Resource Theory of Coherence

Superposition is the first thing quantum mechanics teaches and the last it lets you keep. A qubit prepared in $\tfrac{1}{\sqrt2}(|0\rangle+|1\rangle)$ carries a definite phase relation between two basis states; leave it exposed to its environment and that phase relation bleeds away, leaving a classical coin. The resource theory of coherence is the accounting system for that phase relation — it treats definite superposition relative to a fixed basis as a commodity that can be measured, spent, transferred, and irreversibly lost, exactly as one treats energy in thermodynamics or entanglement in quantum communication. The construction, due to Baumgratz, Cramer and Plenio in 2014 (building on Åberg's 2006 formalization of superposition), imports the architecture that had already tamed entanglement. Fix a preferred orthonormal basis $\{|i\rangle\}$ — computational, energy, or a decoherence-selected pointer basis. Declare the states diagonal in it free (incoherent), because a classical apparatus can prepare them for nothing, and declare the operations that can never manufacture off-diagonal terms free operations. Then any genuine superposition is a resource, and a coherence monotone assigns it a number that free operations can only leave alone or destroy. This article develops the theory to its operational core: the two workhorse measures $C_r(\rho)=S(\Delta\rho)-S(\rho)$ and $C_{\ell_1}(\rho)=\sum_{i\neq j}|\rho_{ij}|$ ; the hierarchy of free operation classes and why it is not one class but many; and the crowning result that $C_r$ is the physical rate at which pure coherence can be distilled.

Key takeaways

The theory fixes a reference basis and calls the diagonal (incoherent) states free; every superposition of basis states is the resource, and the fully dephasing channel Δ is the canonical way to destroy it.
Free operations are the incoherent operations (IO): channels with a Kraus decomposition that maps incoherent states to incoherent states branch-by-branch — flanked by looser (MIO) and stricter (SIO, DIO, PIO) classes that are genuinely inequivalent.
A coherence monotone vanishes exactly on the incoherent set and never increases under the free operations; the two standard choices are the relative entropy of coherence $C_r(\rho)=S(\Delta\rho)-S(\rho)$ and the $\ell_1$ -norm $C_{\ell_1}(\rho)=\sum_{i\neq j}|\rho_{ij}|$ .
The maximally coherent state $|\Psi_d\rangle=\tfrac{1}{\sqrt d}\sum_i|i\rangle$ is the universal unit of the resource, with $C_r=\log_2 d$ and $C_{\ell_1}=d-1$ ; it can generate any state of the same dimension by IO alone.
$C_r$ carries an operational meaning: it equals the distillable coherence, the optimal asymptotic rate of extracting maximally coherent qubits under IO — the coherence analogue of the entropy of entanglement.
Key idea: pick a basis, call its diagonal states free and the operations that never create off-diagonal terms free; then coherence becomes a bona fide resource whose amount is captured by basis-dependent monotones, the foremost being $C_r(\rho)=S(\Delta\rho)-S(\rho)$ , equal to the asymptotic distillable coherence.

Where this lives today

Coherence-as-resource has moved from formalism to laboratory bookkeeping. Photonic and NMR experiments now directly witness and quantify $C_{\ell_1}$ and $C_r$ by interferometric readout of the off-diagonals. The theory is the natural currency of quantum thermodynamics: with the reference basis taken as the energy eigenbasis, coherence between energy levels is a resource thermal operations cannot create for free, setting limits on quantum heat engines, batteries, and the work extractable from a coherently driven qubit — coherent strokes can beat the incoherent bound, but only by spending a budget the theory tracks exactly. In metrology, a probe's coherence lower-bounds its quantum Fisher information for phase estimation. The live open problems are structural: the theory is asymptotically reversible under MIO but not under IO, where bound coherence — costing more to create than can ever be distilled back — genuinely exists, mirroring bound entanglement. Pinning down the exact gaps between MIO, IO, DIO and SIO remains an active frontier a decade on.

The physical picture

Every resource theory partitions the world into the free and the precious, and the whole content of coherence theory is encoded in the density matrix written in the chosen basis. The diagonal entries $\rho_{ii}$ are populations — the classical probabilities of finding the system in each basis state, an interferometer's which-path information. The off-diagonal entries $\rho_{ij}$ are the coherences — the phase relations that make interference fringes visible. A classical apparatus can dial any diagonal for free, so the diagonal is the free part; the off-diagonals are what quantum mechanics adds and decoherence takes away. The dephasing channel Δ erases the off-diagonals while leaving the diagonal untouched — the natural yardstick, since coherence is what Δ destroys.

Figure

The density matrix in the reference basis splits into a free diagonal (populations, which-path information) and a resourceful off-diagonal (coherences, fringe visibility). The dephasing channel Δ deletes the off-diagonals, projecting ρ onto the incoherent set, and the entropy it releases is the relative entropy of coherence C_r = S(Δρ) − S(ρ)

The parallel to entanglement theory is a near-isomorphism: incoherent states play the role of separable states, incoherent operations that of LOCC, the maximally coherent state $|\Psi_d\rangle$ that of the singlet, and distillable coherence mirrors distillable entanglement. The second figure makes the payoff concrete: the two standard measures agree on the qubit unit but diverge as the dimension grows, a reminder that a resource can be scored on more than one honest scale.

Figure

The two standard measures of the maximally coherent state Ψ_d as dimension grows: the relative entropy of coherence C_r = log₂ d in blue rises logarithmically, while the ℓ₁-norm C_ℓ1 = d − 1 in amber rises linearly. They coincide at the qubit unit d = 2 (both equal 1) and split thereafter — at d = 8, C_r = 3 bits but C_ℓ1 = 7

The core equations

The theory is specified by three ingredients — free states, free operations, monotones — closed by an operational theorem, taken here in that order. The first fixes what "free" means for states: a density matrix is incoherent exactly when it is diagonal in the reference basis.

\mathcal{I}=\Big\{\,\rho=\sum_{i} p_i\,|i\rangle\langle i|\ :\ p_i\geq0,\ \textstyle\sum_i p_i=1\,\Big\} \qquad (1)

where: $\{|i\rangle\}$ — the fixed reference orthonormal basis of $\mathbb{C}^d$ , an input to the theory; $p_i$ — classical probabilities, the eigenvalues of the diagonal state; $\mathcal{I}$ — the convex set of incoherent states. Reading: $\mathcal{I}$ is the free set; its convexity matters (mixing free states must stay free), and every resource number below vanishes precisely on it. The dephasing channel projects any state onto $\mathcal{I}$ by deleting off-diagonals — the canonical free way to destroy coherence and the reference map against which it is measured.

\Delta(\rho)=\sum_i |i\rangle\langle i|\,\rho\,|i\rangle\langle i|=\sum_i \rho_{ii}\,|i\rangle\langle i| \qquad (2)

where: $\Delta$ — the completely dephasing channel; $\rho_{ii}=\langle i|\rho|i\rangle$ — a diagonal matrix element. Reading: Δ is CPTP, idempotent ( $\Delta\circ\Delta=\Delta$ ), and fixes exactly the incoherent states ( $\Delta\rho=\rho \iff \rho\in\mathcal{I}$ ) — the coherence-theory analogue of the partial trace, the free operation that annihilates the resource. Free operations are subtler, because "cannot create coherence" can be read at the level of the channel or of its individual outcomes. The original and central choice is the incoherent operations.

\mathcal{E}(\rho)=\sum_n K_n\rho K_n^\dagger,\qquad K_n\,\mathcal{I}\,K_n^\dagger\subseteq\mathcal{I}\ \ \forall n,\qquad \sum_n K_n^\dagger K_n=I \qquad (3)

where: $\{K_n\}$ — a Kraus decomposition of $\mathcal{E}$ ; $K_n\rho K_n^\dagger$ — the unnormalized post-measurement state of outcome $n$ , required incoherent whenever the input was. Reading: the demand is branch-by-branch — no single selective outcome may create coherence — strictly stronger than merely mapping $\mathcal{I}\to\mathcal{I}$ overall (the weaker demand defining the largest class, MIO). Incoherence is a property of a decomposition, not of the channel alone. The first workhorse measure quantifies coherence as distinguishability from the free set, collapsing to a simple closed form.

C_r(\rho)=\min_{\sigma\in\mathcal{I}}D(\rho\,\|\,\sigma)=S(\Delta\rho)-S(\rho) \qquad (4)

where: $D(\rho\|\sigma)=\mathrm{Tr}\,\rho(\log\rho-\log\sigma)$ — quantum relative entropy; $S(\rho)=-\mathrm{Tr}\,\rho\log\rho$ — von Neumann entropy (log base 2, so the unit is bits). Reading: the closest free state to $\rho$ is always its own dephasing $\Delta\rho$ , so the coherence is exactly the entropy Δ releases (Derivation A). The second workhorse measure is geometric, not entropic: add up the moduli of the off-diagonal entries.

C_{\ell_1}(\rho)=\sum_{i\neq j}|\rho_{ij}|=\Big(\sum_{i,j}|\rho_{ij}|\Big)-1 \qquad (5)

where: $\rho_{ij}=\langle i|\rho|j\rangle$ — an off-diagonal matrix element in the reference basis. Reading: $C_{\ell_1}$ is zero for diagonal states and trivial to compute, the everyday diagnostic; it is an IO monotone but, unlike $C_r$ , is not a monotone under all of MIO — a first warning that measure and operation class must be quoted together. The maximally coherent state is the universal unit — the state from which every other state of the same dimension can be reached for free.

|\Psi_d\rangle=\frac{1}{\sqrt d}\sum_{i=0}^{d-1}|i\rangle,\qquad C_r(|\Psi_d\rangle)=\log_2 d,\quad C_{\ell_1}(|\Psi_d\rangle)=d-1 \qquad (6)

where: $|\Psi_d\rangle$ — the uniform superposition, the golden unit of coherence in dimension $d$ . Reading: because $|\Psi_d\rangle\to|\psi\rangle$ by IO for any target, its coherence upper-bounds every state's in that dimension; $\log_2 d$ and $d-1$ are the ceilings, one per measure. Finally, the identity that lifts $C_r$ from definition to physics — the rate at which pure coherence can be distilled.

C_d(\rho)=C_r^{\infty}(\rho)=C_r(\rho)\qquad (\text{distillable coherence under IO}) \qquad (7)

where: $C_d(\rho)$ — the optimal asymptotic rate of maximally coherent qubits $|\Psi_2\rangle$ extractable per copy of $\rho$ under IO; $C_r^{\infty}$ — its regularized value. Reading: Winter and Yang proved this single-letter identity — no regularization needed — establishing $C_r$ as the coherence analogue of the entropy of entanglement, the resource's true operational currency.

Derivation AThe closed form of the relative entropy of coherence

The goal is to prove the second equality in equation (4): that the variational quantity $\min_{\sigma\in\mathcal{I}}D(\rho\|\sigma)$ , an optimization over the entire convex set of incoherent states, collapses to the explicit $S(\Delta\rho)-S(\rho)$ , with the minimizer being $\rho$ 's own dephasing $\Delta\rho$ . This is what makes the relative entropy of coherence computable in closed form rather than as a convex program, and it mirrors the classical fact that the closest diagonal distribution to a given one, in relative entropy, is that distribution itself.

Step 1. Expand: $D(\rho\|\sigma)=-S(\rho)-\mathrm{Tr}(\rho\log\sigma)$ . Why: the first term is fixed by $\rho$ , so the entire dependence on the free state $\sigma$ lives in the cross term $-\mathrm{Tr}(\rho\log\sigma)$ , and minimizing $D$ means minimizing that term alone.

Step 2. For an incoherent $\sigma=\sum_i q_i|i\rangle\langle i|$ , $\log\sigma$ is diagonal, so $\mathrm{Tr}(\rho\log\sigma)=\sum_i\rho_{ii}\log q_i=\mathrm{Tr}(\Delta\rho\,\log\sigma)$ . Why: only the diagonal entries $\rho_{ii}$ ever meet the diagonal $\log\sigma$ ; the coherences are invisible to any incoherent $\sigma$ , so the optimal $\sigma$ can only see $\Delta\rho$ .

Step 3. Minimize the cross term. Writing $p_i=\rho_{ii}$ , Gibbs' inequality $-\sum_i p_i\log q_i\geq H(p)$ gives the optimum at $q_i=p_i$ , i.e. $\sigma=\Delta\rho$ , with value $S(\Delta\rho)$ . Why: cross-entropy is minimized by the true distribution, so the best free state is the one whose diagonal matches $\rho$ 's exactly.

Step 4. Assemble: $C_r(\rho)=S(\Delta\rho)-S(\rho)$ , non-negative since dephasing can only raise entropy, and vanishing iff $\rho=\Delta\rho$ , i.e. iff $\rho\in\mathcal{I}$ . Why: this delivers the closed form and verifies faithfulness (axiom C1) at once — $C_r$ is zero exactly on the free set. *Verified: for $\rho=\tfrac12(I+0.8\,\sigma_x)$ , $S(\rho)=0.4690$ bits and $S(\Delta\rho)=1$ bit give $C_r=0.531$ bits; the minimizer $\Delta\rho=\mathrm{diag}(\tfrac12,\tfrac12)$ was checked against Gibbs' inequality.*

Go deeper · Derivation B · 1 minWhy relative-entropy coherence cannot increase under incoherent operations

Derivation BWhy relative-entropy coherence cannot increase under incoherent operations

The goal is to establish the monotonicity axiom C2 for $C_r$ : that no incoherent operation $\mathcal{E}$ can raise it, $C_r(\mathcal{E}(\rho))\leq C_r(\rho)$ . This is the property that earns $C_r$ the title of monotone — a quantity free operations can only waste. The proof rests on two facts in hand: the closed form of Derivation A, and the data-processing inequality, which says relative entropy is contractive under every CPTP map, so no channel can make two states more distinguishable than they already were.

Step 1. Use the closed form as a variational bound. From Derivation A, $C_r(\rho)=D(\rho\|\Delta\rho)$ , and for the output, $C_r(\mathcal{E}(\rho))=\min_{\sigma\in\mathcal{I}}D(\mathcal{E}(\rho)\|\sigma)$ . Why: writing the input as an exact relative entropy and the output as a minimum lets us bound the output by any convenient incoherent trial state.

Step 2. Choose the trial state $\sigma=\mathcal{E}(\Delta\rho)$ . Because $\mathcal{E}$ is incoherent it maps the incoherent $\Delta\rho$ to an incoherent state, so this $\sigma$ is admissible, giving $C_r(\mathcal{E}(\rho))\leq D(\mathcal{E}(\rho)\|\mathcal{E}(\Delta\rho))$ . Why: the defining property of IO — preserving the free set — is exactly what makes $\mathcal{E}(\Delta\rho)$ a legal competitor in the minimization.

Step 3. Apply data processing. Relative entropy is contractive under any CPTP map, so $D(\mathcal{E}(\rho)\|\mathcal{E}(\Delta\rho))\leq D(\rho\|\Delta\rho)$ . Why: the same channel on both arguments cannot increase their relative entropy — identical noise can only blur the distinction — the one deep inequality the proof leans on.

Step 4. Chain: $C_r(\mathcal{E}(\rho))\leq D(\mathcal{E}(\rho)\|\mathcal{E}(\Delta\rho))\leq D(\rho\|\Delta\rho)=C_r(\rho)$ . Convexity of $C_r$ follows separately from the joint convexity of $D$ . Why: the combined inequality is axiom C2; with C1 from Derivation A it certifies $C_r$ as a full coherence monotone — the abstract requirement is now a theorem. *Verified: the two links use only (i) IO preserving $\mathcal{I}$ and (ii) the data-processing inequality for quantum relative entropy (Wilde, Ch. 11).*

Go deeper · Derivation C · 1 minCoherence of a pure state equals the entropy of its measurement statistics

Derivation CCoherence of a pure state equals the entropy of its measurement statistics

The goal is to show that for a pure state $|\psi\rangle=\sum_i c_i|i\rangle$ the relative entropy of coherence reduces to a classical quantity — the Shannon entropy $H(\{|c_i|^2\})$ of the distribution a basis measurement would produce. This closes the conceptual circle: coherence, an intrinsically quantum notion, is measured on pure states by exactly the classical uncertainty of where the state would be found — and the ceiling $\log_2 d$ falls out in one line.

Step 1. The purity term vanishes: a pure state has $S(|\psi\rangle\langle\psi|)=0$ , its density matrix having a single unit eigenvalue. Why: with $S(\rho)=0$ , the whole of $C_r=S(\Delta\rho)-S(\rho)$ reduces to the entropy of the dephased state, so only Δ's output matters.

Step 3. Take its entropy: $S(\Delta\rho)=-\sum_i|c_i|^2\log_2|c_i|^2=H(\{|c_i|^2\})$ , the entropy of a diagonal state being the Shannon entropy of its diagonal. Why: with the purity term zero, this gives $C_r(|\psi\rangle)=H(\{|c_i|^2\})$ — pure-state coherence is measurement uncertainty, nothing more.

Step 4. Specialize to the uniform superposition $|c_i|^2=1/d$ : $C_r=\log_2 d$ , recovering equation (6). Why: the flat distribution maximizes Shannon entropy at $\log_2 d$ , so $|\Psi_d\rangle$ is literally the maximally coherent pure state, its coherence the ceiling for the whole dimension. *Verified: for $|\Psi_3\rangle$ , $H(\{1/3,1/3,1/3\})=\log_2 3=1.585$ bits, matching equation (6); the pure-state $\ell_1$ value $(\sum_i|c_i|)^2-1=2$ .*

Limiting cases

The standard measures take instructive values in the regimes that bound the theory:

Case	Result	Meaning
Incoherent (diagonal) state	$C_r(\Delta\rho)=0,\ C_{\ell_1}(\Delta\rho)=0$	Faithfulness — every free state carries zero resource, by construction of the theory.
Maximally mixed state $I/d$	$C_r=0,\ C_{\ell_1}=0$	Randomness is not coherence: $I/d$ is diagonal in every basis, so it is free despite maximal entropy.
Pure state $\vert\psi\rangle=\sum_i c_i\vert i\rangle$	$C_r=H(\{\vert c_i\vert^2\}),\ C_{\ell_1}=(\sum_i\vert c_i\vert)^2-1$	$C_r$ becomes the Shannon entropy of the measurement statistics; both vanish iff a single $\vert c_i\vert=1$ .
Maximally coherent qudit $\vert\Psi_d\rangle$	$C_r=\log_2 d,\ C_{\ell_1}=d-1$	The unit resource: maximal on both scales and able to generate any $d$ -dimensional state by IO.
Single qubit $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$	$C_{\ell_1}=\sqrt{r_x^2+r_y^2}=\vert 2\rho_{01}\vert$	Coherence is the equatorial (Bloch-plane) radius; the $z$ -component is pure population and contributes nothing.

Common trapCoherence is an absolute number of a state, independent of basis and operation class

Two traps. First, coherence is irreducibly basis-dependent: the state $|\Psi_d\rangle$ that is maximally coherent in $\{|i\rangle\}$ is perfectly incoherent in its own eigenbasis. So every numerical claim — "this state has coherence $\log_2 d$ " — is implicitly relative to the fixed reference basis, and comparing coherences across different bases is meaningless. Second, the monotone and the operation class must be quoted together: $C_{\ell_1}$ is a bona fide monotone under IO, but it can increase under some maximally incoherent operations (MIO), and asymptotic reversibility holds under MIO but fails under IO. "Incoherent" is likewise a property of a chosen Kraus decomposition, not of the channel as a map.

Worked examples

1 · Coherence of a partially decohered qubit (fully worked)

A qubit is prepared with Bloch vector $\vec r=(0.8,0,0)$ — an equatorial state, partway between pure and maximally mixed. Compute $C_{\ell_1}$ and $C_r$ in the computational basis, and say what $C_r$ means operationally.

Step 1. With $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$ and $\vec r=(0.8,0,0)$ , the off-diagonal is $\rho_{01}=\tfrac12(r_x-ir_y)=0.4$ , so $\rho=\begin{pmatrix}0.5&0.4\\0.4&0.5\end{pmatrix}$ . Populations equal; all the structure is in the coherence.

Step 2. The $\ell_1$ -norm is immediate: $C_{\ell_1}=|\rho_{01}|+|\rho_{10}|=2(0.4)=\mathbf{0.8}$ , equivalently $\sqrt{r_x^2+r_y^2}=0.8$ — the fringe visibility a Mach–Zehnder interferometer would read for this state, 80% of the maximum.

Step 3. For $C_r$ , the eigenvalues of $\rho$ are $\tfrac{1\pm|\vec r|}{2}=\{0.9,0.1\}$ , so $S(\rho)=H(0.9)=-0.9\log_2 0.9-0.1\log_2 0.1=0.4690$ bits. The dephased state $\Delta\rho=\mathrm{diag}(0.5,0.5)$ has $S(\Delta\rho)=1$ bit. Hence $C_r=1-0.4690=\mathbf{0.531}$ bits.

Step 4. Interpret. By equation (7), $C_r$ is the distillable coherence: from $N$ copies, incoherent operations distill about $0.531\,N$ perfect maximally coherent qubits $|\Psi_2\rangle$ asymptotically. Concretely, 1000 copies yield ≈ 531 units of pure coherence — just over half a "coherbit" apiece. The two measures disagree in absolute value (0.8 versus 0.531): different honest scales, and only $C_r$ carries the distillation meaning.

2 · The maximally coherent qutrit as a unit of resource

Evaluate both measures for $|\Psi_3\rangle=\tfrac{1}{\sqrt3}(|0\rangle+|1\rangle+|2\rangle)$ , and compare its distillation value to that of a qubit unit.

Step 1. Every entry of the density matrix equals $1/3$ . The off-diagonals number $3^2-3=6$ , each of modulus $1/3$ , so $C_{\ell_1}=6\cdot\tfrac13=\mathbf{2}=d-1$ , saturating the qutrit ceiling.

Step 2. For $C_r$ , the state is pure so $S(\rho)=0$ , and $\Delta\rho=\mathrm{diag}(\tfrac13,\tfrac13,\tfrac13)$ has $S(\Delta\rho)=\log_2 3=\mathbf{1.585}$ bits (Derivation C). So $C_r(|\Psi_3\rangle)=1.585$ bits.

Step 3. Compare to the qubit unit $|\Psi_2\rangle$ ( $C_r=1$ bit). The qutrit carries $\log_2 3\approx1.585$ times as much: asymptotically, $N$ copies of $|\Psi_3\rangle$ convert by IO into about $1.585\,N$ copies of $|\Psi_2\rangle$ . One maximally coherent qutrit is worth log₂3 ≈ 1.585 maximally coherent qubits — which is why $|\Psi_2\rangle$ , the "coherbit," is the standard reference every resource is priced against.

Physics of the portrait

The resource theory of coherence has a clean origin: a single 2014 paper turned a century-old intuition about superposition into a quantitative theory, by an author who had already done the same for entanglement.

Origins · 1 minMartin Plenio and how this was discovered

Martin Plenio

b. 1968, German physicist · quantum information, resource theories of coherence and entanglement

Martin B. Plenio is a German theoretical physicist whose work runs like a spine through modern quantum resource theory. Shaped in the 1990s Imperial College group around Peter Knight, he became one of the architects of entanglement theory: with Vlatko Vedral he introduced the relative entropy of entanglement in 1997, casting a resource as the distance from a state to the free set — the very template later transplanted to coherence. In 2014, with Tillmann Baumgratz and Marcus Cramer, he published "Quantifying Coherence" (Phys. Rev. Lett. 113, 140401), which fixed the axioms C1–C2, proposed the relative-entropy and $\ell_1$ -norm measures, and launched coherence as a resource theory in its own right — building on Johan Åberg's 2006 formalization of "quantifying superposition." The move was characteristic Plenio: take the geometric-distance idea proven for entanglement and apply it to the more elementary resource of basis superposition. From his chair at Ulm University he has driven the field into quantum thermodynamics, metrology, and the operational hierarchy of incoherent operations. The resource theory of coherence is, in a real sense, entanglement theory's younger sibling — and Plenio is a parent of both.

History

2006

— Åberg formalizes the idea of quantifying superposition relative to a fixed basis

2014

— Baumgratz, Cramer and Plenio fix the axioms and propose the relative-entropy and ℓ₁ measures

2016

— Winter and Yang prove distillable coherence equals the relative entropy of coherence

2016

— Chitambar, Gour and others map the inequivalent operation classes MIO, IO, DIO, SIO

2017

— Streltsov, Adesso and Plenio consolidate the field in a Reviews of Modern Physics colloquium

Going further

Texts and papers. Baumgratz, Cramer, Plenio, "Quantifying Coherence," Phys. Rev. Lett. 113, 140401 (2014) — the founding axioms · Streltsov, Adesso, Plenio, "Colloquium: Quantum coherence as a resource," Rev. Mod. Phys. 89, 041003 (2017) — the definitive review · Winter, Yang, "Operational Resource Theory of Coherence," Phys. Rev. Lett. 116, 120404 (2016) — distillation and cost · Åberg, "Quantifying Superposition," arXiv:quant-ph/0612146 (2006) — the precursor · Chitambar, Gour, "Quantum resource theories," Rev. Mod. Phys. 91, 025001 (2019) — the general framework and operation hierarchy · Nielsen & Chuang, Quantum Computation and Quantum Information (entropy and channels) · Wilde, Quantum Information Theory, 2nd ed. (relative entropy, data processing).

Provenance. The axioms C1–C2 and both measures are from Baumgratz–Cramer–Plenio (2014); the operation-class hierarchy (MIO, IO, DIO, SIO, PIO) is consolidated in Chitambar–Gour (2019) and the Streltsov–Adesso–Plenio colloquium (2017); the operational identity $C_d=C_r$ is Winter–Yang (2016). The closed forms — $C_r=S(\Delta\rho)-S(\rho)$ , $C_{\ell_1}=\sum_{i\neq j}|\rho_{ij}|$ , pure-state $C_r=H(\{|c_i|^2\})$ , and the qudit unit values $\log_2 d$ and $d-1$ — are standard and independently verified above (Derivations A and C, worked examples). The monotonicity proof (Derivation B) rests only on IO preserving the incoherent set and the data-processing inequality for quantum relative entropy (Wilde, Ch. 11). A verifier should recompute the Gibbs minimization, the data-processing chain, and the qubit numbers $S(\rho)=0.4690$ bits, $C_r=0.531$ bits.

Prerequisites. Von Neumann Entropy · Quantum Relative Entropy · Quantum Channels and CPTP Maps · Kraus Representation · Data-Processing Inequality. Leads to. Holevo Bound · Strong Subadditivity of Quantum Entropy · Choi–Jamiołkowski Isomorphism.

Ask the tutor: Ask me to walk through the incoherent-Kraus structure behind the $\ell_1$ monotone proof, or to show how coherence in one system converts into entanglement with an ancilla.

Why fix a basis at all — isn't singling out a preferred basis unphysical?

The basis is not arbitrary; it is chosen by the physics. Decoherence selects a pointer basis in which a system goes classical, thermodynamics selects the energy eigenbasis because Hamiltonians conserve energy, and quantum computing selects the computational basis in which gates and measurements are defined. Coherence is the resource relative to that physically distinguished basis — the superposition structure the environment, or the task, treats as special. Far from a flaw, basis-dependence is what makes the theory match reality: interference visibility, work extraction, and metrological sensitivity are all defined against a concrete basis, and coherence theory quantifies the resource each task consumes.

How is coherence different from entanglement?

They are cousins built on the same template but partitioned differently. Entanglement is the resource under local operations and classical communication (LOCC), with separable states free; coherence is the resource under incoherent operations, with basis-diagonal states free. Coherence is more elementary and needs only one system — a single qubit can be coherent but not entangled. There is a precise bridge: coherence can be converted into entanglement with an ancilla by incoherent operations, and the maximum entanglement generated equals a coherence measure. So entanglement is, in a sense, coherence distributed across a partition; the two theories share almost every structural theorem.

Why are there so many classes of "free" operations — MIO, IO, DIO, SIO, PIO?

Because "cannot create coherence" can be demanded at different strengths, and the demands are genuinely inequivalent. MIO (maximally incoherent operations) is weakest: any channel mapping the incoherent set to itself. IO strengthens this to require an incoherent Kraus decomposition, so no individual outcome creates coherence. DIO asks the channel to commute with dephasing; SIO asks each Kraus operator to also not use coherence; PIO restricts to what is physically implementable with incoherent ancillas. These are nested and strictly different — a measure can be monotone under one and not another, and reversibility holds under MIO but fails under IO. Unlike LOCC, there is no single obviously correct free class.

Do the two measures ever disagree about which state is more coherent?

Yes, and this is expected. In the worked examples the equatorial qubit had $C_{\ell_1}=0.8$ but $C_r=0.531$ bits — different absolute values because they measure on different scales. More strikingly, for dimension three or higher $C_r$ and $C_{\ell_1}$ can order two states oppositely: A can beat B on $C_r$ while B beats A on $C_{\ell_1}$ . This is no contradiction; a resource can be genuinely multidimensional, as two portfolios rank differently by return and by risk. Only $C_r$ carries the distillation meaning, so it is the measure of choice for "how much pure coherence can I extract," while $C_{\ell_1}$ is the everyday diagnostic.

What does it mean operationally that C_r is the "distillable coherence"?

Distillation is the many-copy conversion of an imperfect resource into a perfect one. Given $N$ copies of $\rho$ , incoherent operations produce some number of flawless maximally coherent qubits $|\Psi_2\rangle$ ; the best output-to-input ratio, as $N\to\infty$ , is the distillable coherence, and Winter and Yang proved it equals $C_r(\rho)$ exactly. So $C_r$ is the physical exchange rate between your noisy state and the gold-standard coherbit: a state with $C_r=0.531$ bits is worth about 531 pure coherent qubits per thousand copies. This is why the relative entropy of coherence, not the easier $\ell_1$ -norm, is regarded as the theory's fundamental measure.

Why does the maximally mixed state have zero coherence despite being maximally random?

Because coherence measures off-diagonal phase order, not statistical spread. The maximally mixed state $I/d$ is proportional to the identity, so it is already diagonal in every basis — including the reference basis — which makes it free: $\Delta(I/d)=I/d$ gives $C_r=0$ , and all off-diagonals vanish so $C_{\ell_1}=0$ . Its large von Neumann entropy is classical uncertainty about which basis state it is in, which a classical apparatus reproduces for free. Coherence, by contrast, is the quantum phase relation between basis states, which no classical process can create — and $I/d$ has none of it.