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"artifact_id": "qft.artifact.quantum-information.relative-entropy-distinguishability-recovery-claim-validity-ledger",
"title": "Information measures, operational tasks, and claim boundaries",
"scope": "A theorem-domain comparison of continuum and regulated entropy, one-shot divergences and second-order fluctuations, state and channel distinguishability, correlation measures, recovery guarantees, and continuity bounds; every row retains its regulator and failure controls.",
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"qft.topic.relative-information.petz-rotated-universal-recovery",
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"path": "src/content/docs/quantum-information/relative-entropy-distinguishability-recovery/index.md",
"route": "/quantum-information/relative-entropy-distinguishability-recovery/",
"heading": "What the chapter establishes",
"fragment": "relative-entropy-task-boundaries",
"public_data_path": "public/data/quantum-information/relative-entropy-task-boundaries.json"
},
"columns": [
{
"key": "quantity",
"label": "Quantity",
"definition": "Information measure, operational divergence, norm, or continuity statement being compared."
},
{
"key": "domain",
"label": "Domain",
"definition": "States, algebras, channels, or constrained input sets on which the quantity is defined."
},
{
"key": "topology_or_norm",
"label": "Topology or norm",
"definition": "Topology, norm, modular object, or optimization that determines the comparison."
},
{
"key": "regulator_status",
"label": "Regulator status",
"definition": "Whether the statement is intrinsic or requires a fixed and controlled regulator or resource limit."
},
{
"key": "operational_task",
"label": "Operational task",
"definition": "Decision, correlation, reconstruction, or error-propagation task supported by the quantity."
},
{
"key": "theorem_hypotheses",
"label": "Theorem hypotheses",
"definition": "Algebraic, support, copy, channel, dimensional, or energy assumptions needed by the stated theorem."
},
{
"key": "equality_case",
"label": "Equality case",
"definition": "Condition under which the defining inequality or comparison is saturated."
},
{
"key": "error_or_bound",
"label": "Error or bound",
"definition": "Quantitative error, exponent, contraction, continuity, or recovery bound available under the hypotheses."
},
{
"key": "counterexample_or_failed_control",
"label": "Counterexample or failed control",
"definition": "A concrete way the conclusion fails when its domain, convention, regulator, or resource control is changed."
}
],
"rows": [
{
"id": "araki_relative_entropy",
"quantity": {
"html": "Araki relative entropy S(ω‖φ)",
"text": "Araki relative entropy S(ω‖φ)"
},
"domain": {
"html": "Normal states on one von Neumann algebra; extended value +∞ is allowed",
"text": "Normal states on one von Neumann algebra; extended value +∞ is allowed"
},
"topology_or_norm": {
"html": "Standard-form relative modular operator; no ambient trace is required",
"text": "Standard-form relative modular operator; no ambient trace is required"
},
"regulator_status": {
"html": "Intrinsic target; type-I or lattice values need compatible embeddings and convergence",
"text": "Intrinsic target; type-I or lattice values need compatible embeddings and convergence"
},
"operational_task": {
"html": "Asymmetric comparison with a reference; Stein rate when a separate iid theorem applies",
"text": "Asymmetric comparison with a reference; Stein rate when a separate iid theorem applies"
},
"theorem_hypotheses": {
"html": "Same algebra and state order; support inclusion is necessary, not sufficient, for a finite value in infinite dimensions",
"text": "Same algebra and state order; support inclusion is necessary, not sufficient, for a finite value in infinite dimensions"
},
"equality_case": {
"html": "S=0 exactly for equal states",
"text": "S=0 exactly for equal states"
},
"error_or_bound": {
"html": "Data processing contracts S under a declared physical restriction or channel",
"text": "Data processing contracts S under a declared physical restriction or channel"
},
"counterexample_or_failed_control": {
"html": "Subtracting divergent local entropies, or comparing states on unmatched algebras, does not define this quantity",
"text": "Subtracting divergent local entropies, or comparing states on unmatched algebras, does not define this quantity"
},
"reference_ids": [
"doi:10.2977/prims/1195191148",
"doi:10.2977/prims/1195190105",
"doi:10.1007/s00220-021-04249-x",
"doi:10.1103/physrevd.107.125016",
"doi:10.1063/1.5039973",
"url:https://www.theta.ro/jot/archive/1986-016-002/1986-016-002-010.pdf",
"doi:10.1007/bf01609396"
]
},
{
"id": "regulated_von_neumann_entropy",
"quantity": {
"html": "Regulated von Neumann entropy SΛ(ρA)",
"text": "Regulated von Neumann entropy SΛ(ρA)"
},
"domain": {
"html": "A density operator on one finite-dimensional, regulator-defined type-I subsystem A",
"text": "A density operator on one finite-dimensional, regulator-defined type-I subsystem A"
},
"topology_or_norm": {
"html": "Spectral trace functional −Tr ρA ln ρA; it is not a state-distance topology",
"text": "Spectral trace functional −Tr ρA ln ρA; it is not a state-distance topology"
},
"regulator_status": {
"html": "Regulator dependent; only a complete subtraction, combination, or controlled limit can define a continuum quantity",
"text": "Regulator dependent; only a complete subtraction, combination, or controlled limit can define a continuum quantity"
},
"operational_task": {
"html": "Quantify mixedness, and entanglement for a pure regulated bipartition",
"text": "Quantify mixedness, and entanglement for a pure regulated bipartition"
},
"theorem_hypotheses": {
"html": "Normalized ρA; finite dA≥2, tensor factor, center or edge prescription, logarithm base, and common regulator are fixed",
"text": "Normalized ρA; finite dA≥2, tensor factor, center or edge prescription, logarithm base, and common regulator are fixed"
},
"equality_case": {
"html": "For a pure regulated bipartite state, SΛ(A)=SΛ(Ā); SΛ=0 exactly when ρA is pure",
"text": "For a pure regulated bipartite state, SΛ(A)=SΛ(Ā); SΛ=0 exactly when ρA is pure"
},
"error_or_bound": {
"html": "For common dA and T=‖ρ−σ‖1/2≤1−1/dA, |ΔS|≤T ln(dA−1)+h2(T); also 0≤SΛ≤ln dA",
"text": "For common dA and T=‖ρ−σ‖1/2≤1−1/dA, |ΔS|≤T ln(dA−1)+h2(T); also 0≤SΛ≤ln dA"
},
"counterexample_or_failed_control": {
"html": "A sharp type-III region has no such reduced density matrix; changing the cutoff, tensor factor, center, or edge modes changes the entropy",
"text": "A sharp type-III region has no such reduced density matrix; changing the cutoff, tensor factor, center, or edge modes changes the entropy"
},
"reference_ids": [
"doi:10.1088/1742-5468/2004/06/p06002",
"doi:10.1103/physrevd.89.085012",
"doi:10.1088/1751-8113/40/28/s18"
]
},
{
"id": "smooth_min_relative_entropy",
"quantity": {
"html": "Smooth support min-relative entropy Dminε",
"text": "Smooth support min-relative entropy Dminε"
},
"domain": {
"html": "A finite-dimensional state pair, with the optimized state allowed to be normalized or subnormalized as declared",
"text": "A finite-dimensional state pair, with the optimized state allowed to be normalized or subnormalized as declared"
},
"topology_or_norm": {
"html": "Support divergence −ln Tr(Πρ̃σ), maximized over one declared trace- or purified-distance ε-ball",
"text": "Support divergence −ln Tr(Πρ̃σ), maximized over one declared trace- or purified-distance ε-ball"
},
"regulator_status": {
"html": "State space, smoothing metric, normalization convention, ε, logarithm base, and cutoff remain fixed",
"text": "State space, smoothing metric, normalization convention, ε, logarithm base, and cutoff remain fixed"
},
"operational_task": {
"html": "One-shot support overlap and zero-type-I asymmetric testing, with a controlled approximation",
"text": "One-shot support overlap and zero-type-I asymmetric testing, with a controlled approximation"
},
"theorem_hypotheses": {
"html": "For Dmin=DH0, ρ and σ are normalized, Tr(Qρ)≥1, and the log base and state order agree; a smooth identity is not implied",
"text": "For Dmin=DH0, ρ and σ are normalized, Tr(Qρ)≥1, and the log base and state order agree; a smooth identity is not implied"
},
"equality_case": {
"html": "At ε=0 it reduces to the support Dmin; with the stated normalized testing convention this equals DH0",
"text": "At ε=0 it reduces to the support Dmin; with the stated normalized testing convention this equals DH0"
},
"error_or_bound": {
"html": "ε is a metric radius, not a type-I error probability; any comparison bound must translate metric, normalization, and log base",
"text": "ε is a metric radius, not a type-I error probability; any comparison bound must translate metric, normalization, and log base"
},
"counterexample_or_failed_control": {
"html": "Switching from trace to purified distance, or from normalized to subnormalized smoothing, changes the feasible set and boundary terms",
"text": "Switching from trace to purified distance, or from normalized to subnormalized smoothing, changes the feasible set and boundary terms"
},
"reference_ids": [
"doi:10.1109/tit.2009.2018325",
"doi:10.1109/tit.2013.2276628"
]
},
{
"id": "smooth_max_relative_entropy",
"quantity": {
"html": "Smooth max-relative entropy Dmaxε",
"text": "Smooth max-relative entropy Dmaxε"
},
"domain": {
"html": "A finite-dimensional state pair, with the optimized state allowed to be normalized or subnormalized as declared",
"text": "A finite-dimensional state pair, with the optimized state allowed to be normalized or subnormalized as declared"
},
"topology_or_norm": {
"html": "Smallest domination exponent ρ̃≤eλσ, minimized over one declared trace- or purified-distance ε-ball",
"text": "Smallest domination exponent ρ̃≤eλσ, minimized over one declared trace- or purified-distance ε-ball"
},
"regulator_status": {
"html": "State space, smoothing metric, normalization convention, ε, logarithm base, and cutoff remain fixed",
"text": "State space, smoothing metric, normalization convention, ε, logarithm base, and cutoff remain fixed"
},
"operational_task": {
"html": "One-shot domination and resource-conversion bounds formulated in the chosen smoothing convention",
"text": "One-shot domination and resource-conversion bounds formulated in the chosen smoothing convention"
},
"theorem_hypotheses": {
"html": "Support of every finite candidate lies in the support of σ; the theorem uses the same smoothing ball and state normalization",
"text": "Support of every finite candidate lies in the support of σ; the theorem uses the same smoothing ball and state normalization"
},
"equality_case": {
"html": "At ε=0 it reduces to Dmax; for normalized states Dmax=0 exactly when ρ=σ",
"text": "At ε=0 it reduces to Dmax; for normalized states Dmax=0 exactly when ρ=σ"
},
"error_or_bound": {
"html": "The optimizer certifies ρ̃≤eDmaxεσ; ε controls approximation, not operational failure by itself",
"text": "The optimizer certifies ρ̃≤eDmaxεσ; ε controls approximation, not operational failure by itself"
},
"counterexample_or_failed_control": {
"html": "Support leakage makes the unsmoothed value +∞; with subnormalized smoothing it can be negative, so an undeclared convention can hide the singularity",
"text": "Support leakage makes the unsmoothed value +∞; with subnormalized smoothing it can be negative, so an undeclared convention can hide the singularity"
},
"reference_ids": [
"doi:10.1109/tit.2009.2018325",
"doi:10.1109/tit.2013.2276628"
]
},
{
"id": "relative_entropy_variance",
"quantity": {
"html": "Relative-entropy variance V(ρ‖σ)",
"text": "Relative-entropy variance V(ρ‖σ)"
},
"domain": {
"html": "A finite-dimensional ordered state pair with supp ρ⊆supp σ for the iid theorem stated here",
"text": "A finite-dimensional ordered state pair with supp ρ⊆supp σ for the iid theorem stated here"
},
"topology_or_norm": {
"html": "ρ-weighted centered second moment of log ρ−log σ, in nats² (bits² for base-two logs); it is neither a divergence nor a norm",
"text": "ρ-weighted centered second moment of log ρ−log σ, in nats² (bits² for base-two logs); it is neither a divergence nor a norm"
},
"regulator_status": {
"html": "The state pair, cutoff, copy model, and logarithm convention remain fixed before n→∞",
"text": "The state pair, cutoff, copy model, and logarithm convention remain fixed before n→∞"
},
"operational_task": {
"html": "Second-order √n correction to asymmetric iid hypothesis testing",
"text": "Second-order √n correction to asymmetric iid hypothesis testing"
},
"theorem_hypotheses": {
"html": "Independent tensor powers, collective tests, fixed 0<ε<1, and support inclusion; the nondegenerate Gaussian term additionally needs 0<V<∞",
"text": "Independent tensor powers, collective tests, fixed 0<ε<1, and support inclusion; the nondegenerate Gaussian term additionally needs 0V=0 exactly when (log ρ−log σ−D)√ρ=0; then the Gaussian √n term degenerates",
"text": "V=0 exactly when (log ρ−log σ−D)√ρ=0; then the Gaussian √n term degenerates"
},
"error_or_bound": {
"html": "DHε(ρ⊗n‖σ⊗n)=nD+√(nV) Φ−1(ε)+O(ln n) under the stated hypotheses",
"text": "DHε(ρ⊗n‖σ⊗n)=nD+√(nV) Φ−1(ε)+O(ln n) under the stated hypotheses"
},
"counterexample_or_failed_control": {
"html": "Correlated replicas, failed support inclusion, infinite or zero variance, or a moving regulator invalidates the Gaussian expansion",
"text": "Correlated replicas, failed support inclusion, infinite or zero variance, or a moving regulator invalidates the Gaussian expansion"
},
"reference_ids": [
"doi:10.1214/13-aos1185",
"doi:10.1109/tit.2013.2276628"
]
},
{
"id": "hypothesis_testing_divergence",
"quantity": {
"html": "Hypothesis-testing divergence DHε",
"text": "Hypothesis-testing divergence DHε"
},
"domain": {
"html": "Two states and effects on one declared test algebra",
"text": "Two states and effects on one declared test algebra"
},
"topology_or_norm": {
"html": "Optimization over effects 0≤Q≤1 at fixed type-I error tolerance ε",
"text": "Optimization over effects 0≤Q≤1 at fixed type-I error tolerance ε"
},
"regulator_status": {
"html": "Detector algebra, copy model, ε, energy set, and cutoff remain fixed",
"text": "Detector algebra, copy model, ε, energy set, and cutoff remain fixed"
},
"operational_task": {
"html": "One-shot asymmetric binary decision; iid Stein and second-order analysis",
"text": "One-shot asymmetric binary decision; iid Stein and second-order analysis"
},
"theorem_hypotheses": {
"html": "Independent copies and collective tests for the iid limit; finite relative-entropy variance for the √n term",
"text": "Independent copies and collective tests for the iid limit; finite relative-entropy variance for the √n term"
},
"equality_case": {
"html": "DHε(ρ‖ρ)=−ln(1−ε) in the unnormalized convention used here",
"text": "DHε(ρ‖ρ)=−ln(1−ε) in the unnormalized convention used here"
},
"error_or_bound": {
"html": "nD+√(nV) Φ−1(ε)+O(ln n) only under the stated regularity conditions",
"text": "nD+√(nV) Φ−1(ε)+O(ln n) only under the stated regularity conditions"
},
"counterexample_or_failed_control": {
"html": "A blind detector, correlated repetitions, singular support, or a moving cutoff changes or destroys the claimed rate",
"text": "A blind detector, correlated repetitions, singular support, or a moving cutoff changes or destroys the claimed rate"
},
"reference_ids": [
"doi:10.1109/tit.2009.2018325",
"doi:10.21468/scipostphyscore.4.2.019",
"doi:10.1214/13-aos1185",
"doi:10.1109/18.887855",
"doi:10.1007/s00023-023-01269-2",
"doi:10.1109/tit.2013.2276628",
"doi:10.1103/physrevlett.119.120501"
]
},
{
"id": "bounded_observable_bias",
"quantity": {
"html": "Bounded-observable bias ‖ω−φ‖",
"text": "Bounded-observable bias ‖ω−φ‖"
},
"domain": {
"html": "Normal states on one observable algebra",
"text": "Normal states on one observable algebra"
},
"topology_or_norm": {
"html": "Predual norm; trace norm in a type-I realization",
"text": "Predual norm; trace norm in a type-I realization"
},
"regulator_status": {
"html": "Intrinsic to the chosen detector algebra; restriction can only contract it",
"text": "Intrinsic to the chosen detector algebra; restriction can only contract it"
},
"operational_task": {
"html": "Optimal equal-prior one-shot state discrimination",
"text": "Optimal equal-prior one-shot state discrimination"
},
"theorem_hypotheses": {
"html": "All allowed effects belong to the declared algebra; priors and measurement class are fixed",
"text": "All allowed effects belong to the declared algebra; priors and measurement class are fixed"
},
"equality_case": {
"html": "Identical restricted states give success probability 1/2",
"text": "Identical restricted states give success probability 1/2"
},
"error_or_bound": {
"html": "psucc*=1/2+‖ω−φ‖/4 and 0≤‖ω−φ‖≤2",
"text": "psucc*=1/2+‖ω−φ‖/4 and 0≤‖ω−φ‖≤2"
},
"counterexample_or_failed_control": {
"html": "An orthogonal ultraviolet mode outside the detector algebra can be globally decisive while leaving the restricted bias zero",
"text": "An orthogonal ultraviolet mode outside the detector algebra can be globally decisive while leaving the restricted bias zero"
},
"reference_ids": [
"doi:10.1109/18.761271"
]
},
{
"id": "root_fidelity",
"quantity": {
"html": "Root fidelity F",
"text": "Root fidelity F"
},
"domain": {
"html": "States on one C* algebra; trace formula only in a type-I representation",
"text": "States on one C* algebra; trace formula only in a type-I representation"
},
"topology_or_norm": {
"html": "Algebraic transition probability; Bures or purified-distance topology after fixing the square convention",
"text": "Algebraic transition probability; Bures or purified-distance topology after fixing the square convention"
},
"regulator_status": {
"html": "Common-algebra target; finite-cutoff overlap needs an independently controlled limit",
"text": "Common-algebra target; finite-cutoff overlap needs an independently controlled limit"
},
"operational_task": {
"html": "Symmetric closeness and finite-copy discrimination bounds",
"text": "Symmetric closeness and finite-copy discrimination bounds"
},
"theorem_hypotheses": {
"html": "Root convention F=‖√ρ√σ‖1; do not import formulas written for F2 without translating",
"text": "Root convention F=‖√ρ√σ‖1; do not import formulas written for F2 without translating"
},
"equality_case": {
"html": "F=1 exactly for equal states; F=0 for orthogonal supports",
"text": "F=1 exactly for equal states; F=0 for orthogonal supports"
},
"error_or_bound": {
"html": "1−F≤T≤√(1−F2)",
"text": "1−F≤T≤√(1−F2)"
},
"counterexample_or_failed_control": {
"html": "Squaring the wrong convention changes exponents; orthogonal global states can restrict to identical local states",
"text": "Squaring the wrong convention changes exponents; orthogonal global states can restrict to identical local states"
},
"reference_ids": [
"doi:10.1016/0034-4877(76)90060-4",
"doi:10.1103/physrevlett.115.260501",
"doi:10.1109/18.761271"
]
},
{
"id": "chernoff_coefficient",
"quantity": {
"html": "Chernoff coefficient Q and exponent ξQCB",
"text": "Chernoff coefficient Q and exponent ξQCB"
},
"domain": {
"html": "Finite-dimensional density operators for the theorem stated here; trace-class coefficients require their own operational limit theorem",
"text": "Finite-dimensional density operators for the theorem stated here; trace-class coefficients require their own operational limit theorem"
},
"topology_or_norm": {
"html": "Noncommutative s-overlap Tr ρsσ1−s",
"text": "Noncommutative s-overlap Tr ρsσ1−s"
},
"regulator_status": {
"html": "Copy model and measurement class fixed; occupation and mode cutoffs are removed only in a justified order",
"text": "Copy model and measurement class fixed; occupation and mode cutoffs are removed only in a justified order"
},
"operational_task": {
"html": "Optimal symmetric iid discrimination exponent",
"text": "Optimal symmetric iid discrimination exponent"
},
"theorem_hypotheses": {
"html": "Independent copies, fixed nonzero priors, and unrestricted collective measurements",
"text": "Independent copies, fixed nonzero priors, and unrestricted collective measurements"
},
"equality_case": {
"html": "If at least one state is pure, infs Qs=F2; Q1/2 is not generally F",
"text": "If at least one state is pure, infs Qs=F2; Q1/2 is not generally F"
},
"error_or_bound": {
"html": "lim −ln pe,n*/n=−ln inf0<s<1 Qs under those hypotheses",
"text": "lim −ln pe,n*/n=−ln inf0s=1/2, using correlated copies, or interchanging copy and regulator limits can give the wrong exponent",
"text": "Assuming s=1/2, using correlated copies, or interchanging copy and regulator limits can give the wrong exponent"
},
"reference_ids": [
"doi:10.1103/physrevlett.98.160501",
"doi:10.1103/physreva.77.032311",
"doi:10.1214/08-aos593"
]
},
{
"id": "mutual_information",
"quantity": {
"html": "Mutual information I(A:B)",
"text": "Mutual information I(A:B)"
},
"domain": {
"html": "A joint state on commuting region algebras with a defined product reference",
"text": "A joint state on commuting region algebras with a defined product reference"
},
"topology_or_norm": {
"html": "Relative entropy to the product state",
"text": "Relative entropy to the product state"
},
"regulator_status": {
"html": "Can be intrinsic for suitably separated regions; touching regions require a common ultraviolet prescription",
"text": "Can be intrinsic for suitably separated regions; touching regions require a common ultraviolet prescription"
},
"operational_task": {
"html": "Total correlations visible to the two chosen algebras",
"text": "Total correlations visible to the two chosen algebras"
},
"theorem_hypotheses": {
"html": "Region algebras, product reference, state restriction, and separation are fixed",
"text": "Region algebras, product reference, state restriction, and separation are fixed"
},
"equality_case": {
"html": "I=0 exactly for a product state when the comparison is defined",
"text": "I=0 exactly for a product state when the comparison is defined"
},
"error_or_bound": {
"html": "Positivity and Pinsker-type bounds control bounded connected correlations",
"text": "Positivity and Pinsker-type bounds control bounded connected correlations"
},
"counterexample_or_failed_control": {
"html": "Mutual information is not automatically distillable entanglement, and finiteness can fail as the separation closes",
"text": "Mutual information is not automatically distillable entanglement, and finiteness can fail as the separation closes"
},
"reference_ids": [
"doi:10.1103/physrevlett.100.070502",
"doi:10.1007/bf01388641",
"doi:10.1007/s12188-016-0130-9",
"doi:10.1016/j.aim.2018.08.015",
"doi:10.1088/1742-5468/2009/11/p11001",
"doi:10.1088/1751-8113/42/50/504007",
"doi:10.1088/1751-8113/46/28/285402",
"doi:10.3390/axioms5010005",
"doi:10.48550/arxiv.2110.05823",
"doi:10.1007/s00220-019-03367-x"
]
},
{
"id": "conditional_mutual_information",
"quantity": {
"html": "Conditional mutual information I(A:C|B)",
"text": "Conditional mutual information I(A:C|B)"
},
"domain": {
"html": "A compatible ordered tripartite system or an explicitly applicable algebraic extension",
"text": "A compatible ordered tripartite system or an explicitly applicable algebraic extension"
},
"topology_or_norm": {
"html": "Entropy difference; equivalently tied to recovery fidelity by theorem-specific bounds",
"text": "Entropy difference; equivalently tied to recovery fidelity by theorem-specific bounds"
},
"regulator_status": {
"html": "All four entropies use one state, regulator, center choice, and region prescription",
"text": "All four entropies use one state, regulator, center choice, and region prescription"
},
"operational_task": {
"html": "Residual A–C correlation when B is retained",
"text": "Residual A–C correlation when B is retained"
},
"theorem_hypotheses": {
"html": "The subsystem order and recovery direction are fixed; finite-dimensional statements are not silently exported to type-III factors",
"text": "The subsystem order and recovery direction are fixed; finite-dimensional statements are not silently exported to type-III factors"
},
"equality_case": {
"html": "I=0 exactly characterizes a quantum Markov chain in finite dimensions",
"text": "I=0 exactly characterizes a quantum Markov chain in finite dimensions"
},
"error_or_bound": {
"html": "I≥−2 ln F(ρ, recovered ρ) for an appropriate recovery channel",
"text": "I≥−2 ln F(ρ, recovered ρ) for an appropriate recovery channel"
},
"counterexample_or_failed_control": {
"html": "Small I does not make the recovery local, causal, unique, energy bounded, or dynamically Markovian",
"text": "Small I does not make the recovery local, causal, unique, energy bounded, or dynamically Markovian"
},
"reference_ids": [
"doi:10.1063/1.1666274",
"doi:10.1007/bf01646092",
"doi:10.1007/s00220-004-1049-z",
"doi:10.1007/s00220-007-0362-8",
"doi:10.1007/s00220-015-2466-x",
"doi:10.1070/sm8561",
"doi:10.1088/0264-9381/21/9/011",
"doi:10.1088/1742-5468/2004/06/p06002",
"doi:10.1088/1751-8113/42/50/504007",
"doi:10.1088/1751-8121/aa7eaa",
"doi:10.1103/physrevd.89.085012"
]
},
{
"id": "petz_universal_recovery",
"quantity": {
"html": "Petz and universal recovery",
"text": "Petz and universal recovery"
},
"domain": {
"html": "A state pair, reference state, and channel between fixed operator algebras",
"text": "A state pair, reference state, and channel between fixed operator algebras"
},
"topology_or_norm": {
"html": "Relative-entropy loss measured against fidelity or another stated recovery metric",
"text": "Relative-entropy loss measured against fidelity or another stated recovery metric"
},
"regulator_status": {
"html": "Supports, modular powers, inverses, and continuum domains need explicit control",
"text": "Supports, modular powers, inverses, and continuum domains need explicit control"
},
"operational_task": {
"html": "Reconstruct information lost through a channel or restriction",
"text": "Reconstruct information lost through a channel or restriction"
},
"theorem_hypotheses": {
"html": "Normal completely positive channel, declared adjoint, reference state, and theorem-specific support conditions",
"text": "Normal completely positive channel, declared adjoint, reference state, and theorem-specific support conditions"
},
"equality_case": {
"html": "Equality in data processing is equivalent to exact recovery of the specified pair under the standard hypotheses",
"text": "Equality in data processing is equivalent to exact recovery of the specified pair under the standard hypotheses"
},
"error_or_bound": {
"html": "Rotated or universal maps give quantitative fidelity bounds for nonzero information loss",
"text": "Rotated or universal maps give quantitative fidelity bounds for nonzero information loss"
},
"counterexample_or_failed_control": {
"html": "An abstract recovery map need not be unique, local, causal, implementable, or finite energy",
"text": "An abstract recovery map need not be unique, local, causal, implementable, or finite energy"
},
"reference_ids": [
"doi:10.1007/bf01212345",
"doi:10.1093/qmath/39.1.97",
"doi:10.1007/s00220-004-1049-z",
"doi:10.1070/sm8561",
"doi:10.1007/s00023-018-0716-0",
"doi:10.1007/s11005-024-01775-2",
"doi:10.1007/s00220-015-2466-x",
"doi:10.1007/s00220-021-04143-6",
"doi:10.1063/1.5093326",
"doi:10.1088/1751-8121/aaad26",
"doi:10.1098/rspa.2015.0623",
"doi:10.1103/physrevlett.98.100502"
]
},
{
"id": "energy_constrained_diamond_norm",
"quantity": {
"html": "Energy-constrained diamond norm ‖Φ−Ψ‖⋄,E",
"text": "Energy-constrained diamond norm ‖Φ−Ψ‖⋄,E"
},
"domain": {
"html": "Normal channels tested on input–ancilla states obeying a stated mean-energy constraint",
"text": "Normal channels tested on input–ancilla states obeying a stated mean-energy constraint"
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"topology_or_norm": {
"html": "Trace norm optimized on an energy set; weaker than the unconstrained diamond norm",
"text": "Trace norm optimized on an energy set; weaker than the unconstrained diamond norm"
},
"regulator_status": {
"html": "Input Hamiltonian, budget E, ancilla, and cutoff remain explicit",
"text": "Input Hamiltonian, budget E, ancilla, and cutoff remain explicit"
},
"operational_task": {
"html": "Single-use channel discrimination with physically admissible probes",
"text": "Single-use channel discrimination with physically admissible probes"
},
"theorem_hypotheses": {
"html": "Same Hamiltonian and energy set for both channels; allowed ancillary resources are declared",
"text": "Same Hamiltonian and energy set for both channels; allowed ancillary resources are declared"
},
"equality_case": {
"html": "Equal channels give zero; perfectly distinguishable admissible outputs attain two",
"text": "Equal channels give zero; perfectly distinguishable admissible outputs attain two"
},
"error_or_bound": {
"html": "psucc*=1/2+‖Φ−Ψ‖⋄,E/4 and the norm is monotone in E",
"text": "psucc*=1/2+‖Φ−Ψ‖⋄,E/4 and the norm is monotone in E"
},
"counterexample_or_failed_control": {
"html": "Distinct pure-loss channels have unconstrained diamond distance two even when low-energy probes see only a small difference",
"text": "Distinct pure-loss channels have unconstrained diamond distance two even when low-energy probes see only a small difference"
},
"reference_ids": [
"doi:10.1134/s0032946018010027",
"doi:10.1088/1367-2630/10/8/083030",
"doi:10.1038/s41467-025-64872-3",
"doi:10.48550/arxiv.1712.10267",
"url:https://shop.elsevier.com/books/quantum-detection-and-estimation-theory/helstrom/978-0-12-340050-5"
]
},
{
"id": "entropy_continuity",
"quantity": {
"html": "Entropy continuity",
"text": "Entropy continuity"
},
"domain": {
"html": "Finite dimension, or an infinite-dimensional energy set with suitable Gibbs spectral control",
"text": "Finite dimension, or an infinite-dimensional energy set with suitable Gibbs spectral control"
},
"topology_or_norm": {
"html": "Trace-distance perturbation converted into an entropy bound",
"text": "Trace-distance perturbation converted into an entropy bound"
},
"regulator_status": {
"html": "Dimension or Hamiltonian and energy budget are part of the statement",
"text": "Dimension or Hamiltonian and energy budget are part of the statement"
},
"operational_task": {
"html": "Propagate state-approximation error to an entropy error bar",
"text": "Propagate state-approximation error to an entropy error bar"
},
"theorem_hypotheses": {
"html": "For the finite-dimensional sharp bound, δ≤1−1/d; infinite-dimensional variants require a finite partition function in the relevant range",
"text": "For the finite-dimensional sharp bound, δ≤1−1/d; infinite-dimensional variants require a finite partition function in the relevant range"
},
"equality_case": {
"html": "The finite-dimensional Fannes–Audenaert bound is sharp for an extremal commuting pair",
"text": "The finite-dimensional Fannes–Audenaert bound is sharp for an extremal commuting pair"
},
"error_or_bound": {
"html": "|S(ρ)−S(σ)|≤δ ln(d−1)+h2(δ)",
"text": "|S(ρ)−S(σ)|≤δ ln(d−1)+h2(δ)"
},
"counterexample_or_failed_control": {
"html": "At fixed trace distance, sending weight into a growing high-energy subspace makes the entropy difference grow like δ ln d",
"text": "At fixed trace distance, sending weight into a growing high-energy subspace makes the entropy difference grow like δ ln d"
},
"reference_ids": [
"doi:10.1088/1751-8113/40/28/s18",
"doi:10.1007/s00220-016-2609-8"
]
}
],
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