Resource Theories

A quantum resource theory describes what an agent can accomplish when some states, transformations, or auxiliary systems are free and everything else must be accounted for. Given a resource claim, the task is to specify the operational restriction, conversion regime, success and error convention, side resources, and a monotone, witness, protocol, or complete criterion that licenses the conclusion. This page supplies that cross-theory grammar for finite-dimensional state resources and shows how to distinguish a bound from a protocol and an exact one-shot claim from probabilistic, asymptotic, or catalytic claims.

The general framework does not replace the canonical mathematics of states, channels, measurements, or specialist resource theories. It does not rederive LOCC protocols, coherence-operation classifications, thermal operations, stabilizer algebra, magic-state factories, or entanglement-distillation protocols. Instead, it makes the assumptions of such claims explicit and routes the remaining theorem or protocol question to its accepted owner.

Required background. Density Operators for Quantum Information supplies normalized and subnormalized states, state metrics, and subsystem bookkeeping. Quantum Channels and Noise supplies completely positive trace-preserving maps, composition, and channel representations.

Helpful background. Quantum Entropy supplies entropy and relative-entropy conventions; Entanglement Measures distinguishes measures by state class and task; LOCC Preview fixes the entanglement-specific operation class; and Thermal Operations Preview shows why a Hamiltonian, temperature, and work-storage model belong in a thermodynamic resource claim.

Declare the Resource-Theory Problem

A resource is not an intrinsic label attached to a state. It is a capability relative to an operational restriction and a task. The same density operator can be free in one theory, resourceful in another, and irrelevant to a third. For example, a state diagonal in one declared basis is free for basis-relative coherence, while the same state may still be athermally populated relative to a Hamiltonian.

A well-posed resource question follows a disciplined chain:

object and task
  -> free objects and transformations
  -> input, target, and side resources
  -> deterministic, selective, asymptotic, or catalytic regime
  -> success, error, and rate conventions
  -> monotone, witness, complete criterion, or protocol
  -> licensed conclusion and specialist handoff

The first declaration is the object type. A density operator, channel, measurement, instrument, process, and nonlocal box are not interchangeable resources. Their admissible free transformations and operational distances differ. This page develops finite-dimensional state-resource formulas; a channel or measurement resource requires free supermaps or simulation maps and a process-level distance.

The endpoint is not always a yes-or-no conversion theorem. A completed audit should return exactly one of five outcomes:

• the claim is ill posed because a required field is missing;
• a declared criterion rules the conversion out;
• a declared protocol achieves the conversion;
• the tested monotone supplies only a bound and does not decide achievability; or
• a specialist theorem, optimization, or protocol analysis is still required.

This discipline prevents a numerical monotone from being mistaken for a universal amount of “quantumness,” or a favorable postselected branch from being mistaken for a deterministic protocol.

Write the Ten-Field Resource Record

Use the following ten labels in order. Complete every field or write N/A and explain why it does not apply.

1. Resource claim and operational task — State what is to be detected, converted, distilled, diluted, simulated, discriminated, or benchmarked, and which agent is restricted.
2. Object type, systems, and representation — Identify states, channels, measurements, instruments, processes, or boxes; name Hilbert spaces, subsystem order, basis, Hamiltonian, symmetry, or other defining data.
3. Free objects — Declare the free set on every relevant system, including its dimensional and closure assumptions.
4. Free transformations, closure, and side-information rules — Name the transformation class and say whether identity, composition, tensor products, free ancillas, classical records, conditioning, and discarding are allowed.
5. Input, target, composition, and auxiliary resources — Give the input and target, the copy and tensor-product model, and every catalyst, battery, reference frame, communication channel, shared random variable, or borrowed resource.
6. Conversion mode and regime — Distinguish exact from approximate, deterministic from probabilistic, one-shot from finite-block or iid asymptotic, and ordinary from catalytic conversion.
7. Error, success, normalization, and distance — State whether outputs are normalized or subnormalized, report every branch probability, choose a state or process distance, and declare the error tolerance and order of limits.
8. Monotone, witness, or complete criterion with units — Name the quantity or theorem, its assumptions, normalization, logarithm base, and units; say whether it is only necessary, only sufficient, or complete for the declared task.
9. Numerical or evidence provenance, convergence, and alternatives — Identify analytic inputs, synthetic data, experimental data, code, precision, uncertainty convention, solver residuals, convergence tests, and competing diagnostics.
10. Licensed conclusion, stopping rule, and canonical handoff — State only what the evidence establishes, identify what would falsify or stop the analysis, and route any remaining specialist question.

A free-state set alone does not complete this record. Neither does the value of one monotone. In particular, different physically motivated operation classes can preserve the same free states while supporting different conversions.

Objects, Free Sets, and Free Transformations

Let $\mathcal D(\mathcal H_S)$ be the density operators on a finite-dimensional system $S$, and let

$$\mathcal F_S\subseteq\mathcal D(\mathcal H_S)$$

be the declared free states. Write $\mathfrak O(S\!\to\!T)$ for the free transformations from $S$ to $T$. A free channel must at least preserve the free set:

$$\Lambda(\mathcal F_S)\subseteq\mathcal F_T.$$

This resource-nongeneration condition is necessary for many state-resource theories, but it does not uniquely identify a physically meaningful operation class. LOCC is generally smaller than the maximal set of entanglement-nongenerating channels. Likewise, MIO, IO, DIO, and SIO preserve different operational structures even though they are all used in coherence theory. Thermal operations and Gibbs-preserving maps also need not coincide.

The record must therefore state which closures are available. Identity and composition are needed for the exact conversion relation below to be a preorder. Tensor-product closure determines whether independent free procedures remain free. Free ancillas, classical records, conditioning, and discarding control what selective protocols can do. A reference frame, work battery, coherence reservoir, or communication channel must not enter silently through an implementation convention.

Convexity is also an assumption, not a default. If free randomization is allowed and its record may be forgotten, the free set and transformation class are often convex. If the random choice itself is costly or unavailable, convex mixtures need not be free. Topological closure matters when an optimum is approached but not attained.

For measurement and instrument resources, outcome labels and postmeasurement states are part of the object. Generalized Measurements and Instruments owns the POVM–instrument distinction. This page uses that distinction when it audits selective operations; it does not duplicate the measurement theory.

Exact and Approximate Conversion Preorders

For exact deterministic state conversion, define

$$\rho\succeq_{\mathfrak O}\sigma \quad\Longleftrightarrow\quad \exists\,\Lambda\in\mathfrak O(S\!\to\!T) \ \text{such that}\ \Lambda(\rho)=\sigma.$$

If identity maps are free, then $\rho\succeq_{\mathfrak O}\rho$. If free maps are closed under composition, then

$$\rho\succeq_{\mathfrak O}\sigma, \quad \sigma\succeq_{\mathfrak O}\tau \quad\Longrightarrow\quad \rho\succeq_{\mathfrak O}\tau.$$

The relation is therefore reflexive and transitive, but generally not antisymmetric or total. Two objects can be incomparable, and two distinct objects can be mutually interconvertible.

Declare the trace norm by

$$\lVert X\rVert_1 = \operatorname{Tr}\sqrt{X^\dagger X}.$$

For normalized state outputs, a deterministic approximate conversion error is

$$\begin{aligned} \varepsilon_{\mathfrak O}(\rho\!\to\!\sigma) &= \inf_{\Lambda\in\mathfrak O} T\!\left(\Lambda(\rho),\sigma\right),\\ T(\omega,\sigma) &= \frac12\lVert\omega-\sigma\rVert_1. \end{aligned}$$

The factor $1/2$ fixes the conventional state trace distance. This formula must not be copied unchanged to a subnormalized success branch, channel, measurement, or instrument. A selective branch requires its success probability and an explicit convention for conditional versus subnormalized error. Channel and measurement resources require a declared free supermap or simulability relation and an operationally appropriate process distance.

Approximate convertibility also depends on the order of limits. Allowing an auxiliary dimension to diverge before sending the output error to zero can produce a different claim from fixing the auxiliary system first. This issue is central for approximate catalysts and embezzling constructions.

Monotones, Witnesses, and Complete Criteria

A deterministic resource monotone $M$ for the declared free operations satisfies

$$M(\Lambda(\rho))\leq M(\rho) \qquad \text{for every }\Lambda\in\mathfrak O.$$

Thus $M(\rho)<M(\sigma)$ rules out $\rho\to\sigma$. The converse generally fails: $M(\rho)\geq M(\sigma)$ does not construct a free protocol, and equality of one monotone does not establish interconvertibility. A complete family of monotones or a task-specific theorem is stronger than one scalar diagnostic. A witness can certify that an object lies outside the free set without quantifying a conversion rate.

Throughout this page logarithms are base $2$. For a declared free-state set, the relative-entropy resource is

$$\begin{aligned} D_{\mathcal F}(\rho) &= \inf_{\tau\in\mathcal F}D(\rho\Vert\tau),\\ D(\rho\Vert\tau) &= \operatorname{Tr}\!\left[ \rho\left(\log_2\rho-\log_2\tau\right) \right]. \end{aligned}$$

Set $D(\rho\Vert\tau)=+\infty$ unless $\operatorname{supp}\rho\subseteq\operatorname{supp}\tau$. If $\Lambda$ preserves the free set, then data processing gives

$$\begin{aligned} D_{\mathcal F}(\Lambda\rho) &= \inf_{\omega\in\mathcal F} D(\Lambda\rho\Vert\omega)\\ &\leq \inf_{\tau\in\mathcal F} D(\Lambda\rho\Vert\Lambda\tau)\\ &\leq D_{\mathcal F}(\rho). \end{aligned}$$

The middle line restricts the output optimization to images of free inputs. This proof uses data processing and free-set preservation; an arbitrary geometric distance to $\mathcal F$ is not automatically monotone.

Another common quantity is the generalized robustness

$$R_{\mathcal F}^{\mathrm g}(\rho) = \inf\left\{ s\geq0: \exists\tau\in\mathcal D(\mathcal H),\ \frac{\rho+s\tau}{1+s}\in\mathcal F \right\}.$$

The mixing state $\tau$ is arbitrary here. Standard or free robustness restricts the mixing state and is a different quantity. Witness duals and operational discrimination interpretations require their own finite-dimensional, closedness, and convexity hypotheses.

Vanishing, faithfulness, convexity, deterministic monotonicity, strong monotonicity, additivity, asymptotic continuity, and computability are separate properties. No single checklist item should be inferred from another without proof.

Selective Operations, Success Probability, and Strong Monotonicity

Let $\{\Lambda_x\}_x$ be a free selective instrument whose sum is trace preserving. For input $\rho$, define the subnormalized branch $\Lambda_x(\rho)$, its probability, and its normalized conditional state by

$$\begin{aligned} p_x&=\operatorname{Tr}\Lambda_x(\rho),\\ \rho_x&=\frac{\Lambda_x(\rho)}{p_x} \qquad (p_x>0). \end{aligned}$$

When $M$ is strongly monotone for this instrument class,

$$M(\rho)\geq\sum_x p_xM(\rho_x).$$

Deterministic monotonicity of the averaged channel $\sum_x\Lambda_x$ does not by itself prove this branch-averaged inequality. Convexity also does not replace the needed selective-operation statement. The transformation class and monotone must support the exact claim being made.

A selected branch can have more resource than the input without violating strong monotonicity. The probability of that branch and every failure branch are part of the ledger. Reporting only the accepted state replaces a probabilistic protocol by a fictitious deterministic one.

Normalization conventions are equally important. If $\widetilde\rho_x=\Lambda_x(\rho)$ is subnormalized, then $\operatorname{Tr}\widetilde\rho_x=p_x$. A monotone defined only on normalized states is evaluated on $\rho_x$, not silently on $\widetilde\rho_x$. Conversely, an optimization written for subnormalized operators should not be read as an ordinary state-distance statement.

One-Shot, Asymptotic, and Catalytic Regimes

A one-shot conversion uses one specified input and returns one specified output, perhaps with a declared error or success probability. A finite-block statement fixes a finite number of copies. An iid asymptotic statement concerns a sequence of transformations and an order of limits.

For vanishing trace-distance error, define the iid asymptotic rate by the existence of free maps:

$$\begin{aligned} r_{\mathfrak O}(\rho\!\to\!\sigma) = \sup\bigg\{r\geq0: &\ \exists\{\Lambda_n\},\ \Lambda_n\in\mathfrak O_n,\\ &\lim_{n\to\infty} \frac12\left\lVert \Lambda_n(\rho^{\otimes n}) -\sigma^{\otimes\lfloor rn\rfloor} \right\rVert_1=0 \bigg\}. \end{aligned}$$

The copy model, error metric, floor convention, and order of limits are part of the definition. Define

$$M^\infty(\rho) = \lim_{n\to\infty} \frac{M(\rho^{\otimes n})}{n}$$

only when the limit exists. A converse for vanishing-error conversion needs both a declared tensor-power scaling condition—additivity or a suitable regularization—and a continuity bound strong enough to make the output approximation contribute only $o(n)$. Neither ingredient replaces the other. Such a converse does not establish an achievable protocol or general asymptotic reversibility.

For a fixed declared catalyst $\omega_C$, exact uncorrelated catalytic conversion requires a free map on the joint systems:

$$\begin{aligned} \rho\succeq_{\mathfrak O}^{\,\omega_C}\sigma \quad\Longleftrightarrow\quad &\exists\,\Lambda\in \mathfrak O(S\!\otimes\!C\to T\!\otimes\!C)\ \text{such that}\\ &\Lambda(\rho\otimes\omega_C) =\sigma\otimes\omega_C. \end{aligned}$$

A claim that some catalyst exists must also quantify over the admissible catalyst system $C$ and state $\omega_C$; the catalyst is an accounted auxiliary resource, not a free state by default. Exact product return is different from returning only the catalyst marginal while allowing correlations, and both differ from approximate return. An approximate catalytic claim must specify catalyst dimension, return distance, permitted correlations, repeat-use rule, and cumulative error. Otherwise a growing catalyst or hidden reference frame can be relabeled as free assistance.

Compare Resource Theories and Route to Canonical Owners

The general ledger remains the same across theories, but the defining operational data change.

Theory	Declaration and canonical handoff
entanglement	Fix the subsystem partition, separable free states, and an operation class such as one-way or two-way LOCC. LOCC Preview owns that operation class, Entanglement Measures owns measure selection, and Entanglement Distillation owns distillation protocols.
basis-relative coherence	Fix the preferred basis, incoherent states, and one of MIO, IO, DIO, or SIO. Coherence and Preferred Bases owns the basis and decoherence interpretation.
athermality	Fix the Hamiltonian, inverse temperature, Gibbs state, energy-conservation model, and any battery or clock. Thermal Operations Preview owns thermal operations and their physical assumptions.
magic	Fix qudit dimension, stabilizer states, free operations, architecture, and target non-Clifford resource. Magic-State Distillation owns factory protocols and implementation-level costs.
asymmetry	Fix a symmetry group, its representation, symmetric states, covariant transformations, and the reference-frame task. No mature dedicated general page currently owns this theory; use the primary reference by Gour and Spekkens and do not route to a scaffold.

For channels, measurements, and instruments, declare free supermaps or simulation procedures instead of importing the state formulas unchanged. Quantum Channels and Noise owns CPTP, Kraus, Choi, and composition theory, while Generalized Measurements and Instruments owns outcome and backaction structure.

For orientation rather than a specialist theorem, use the Information-Theoretic Foundations chapter guide, the Quantum Information and Computation overview, or What Is Quantum Information?. The Quantum Information roadmap sequences prerequisites, and the Quantum Information mathematics crosswalk routes needed mathematical repairs.

The owner firewall is strict: this page owns the declaration grammar, regime comparison, and cross-theory audit. Density operators, channel theory, entropy derivations, specialist measures, operation-class theorems, physical noise mechanisms, and concrete distillation or factory protocols remain with the linked canonical pages.

Validate a Resource Claim Numerically or Experimentally

A numerical value is useful only when its provenance and decision role are recorded. Use the following sequence.

1. Validate the object. Check dimensions, subsystem order, Hermiticity, trace, positivity, and normalization. For a branch, check that its trace equals the reported probability.
2. Validate the transformation. Test complete positivity and trace preservation or nonincrease as appropriate, then test the declared free-operation constraints. Preserving free states numerically is not proof that a map belongs to a smaller physical class.
3. Validate the criterion. State whether a monotone, witness, semidefinite program, majorization test, or explicit protocol is necessary, sufficient, or complete for the exact regime.
4. Validate optimization evidence. Archive solver, precision, tolerances, primal and dual residuals, convergence under tighter settings, and at least one independent analytic or numerical check.
5. Validate uncertainty. State whether uncertainties are standard uncertainties, confidence intervals, posterior intervals, or deterministic tolerances. Preserve covariance when several reported entries depend on one parameter.
6. Validate alternatives. Compare another monotone, witness, protocol, or free-operation class when the conclusion could change under that choice.
7. Stop at the licensed claim. A witness establishes nonfreeness, a converse establishes a bound, and a protocol establishes achievability under its declared assumptions. None automatically establishes the other two.

The two audits below are explicitly SYNTHETIC pedagogical calculations. They contain no experimental observations and make no hardware-performance claim. The first propagates a supplied standard uncertainty through an analytic qubit channel. The second uses exact-by-construction Schmidt data and reports rounded decimal summaries only for readability.

Worked Audit: Qubit Coherence under Dephasing

SYNTHETIC audit. The supplied parameter $\eta=0.600\pm0.010$ is a constructed input with one standard uncertainty, not a fitted experimental result. The same uncertain parameter controls both Hermitian-conjugate off-diagonal entries, so their errors are fully correlated.

1. Resource claim and operational task — Audit whether one declared DIO channel reduces, preserves, or removes the basis-relative coherence of an input qubit. This is a deterministic state-conversion calculation, not a model of an environmental mechanism.

2. Object type, systems, and representation — The object is a normalized qubit density operator in the computational basis. The input is

   $$\rho_+ = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

3. Free objects — Free states are the density operators diagonal in the computational basis. The dephasing map $\Delta$ removes all off-diagonal entries.

4. Free transformations, closure, and side-information rules — Free transformations are DIO channels satisfying $\Delta\Lambda=\Lambda\Delta$. The tested channel is

   $$\Phi_\eta(\rho) = \begin{pmatrix} \rho_{00}&\eta\rho_{01}\\ \eta\rho_{10}&\rho_{11} \end{pmatrix}.$$

   It is CPTP for $|\eta|\leq1$ and commutes with $\Delta$. Identity, composition, free diagonal ancillas, and discarding are admitted; no branch record is used.

5. Input, target, composition, and auxiliary resources — The input is $\rho_+$ and the target is the actual channel output $\Phi_\eta(\rho_+)$. There is one channel use and no catalyst, reference frame, postselection, borrowed coherence, or asymptotic block.

6. Conversion mode and regime — The claim is deterministic, one-shot, and exact conditional on the supplied value of $\eta$. Parameter uncertainty is propagated separately; it is not an approximation error between target states.

7. Error, success, normalization, and distance — Success probability is $1$ and both states have unit trace. The output is

   $$\Phi_\eta(\rho_+) = \begin{pmatrix} 0.5&0.300\pm0.005\\ 0.300\pm0.005&0.5 \end{pmatrix}.$$

   No target-distance tolerance is invoked. The paired off-diagonal uncertainties are common rather than independent.

8. Monotone, witness, or complete criterion with units — Use the relative entropy of coherence

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

   in bits and, separately, the dimensionless $l_1$ coherence

   $$C_{l_1}(\rho)=\sum_{i\ne j}|\rho_{ij}|.$$

   Here $C_r$ is the DIO monotone used for the resource claim. The $l_1$ quantity is reported as a separate coherence diagnostic and is a standard monotone for narrower incoherent-operation classes; it is not invoked as a criterion for arbitrary DIO conversion. The two quantities have neither interchangeable units nor a shared complete-conversion interpretation.

9. Numerical or evidence provenance, convergence, and alternatives — The eigenvalues are

   $$\lambda_+=0.800\pm0.005, \qquad \lambda_-=0.200\mp0.005,$$

   where the opposite signs emphasize anticorrelation. With

   $$h_2(p)=-p\log_2p-(1-p)\log_2(1-p),$$

   the analytic results are

   $$\begin{aligned} C_r(\rho_+)&=1\ \text{bit},\\ C_r(\Phi_\eta(\rho_+)) &=1-h_2(0.8)\\ &=0.278072\pm0.010\ \text{bits},\\ C_{l_1}(\Phi_\eta(\rho_+)) &=0.600\pm0.010. \end{aligned}$$

   At $\eta=0.6$, $dC_r/d\eta=1$, which gives the displayed first-order standard uncertainty. The calculation is analytic; an archived implementation should record the formula, basis, channel class, floating-point precision, and residuals from trace and eigenvalue checks. The Dephasing Channel owns the channel model, while Coherence and Preferred Bases owns the physical basis interpretation.

10. Licensed conclusion, stopping rule, and canonical handoff — The declared DIO channel decreases computational-basis coherence from $1$ bit to $0.278072\pm0.010$ bits and does not remove it. Stop if the channel is not CPTP or DIO over the uncertainty range, or if the basis changes. The calculation does not license basis-independent “quantumness,” a coherence-conversion rate, or an environmental decoherence mechanism.

Worked Audit: Pure-State Entanglement Conversion

SYNTHETIC audit. The amplitudes and Schmidt vectors below are exact-by-construction. The decimals are rounded summaries, not measured estimates, and carry no experimental uncertainty.

Use the states

$$\begin{aligned} |\psi\rangle &=\sqrt{0.9}|00\rangle+\sqrt{0.1}|11\rangle,\\ |\Phi^+\rangle &=\frac{|00\rangle+|11\rangle}{\sqrt2}, \end{aligned}$$

with nonincreasing Schmidt vectors $\lambda^\psi=(0.9,0.1)$ and $\lambda^\Phi=(0.5,0.5)$. Each field below separates deterministic one-copy, probabilistic one-copy, and iid asymptotic claims.

1. Resource claim and operational task — Deterministic: decide whether one $|\psi\rangle$ can become one Bell pair with certainty. Probabilistic: maximize the success probability of exact one-copy conversion. Asymptotic: determine the vanishing-error Bell-pair distillation rate from iid copies of $|\psi\rangle$.

2. Object type, systems, and representation — In all three regimes the objects are exact normalized bipartite pure states on two qubits, represented by their declared $A|B$ Schmidt vectors. The subsystem split and vector ordering are fixed.

3. Free objects — Separable bipartite states are free in all three records. This pure-state calculation does not extend the free set to PPT or bound-entangled states.

4. Free transformations, closure, and side-information rules — The class is exact two-way LOCC with local free ancillas, shared randomness, classical records and messages, composition, and discarding. There is no quantum communication. The selective record is retained in the probabilistic regime; it is absent from the deterministic output and included in the asymptotic protocol accounting.

5. Input, target, composition, and auxiliary resources — Deterministic and probabilistic inputs are one $|\psi\rangle$ and the successful target is one $|\Phi^+\rangle$. The asymptotic input is $|\psi\rangle^{\otimes n}$ and its target is $\lfloor rn\rfloor$ Bell pairs. No catalyst, borrowed entanglement, hidden postselection, or quantum side channel is allowed.

6. Conversion mode and regime — Deterministic: exact, one-copy, success probability $1$. Probabilistic: exact on the accepted branch, one-copy, with all branches reported. Asymptotic: deterministic overall conversion with iid inputs and trace-distance error vanishing after the copy limit.

7. Error, success, normalization, and distance — Deterministic conversion has zero allowed state error and fails the complete criterion. Probabilistic conversion has normalized conditional branch states and optimal success probability $p_{\max}=0.20$; its failure probability is $0.80$. The asymptotic statement uses normalized outputs and vanishing trace-distance error, with $n\to\infty$ taken at fixed source state.

8. Monotone, witness, or complete criterion with units — Deterministic conversion uses Nielsen’s complete pure-state majorization criterion,

   $$|\psi\rangle\longrightarrow|\Phi^+\rangle \quad\Longleftrightarrow\quad \lambda^\psi\prec\lambda^\Phi.$$

   It fails because the first partial sum would require $0.9\leq0.5$. Probabilistic conversion uses Vidal’s complete tail-ratio formula,

   $$\begin{aligned} E_k(\psi)&=\sum_{i=k}^{d}\lambda_i^\psi,\\ p_{\max} &=\min_k\frac{E_k(\psi)}{E_k(\Phi)}\\ &=\min\left(1,\frac{0.1}{0.5}\right)=0.20. \end{aligned}$$

   The iid pure-state distillation rate is the entanglement entropy,

   $$h_2(0.1) =0.468996\ \text{ebits per input copy}.$$

9. Numerical or evidence provenance, convergence, and alternatives — Schmidt data and the majorization inequalities are exact. The displayed entropy is the base-$2$ analytic value rounded to six decimals. An independent check diagonalizes either reduced state and returns eigenvalues $(0.9,0.1)$. No optimizer or experimental data are used. Entanglement Measures owns the choice and interpretation of entanglement quantities.

10. Licensed conclusion, stopping rule, and canonical handoff — Exact deterministic one-copy conversion is impossible under the declared LOCC class; optimal exact one-copy success is $20\%$; and the iid vanishing-error pure-state distillation rate is $0.468996$ ebits per input copy. Stop and re-audit if the state is mixed, the operation class changes, a finite-block yield is requested, or auxiliary entanglement is admitted. LOCC Preview owns the operation theorem, and Entanglement Distillation owns concrete multi-copy protocols.

Common Failure Modes and Canonical Handoffs

Calling a state “quantum” without a task. A resource is relative to a restriction and an operational goal. There is no task-independent scalar called simply “quantumness.”

Changing the object type silently. State, channel, measurement, instrument, process, and box resources require different free transformations and distances. Route channel structure to Quantum Channels and Noise and outcome-plus-backaction structure to Generalized Measurements and Instruments.

Inferring operations from free states. Free states do not determine free operations. Maximal resource-nongenerating maps can be strictly larger than a physically implementable class.

Assuming closure. Declare identity, composition, tensor products, free ancillas, classical records, conditioning, and discarding. Without identity and composition closure, the advertised conversion relation need not be a preorder.

Treating one monotone as complete. A preorder need not be total, and equal values of one monotone need not imply interconvertibility. Vanishing, faithfulness, convexity, additivity, asymptotic continuity, strong monotonicity, and computability are distinct properties.

Dropping unsuccessful branches. Deterministic monotonicity is weaker than branch-averaged strong monotonicity. Keep normalized conditional states distinct from subnormalized branch maps, and report success and failure probabilities.

Mixing conversion regimes. Exact, approximate, deterministic, probabilistic, one-shot, finite-block, iid-asymptotic, and catalytic statements are not interchangeable. State the error metric, success convention, rate unit, reference resource, copy model, and order of limits.

Assuming an asymptotic simplification. A regularized limit, single-letter formula, and asymptotic reversibility each require proof. The Brandão–Gour reversible framework assumes a maximal asymptotically resource-nongenerating class and additional hypotheses; it is not a theorem about ordinary LOCC, DIO, thermal, or stabilizer operations.

Losing mathematical hypotheses. Preserve the support condition and logarithm base in relative entropy. Distinguish generalized from standard or free robustness, and invoke convex duality only under stated finite-dimensional, closedness, and convexity hypotheses.

Treating catalysts as free by name. Distinguish exact product return, correlated marginal return, and approximate return. Record catalyst dimension, return error, correlations, and repeat-use rules so a hidden reference frame or embezzling limit is not concealed.

Omitting side resources. Count classical communication, shared randomness, postselection, reference frames, work batteries, coherence reservoirs, borrowed entanglement, and quantum communication whenever they are present.

Omitting defining physical data. Basis, subsystem partition, symmetry group, Hamiltonian, temperature, and computational architecture are part of the theory. They are not cosmetic labels.

Conflating specialist operation classes. LOCC, separable operations, and PPT-preserving operations differ; MIO, IO, DIO, and SIO differ; thermal operations, Gibbs-preserving maps, and master equations differ; and stabilizer free operations depend on dimension and architecture. Use the specialist handoffs in the comparison section.

Extrapolating beyond finite dimensions. Infinite-dimensional theories require topological closure and energy constraints. The finite-dimensional state formulas on this page do not transfer automatically.

Turning a bound into an implementation. A witness or monotone bound is not automatically an achievable rate, finite protocol, or hardware estimate. Magic-State Distillation and Entanglement Distillation own their concrete protocols and costs.

Exercises

1. Repair an underspecified resource claim

Repair the statement “this state contains quantumness” by filling all ten resource-record fields and naming the first unresolved field.

Solution

The first unresolved field is Resource claim and operational task: no restricted agent or task has been stated. The object itself is also unspecified, so the original sentence licenses no resource conclusion.

One possible repair is: “For the qubit state $\rho_+$ in the computational basis, quantify basis-relative coherence under deterministic DIO conversion.” The remaining record is then: object $\rho_+$ on one qubit; diagonal free states; DIO channels with identity, composition, free diagonal ancillas, and discarding; no auxiliary resource; deterministic one-shot conversion; normalized outputs with trace distance; $C_r$ in bits as a monotone; analytic synthetic provenance with an $l_1$ alternative; and a conclusion limited to DIO coherence. This is one repaired question, not information contained in the original sentence. If the task, basis, or operation class cannot be supplied, fields downstream are N/A and the correct verdict is “ill posed.”

2. Test the preorder assumptions

Prove reflexivity and transitivity of $\succeq_{\mathfrak O}$ from identity and composition closure. Then give a declared transformation set without composition closure for which calling the relation a preorder is invalid.

Solution

If $\operatorname{id}_S\in\mathfrak O(S\!\to\!S)$, then $\operatorname{id}_S(\rho)=\rho$, so $\rho\succeq_{\mathfrak O}\rho$. If $\Lambda(\rho)=\sigma$ and $\Gamma(\sigma)=\tau$, composition closure gives $\Gamma\circ\Lambda\in\mathfrak O$ and

$$(\Gamma\circ\Lambda)(\rho)=\tau,$$

which proves transitivity.

For a counterexample, consider three perfectly distinguishable classical states $0,1,2$ embedded as diagonal quantum states. Let $f$ be a measure-and-prepare channel with $f(0)=1$, $f(1)=2$, and $f(2)=2$, and declare $\mathfrak O=\{\operatorname{id},f\}$ while excluding $f^2$. Then $0\succeq1$ and $1\succeq2$, but neither admitted map sends $0$ to $2$. The relation is not transitive, exactly because the declared set is not closed under composition.

3. Recompute the coherence audit

Recompute the output matrix, its eigenvalues, $C_r=0.278072\pm0.010$ bits, and $C_{l_1}=0.600\pm0.010$. Explain why the two numbers are not interchangeable.

Solution

Multiplying each off-diagonal entry of $\rho_+$ by $\eta=0.600\pm0.010$ gives

$$\Phi_\eta(\rho_+) = \begin{pmatrix} 0.5&0.300\pm0.005\\ 0.300\pm0.005&0.5 \end{pmatrix}.$$

Its eigenvalues are $(1\pm\eta)/2$, hence $0.800\pm0.005$ and $0.200\mp0.005$. Since dephasing leaves the maximally mixed diagonal state,

$$C_r=1-h_2(0.8)=0.278071905\ldots\ \text{bits}.$$

At $\eta=0.6$, $dC_r/d\eta=1$, so first-order propagation gives standard uncertainty $0.010$ bit. The $l_1$ sum contains two entries of magnitude $\eta/2$, giving $C_{l_1}=\eta=0.600\pm0.010$. Relative-entropy coherence is measured in bits and has an information-theoretic definition; $l_1$ coherence is dimensionless and has a different operational meaning. Their numerical values cannot be compared as though they were two estimates of one quantity.

4. Prove relative-entropy monotonicity

Prove $D_{\mathcal F}(\Lambda\rho)\leq D_{\mathcal F}(\rho)$ from data processing and free-set preservation, retaining the support condition.

Solution

For every free $\tau\in\mathcal F$, preservation gives $\Lambda\tau\in\mathcal F$. Therefore

$$\begin{aligned} D_{\mathcal F}(\Lambda\rho) &=\inf_{\omega\in\mathcal F}D(\Lambda\rho\Vert\omega)\\ &\leq D(\Lambda\rho\Vert\Lambda\tau)\\ &\leq D(\rho\Vert\tau), \end{aligned}$$

where the last line is quantum relative-entropy data processing. Taking the infimum over free $\tau$ proves the claim. The definition uses $D(\rho\Vert\tau)=+\infty$ when $\operatorname{supp}\rho\nsubseteq\operatorname{supp}\tau$; CPTP maps can restore support overlap, so the extended-real data-processing inequality remains the correct statement.

5. Audit a successful branch

A local filter returns a Bell pair with probability $0.20$ and a product failure state otherwise. Compute the average output entanglement and compare it with the $0.468996$-ebit input.

Solution

The accepted Bell branch contains one ebit and the product failure branch contains zero. The branch-averaged output is

$$0.20(1\ \text{ebit})+0.80(0\ \text{ebits}) =0.20\ \text{ebits}.$$

Strong monotonicity requires $0.468996\geq0.20$, which holds. The one-ebit accepted branch is therefore compatible with monotonicity; reporting it without the $20\%$ probability and failure state would be the error.

6. Separate three entanglement regimes

For the synthetic entanglement audit, perform the Nielsen partial-sum test, Vidal tail-ratio calculation, and entropy calculation, then label the three regimes.

Solution

For deterministic exact one-copy LOCC, Nielsen’s criterion requires $\lambda^\psi\prec\lambda^\Phi$. The first partial sum fails:

$$0.9\nleq0.5.$$

Thus deterministic one-copy conversion is impossible. For probabilistic exact one-copy LOCC, Vidal’s tail ratios are $1$ for the full sum and $0.1/0.5=0.2$ for the second tail, so $p_{\max}=0.20$. For iid asymptotic pure-state distillation with vanishing trace-distance error,

$$h_2(0.1)=0.468995593\ldots$$

Bell pairs are obtained per input copy. These are respectively an impossibility result, an optimal success probability, and an asymptotic rate. None is a finite-block yield.

7. Derive a monotone converse bound

Given $M^\infty(\rho)=0.4$ and $M^\infty(\sigma)=0.8$, derive the converse bound $r\leq0.5$ under explicitly supplied assumptions. Explain why it does not prove achievability.

Solution

Assume $M$ is monotone for the admissible asymptotic maps, has the normalization used in the supplied values, and obeys the tensor-power scalings

$$\begin{aligned} M(\rho^{\otimes n}) &=nM^\infty(\rho)+o(n),\\ M(\sigma^{\otimes m_n}) &=m_nM^\infty(\sigma)+o(n). \end{aligned}$$

Additivity can supply these scalings, but it does not control approximation error. Separately assume an asymptotic-continuity bound that turns the vanishing trace-distance error into an $o(n)$ correction. Monotonicity and continuity then give

$$\begin{aligned} M(\rho^{\otimes n}) &\geq M\!\left(\Lambda_n(\rho^{\otimes n})\right)\\ &\geq M(\sigma^{\otimes m_n})-o(n), \end{aligned}$$

and hence

$$nM^\infty(\rho)+o(n) \geq m_nM^\infty(\sigma)-o(n).$$

Dividing by $nM^\infty(\sigma)$ and taking the limit superior yields

$$r =\limsup_{n\to\infty}\frac{m_n}{n} \leq\frac{0.4}{0.8}=0.5.$$

This is a converse: rates above $0.5$ are excluded. It supplies no sequence of free maps attaining $0.5$ or any positive rate, so it does not prove achievability or reversibility.

8. Audit a catalytic conversion

Fill all ten fields for a proposed catalytic conversion. Distinguish exact product return, exact marginal return with correlations, and approximate return; declare catalyst dimension and error, hidden side resources, repeat-use rule, specialist owner, and a stopping test for embezzling.

Solution

Consider exact pure-state bipartite LOCC conversion with Schmidt vectors

$$p=(0.4,0.4,0.1,0.1), \qquad q=(0.5,0.25,0.25,0),$$

and a two-dimensional catalyst with Schmidt vector $c=(0.6,0.4)$.

1. Resource claim and operational task — Decide whether $|\psi_p\rangle$ can become $|\phi_q\rangle$ by deterministic LOCC when the declared catalyst is returned, and distinguish three return conventions.

2. Object type, systems, and representation — The objects are finite-dimensional bipartite pure states represented by nonincreasing Schmidt vectors. The principal systems have local dimension $4$; the catalyst has local dimension $2$.

3. Free objects — Separable bipartite states are free. The catalyst is not free and must be returned under the declared convention.

4. Free transformations, closure, and side-information rules — Exact two-way LOCC is allowed, with local free ancillas, classical communication and records, composition, and discarding. No quantum communication, borrowed entanglement, or undeclared reference frame is allowed.

5. Input, target, composition, and auxiliary resources — The input is $|\psi_p\rangle\otimes|\chi_c\rangle$ and the exact product-return target is $|\phi_q\rangle\otimes|\chi_c\rangle$. One catalyst is available; repeated use is licensed only if the return convention leaves the same usable catalyst.

6. Conversion mode and regime — The primary claim is exact deterministic one-shot product-return catalysis. Exact marginal return with catalyst–target correlations is a separate regime. Approximate return with trace-distance error at most $\varepsilon$ is a third regime and is not inferred from the exact calculation.

7. Error, success, normalization, and distance — Product return has success probability $1$ and zero error. Marginal return requires the catalyst marginal to equal $|\chi_c\rangle\langle\chi_c|$ while any allowed correlation is reported; for a pure catalyst marginal, exact equality in fact forces product form. Approximate return must state trace distance, $\varepsilon$, catalyst dimension $2$, target error, correlations, and cumulative error under reuse.

8. Monotone, witness, or complete criterion with units — Nielsen majorization is complete for the exact pure-state LOCC records. Direct conversion fails because the second cumulative sum gives $0.8>0.75$. After tensoring, the sorted vectors are

   $$\begin{aligned} p\otimes c &=(0.24,0.24,0.16,0.16,\\ &\qquad 0.06,0.06,0.04,0.04),\\ q\otimes c &=(0.30,0.20,0.15,0.15,\\ &\qquad 0.10,0.10,0,0). \end{aligned}$$

   Their cumulative sums satisfy $p\otimes c\prec q\otimes c$, so exact product-return conversion is possible. Majorization is dimensionless.

9. Numerical or evidence provenance, convergence, and alternatives — All vectors and partial sums are exact decimals. The cumulative sums are

   $$\begin{aligned} P&=(0.24,0.48,0.64,0.80,\\ &\qquad 0.86,0.92,0.96,1),\\ Q&=(0.30,0.50,0.65,0.80,\\ &\qquad 0.90,1,1,1). \end{aligned}$$

   No optimizer or experimental data is involved. An independent check verifies normalization and every majorization inequality. Approximate or correlated variants require new evidence rather than rounding these exact vectors.

10. Licensed conclusion, stopping rule, and canonical handoff — The declared two-dimensional catalyst enables the exact product-return pure-state LOCC conversion. Do not extend that verdict to correlated or approximate return without specifying their maps and errors. Stop an embezzling claim if catalyst dimension grows with $1/\varepsilon$, catalyst resource changes, correlations accumulate, or repeated-use error is uncontrolled. Route LOCC structure to LOCC Preview and use the primary catalyst literature for the specialist theorem.

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