trureturingGitHub MATHEMATICAL ATLAS

Quantum / CONCEPT

Finite Kraus Quantum Channels

Finite Kraus realizations in the canonical completely positive channel interface.

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DIRECT PREREQUISITES1
DIRECT CONSEQUENCES42
PROOF DEPTH1
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THEOREM

Kraus completeness gives a canonical quantum channel

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Finite Density States and Quantum ChannelsFinite Density States andQuantum ChannelsFinite Kraus Quantum ChannelsFinite Kraus Quantum ChannelsRelative entropy coherence is at most l1 coherenceRelative entropy coherence isat most l1 coherenceOrdered GNS ProjectionsOrdered GNS ProjectionsSteering does not force atemporalitySteering does not forceatemporalityFinite Shifted Record RealizationFinite Shifted RecordRealizationThe three-state root-fidelity Holevo boundThe three-state root-fidelityHolevo boundDisjoint quantum ports and topological-order independenceDisjoint quantum ports andtopological-order independenceFinite Sector Channel OptimalityFinite Sector ChannelOptimalityPhysical Sector Channel ConstructionPhysical Sector ChannelConstructionSequential Occupation HistorySequential Occupation HistoryFinite Kraus RepresentationsFinite Kraus RepresentationsFinitePreparationChannelFinitePreparationChannelQubit Generalized Fidelity and Failure of Data ProcessingQubit Generalized Fidelity andFailure of Data ProcessingConditional min-entropy of noisy unitariesConditional min-entropy ofnoisy unitariesRefutation of completely contractive CMI data processingRefutation of completelycontractive CMI data processingTrace-norm CMI contraction conjectureTrace-norm CMI contractionconjecturePurify-and-clone is not optimal for one-site worst-case riskPurify-and-clone is not optimalfor one-site worst-case riskThe quantum coherence fraction can increase under incoherent operationsThe quantum coherence fractioncan increase under incoherentoperationsCyclic Refractory Quantum GeneratorCyclic Refractory QuantumGeneratorDiagonal Kraus Instrument FactorizationDiagonal Kraus InstrumentFactorizationThe Eventual-Click Doob InstrumentThe Eventual-Click DoobInstrumentExact Conditional Preparation CostExact Conditional PreparationCostVisible Adaptive Physical Record LawVisible Adaptive PhysicalRecord LawBroadcasting channels need have neither extremumBroadcasting channels need haveneither extremumThe gcd-three equator formula fails at dimension sixThe gcd-three equator formulafails at dimension sixControlled-rotation assemblages do not obey the proposed LOSR angle orderingControlled-rotation assemblagesdo not obey the proposed LOSRangle orderingThe overlap-to-purity part of Kopel's equality theoremThe overlap-to-purity part ofKopel's equality theoremThe least trace-distance contraction coefficient over a classical channelThe least trace-distancecontraction coefficient over aclassical channelThe Choi spectrum of unital product-unitary-equivariant mapsThe Choi spectrum of unitalproduct-unitary-equivariantmapsA qubit pure-program obstructionA qubit pure-programobstructionPure targets prevent general square-root savingsPure targets prevent generalsquare-root savingsPauli-weight expansion and alternating truncation bounds for qubit SIC k-fidelitiesPauli-weight expansion andalternating truncation boundsfor qubit SIC k-fidelitiesOperational sparse-probe compression boundOperational sparse-probecompression boundThe k-positive region of the diagonal-perturbed Tomiyama mapsThe k-positive region of thediagonal-perturbed TomiyamamapsWassersteinComplexityNonconvexWassersteinComplexityNonconvexKrausCompletionKrausCompletionRetainedLocalProtocolRetainedLocalProtocolPositive refinement of a coarse qubit mapPositive refinement of a coarsequbit mapRelative quantumness can increase under a commutativity-preserving channelRelative quantumness canincrease under a commutativity-preserving channelA non-extreme minimum-rank superchannel extensionA non-extreme minimum-ranksuperchannel extensionFlat-pair endpoint homomorphismsFlat-pair endpointhomomorphismsPositive Kraus operators with an entanglement advantagePositive Kraus operators withan entanglement advantageSchur-channel mixed-unitary distance equalitySchur-channel mixed-unitarydistance equality
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Lean sourceResearch workspace
Status
Closed
Depth
1
Layer
D5/S3
Domain
Quantum
Node ID
D5/S3/Quantum/Foundation/FiniteKrausChannel
Repository path
D5/S3/Quantum/Foundation/FiniteKrausChannel.lean
Truth release
sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608
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b52262a023c4db9bb30a128ebfd80ea4ef4c1a53
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37a24ddf3b9983bdabf0acbfd9b32b06c35c28d3

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Dependency edges, status, and source coordinates come from the verified release and certified topology. Blueprint exposition is labeled separately. A fallback title carries no additional mathematical authority.