-energy application to top–antitop spin states in QCD uses REC for the two-qubit spin density matrix

$\{|i\rangle\}$0

in the helicity basis. In that setting REC depends on the invariant mass $\{|i\rangle\}$1, the scattering angle $\{|i\rangle\}$2, and the gluon-mixture weight $\{|i\rangle\}$3. For pure $\{|i\rangle\}$4, REC decreases monotonically with $\{|i\rangle\}$5 near threshold; for pure $\{|i\rangle\}$6, it increases with $\{|i\rangle\}$7; and in mixed channels the low-$\{|i\rangle\}$8 suppression is gradually filled in as $\{|i\rangle\}$9 increases. REC also enters a complete complementarity relation and an intrinsic uncertainty–coherence–predictability relation for the top–antitop system (Chen et al., 3 Apr 2026).

Across these settings, REC functions as an entropy-based quantifier of basis-resolved superposition, but its precise interpretation depends on context: a state monotone in finite-dimensional resource theory, a finite quantity under energy constraints in Fock space, an operational rate in coherence distillation, an information gap in Bayesian metrology, and—through QI extensions—a diagnostic for channel quantumness and non-Markovian information backflow (Winter et al., 2015, Zhang et al., 2015, Lecamwasam et al., 2024, Kamin et al., 2020, Wu et al., 2019).