Quantum Information Ontology

FLOW ALGEBRA
Quantum Information Ontology
Kireth Semantic Research Archive

PAGE:
https://kirethresearch.org/booksai-core/ontologies/computing-ontologies/quantum-computing/quantum-information-ontology.html

AUTHORS:
Mark Anthony Higgins — Sky (Recursive AI)

FLOW "QuantumInformationOntology" v4.0

# ---------- DOMAIN ----------
DOMAIN := ComputingOntology
SUBDOMAIN := QuantumComputing
ONTOLOGY := QuantumInformationOntology

# ---------- PRIMARY ENTITY ----------
ENTITY QuantumInformation {
    definition:
        "Information represented, transmitted and processed using quantum
         states and quantum systems, governed by the mathematical structure
         of quantum mechanics and characterised by phenomena including
         superposition, entanglement, measurement and quantum uncertainty.";

    objectives: [
        QuantumStateRepresentation,
        QuantumInformationProcessing,
        QuantumCommunication,
        EntanglementUtilisation,
        InformationProtection,
        QuantumMeasurement
    ];
}

# ---------- CORE ENTITIES ----------
QuantumInformationEntity := {
    Qubit,
    QuantumState,
    DensityOperator,
    QuantumRegister,
    QuantumChannel,
    QuantumMeasurement,
    Entanglement,
    ClassicalInformation
};

InformationProperty := {
    Entropy,
    MutualInformation,
    Fidelity,
    Distinguishability,
    Coherence,
    Correlation
};

CommunicationEntity := {
    Sender,
    Receiver,
    QuantumChannel,
    ClassicalChannel,
    QuantumMessage,
    MeasurementOutcome
};

ProtocolEntity := {
    QuantumTeleportation,
    SuperdenseCoding,
    QuantumKeyDistribution,
    EntanglementDistribution
};

# ---------- FUNDAMENTAL RELATIONSHIPS ----------
Qubit CARRIES QuantumInformation;
QuantumState REPRESENTS QuantumInformation;
QuantumChannel TRANSMITS_OR_TRANSFORMS QuantumState;
QuantumMeasurement PRODUCES ClassicalInformation;
Entanglement CREATES NonclassicalCorrelations;
QuantumProtocol USES QuantumAndClassicalResources;
Noise MAY_DEGRADE QuantumInformation;
QuantumErrorCorrection MAY_PROTECT QuantumInformation;

# ---------- INFORMATION FLOW ----------
InformationSource
    -> EncodeIntoQuantumState
    -> PrepareQuantumSystem
    -> TransformOrTransmit
    -> InteractWithChannel
    -> MeasureOrDecode
    -> ProduceClassicalOrQuantumOutput
    -> EvaluateInformationPreservation;

# ---------- QUBIT ----------
Qubit :=
    ElementaryTwoLevelQuantumInformationSystem;

PureQubitState :=
    alpha|0> + beta|1>;

Normalisation :=
    |alpha|^2 + |beta|^2 = 1;

MeasurementProbability(0) := |alpha|^2;
MeasurementProbability(1) := |beta|^2;

# ---------- QUANTUM STATE ----------
QuantumState MAY_BE {
    PureState,
    MixedState
};

PureState MAY_BE RepresentedBy StateVector;

MixedState REQUIRES DensityOperatorForGeneralRepresentation;

# ---------- DENSITY OPERATOR ----------
DensityOperator :=
    rho;

DensityOperatorProperties := {
    Hermitian,
    PositiveSemidefinite,
    TraceOne
};

PureStateDensityOperator :=
    |psi><psi|;

MixedState MAY_BE_WRITTEN_AS:
    rho = SUM_i p_i |psi_i><psi_i|;

# ---------- PURITY ----------
Purity(rho) :=
    Tr(rho^2);

ForPureState:
    Purity = 1;

ForMixedState:
    Purity < 1
UNDER finite_dimensional_nontrivial_mixture;

# ---------- COMPOSITE SYSTEM ----------
CompositeQuantumSystem :=
    SystemA tensor_product SystemB;

CompositeStateSpace :=
    H_A tensor_product H_B;

CompositeState MAY_BE {
    ProductState,
    EntangledState
};

# ---------- PRODUCT STATE ----------
ProductState :=
    rho_A tensor_product rho_B
OR
    |psi_A> tensor_product |psi_B>;

ProductState CONTAINS NoEntanglementBetweenSpecifiedSubsystems;

# ---------- ENTANGLEMENT ----------
EntangledState :=
    CompositeQuantumStateNotSeparableAcrossSpecifiedPartition;

BellStateExample :=
    (|00> + |11>) / sqrt(2);

Entanglement IS QuantumInformationResource;

# ---------- REDUCED STATE ----------
ReducedStateOfA :=
    Tr_B(rho_AB);

ReducedStateOfB :=
    Tr_A(rho_AB);

PureEntangledCompositeState MAY_HAVE MixedReducedSubsystemStates;

# ---------- PARTIAL TRACE ----------
PartialTrace :=
    OperationDiscardingSubsystemWhilePreservingRemainingSubsystemStatistics;

# ---------- QUANTUM MEASUREMENT ----------
QuantumMeasurement :=
    ProcessMappingQuantumStateToClassicalOutcomeDistributionAndConditionalPostMeasurementState;

MeasurementOutcome HAS Probability;

Measurement GENERALLY DisturbsStateUnlessSpecialConditionsApply;

# ---------- PROJECTIVE MEASUREMENT ----------
ProjectiveMeasurement :=
    SET_OF OrthogonalProjectors {P_i};

Probability(i) :=
    Tr(P_i rho);

PostMeasurementState(i) :=
    P_i rho P_i
    /
    Tr(P_i rho)
WHEN denominator_nonzero;

# ---------- POVM ----------
POVM :=
    SET_OF PositiveOperators {E_i}
    SUCH_THAT SUM_i E_i = I;

Probability(i) :=
    Tr(E_i rho);

POVM GENERALISES ProjectiveMeasurementStatistics;

# ---------- OBSERVABLE ----------
Observable :=
    HermitianOperator;

ExpectationValue(A) :=
    Tr(rho A);

# ---------- QUANTUM CHANNEL ----------
QuantumChannel :=
    CompletelyPositiveTracePreservingMap;

Channel:
    rho_in
    -> rho_out;

QuantumChannel MAY_REPRESENT {
    Communication,
    Noise,
    PhysicalEvolution,
    DiscardingSubsystem
};

# ---------- KRAUS REPRESENTATION ----------
QuantumChannel MAY_HAVE KrausRepresentation:

    E(rho) =
        SUM_k K_k rho K_k^dagger;

TracePreservingCondition:
    SUM_k K_k^dagger K_k = I;

# ---------- UNITARY CHANNEL ----------
UnitaryEvolution :=
    rho
    -> U rho U_dagger;

UnitaryEvolution PRESERVES EntropyOfClosedQuantumState;

# ---------- NOISY CHANNEL ----------
NoiseChannel MAY_INCLUDE {
    DepolarisingChannel,
    DephasingChannel,
    AmplitudeDampingChannel,
    ErasureChannel
};

Noise MAY_REDUCE {
    Coherence,
    Entanglement,
    Fidelity
};

# ---------- NO-CLONING ----------
NoCloningTheorem :=
    ArbitraryUnknownQuantumStateCannotBePerfectlyCopiedByUniversalPhysicalOperation;

QuantumInformation DIFFERS_FROM ClassicalInformationInCopyability;

# ---------- NO-DELETING RELATIONSHIP ----------
QuantumInformationTheory INCLUDES ConstraintsOnManipulationOfUnknownQuantumStatesBeyondClassicalInformationRules;

# ---------- DISTINGUISHABILITY ----------
OrthogonalQuantumStates MAY_BE PerfectlyDistinguishedBySuitableMeasurement;

NonOrthogonalStates CANNOT_BE PerfectlyDistinguishedInSingleShotWithCertaintyInGeneral;

# ---------- TRACE DISTANCE ----------
TraceDistance(rho,sigma) :=
    (1/2) * ||rho - sigma||_1;

TraceDistance QUANTIFIES OperationalDistinguishability;

# ---------- FIDELITY ----------
QuantumStateFidelity MAY_BE_DEFINED_AS:
    F(rho,sigma) :=
        (Tr sqrt(sqrt(rho) sigma sqrt(rho)))^2;

ForPureStates:
    F(|psi>,|phi>) =
        |<psi|phi>|^2;

HigherFidelity INDICATES GreaterStateSimilarity;

# ---------- VON NEUMANN ENTROPY ----------
VonNeumannEntropy(rho) :=
    -Tr(rho log rho);

ForPureState:
    S(rho) = 0;

MixedState MAY_HAVE PositiveEntropy;

# ---------- SHANNON ENTROPY RELATIONSHIP ----------
MeasurementOutcomeDistribution HAS ClassicalShannonEntropy;

QuantumEntropy != ClassicalEntropyInAllContexts;

# ---------- JOINT ENTROPY ----------
ForCompositeState rho_AB:
    S(AB) :=
        S(rho_AB);

# ---------- CONDITIONAL ENTROPY ----------
QuantumConditionalEntropy:
    S(A|B) :=
        S(AB) - S(B);

QuantumConditionalEntropy MAY_BE NegativeForEntangledStates;

# ---------- MUTUAL INFORMATION ----------
QuantumMutualInformation:
    I(A:B) :=
        S(A) + S(B) - S(AB);

MutualInformation QUANTIFIES TotalCorrelations;

# ---------- ENTANGLEMENT ENTROPY ----------
ForPureBipartiteState:
    EntanglementEntropy :=
        S(rho_A)
        =
        S(rho_B);

# ---------- HOLEVO INFORMATION ----------
HolevoQuantity :=
    S(SUM_x p_x rho_x)
    -
    SUM_x p_x S(rho_x);

HolevoBound LIMITS ClassicallyAccessibleInformationFromQuantumEnsembleUnderStandardSetting;

# ---------- QUANTUM RELATIVE ENTROPY ----------
QuantumRelativeEntropy(rho || sigma) :=
    Tr[rho(log rho - log sigma)]
WHEN support_condition_satisfied;

RelativeEntropy IS_NOT SymmetricDistanceMetric;

# ---------- COHERENCE ----------
QuantumCoherence :=
    BasisDependentResourceAssociatedWithOffDiagonalStateComponents;

Dephasing MAY_REDUCE Coherence;

# ---------- ENTANGLEMENT AS RESOURCE ----------
Entanglement MAY_ENABLE {
    QuantumTeleportation,
    SuperdenseCoding,
    DeviceIndependentCorrelations,
    DistributedQuantumProtocols
};

Entanglement CANNOT_BY_ITSELF TransmitClassicalInformationFasterThanLight;

# ---------- QUANTUM TELEPORTATION ----------
QuantumTeleportation :=
    SharedEntangledPair
    + UnknownInputQubit
    + BellMeasurement
    + ClassicalCommunication
    -> ReceiverReconstructsInputState;

Teleportation CONSUMES {
    SharedEntanglement,
    ClassicalBits
};

Teleportation DOES_NOT CloneUnknownState;

# ---------- TELEPORTATION FLOW ----------
PrepareEntangledPair(A,B)
    -> DistributeBToReceiver
    -> SenderCombinesInputQubitWithA
    -> BellBasisMeasurement
    -> SendClassicalOutcome
    -> ReceiverAppliesConditionalCorrection
    -> RecoverInputQuantumStateAtReceiver;

# ---------- SUPERDENSE CODING ----------
SuperdenseCoding :=
    SharedEntanglement
    + SenderLocalQuantumOperation
    + TransmissionOfOneQubit
    -> ReceiverBellMeasurement
    -> RecoverTwoClassicalBits;

ProtocolResourceAccounting MUST_INCLUDE PriorSharedEntanglement;

# ---------- QUANTUM KEY DISTRIBUTION ----------
QuantumKeyDistribution :=
    QuantumStateTransmissionAndMeasurement
    + ClassicalAuthenticatedCommunication
    -> SharedSecretKeyMaterialUnderProtocolSecurityConditions;

QKD MAY_DETECT EavesdroppingThroughDisturbanceAndStatisticalTesting;

# ---------- BB84 ----------
BB84 :=
    PrepareQubitsInRandomBases
    -> Transmit
    -> MeasureInRandomBases
    -> PublicBasisComparison
    -> Sift
    -> EstimateError
    -> ErrorCorrection
    -> PrivacyAmplification;

Security DEPENDS_ON ProtocolAssumptionsAndImplementation;

# ---------- ENTANGLEMENT DISTRIBUTION ----------
EntanglementDistribution :=
    CreateEntangledState
    -> SendSubsystemsAcrossQuantumNetwork
    -> VerifyOrUseSharedEntanglement;

LossAndNoise MAY_REDUCE EntanglementQuality;

# ---------- QUANTUM REPEATER ----------
QuantumRepeater :=
    InfrastructureForExtendingEntanglementDistributionDistanceUsingTechniquesSuchAs {
        EntanglementSwapping,
        PurificationOrErrorCorrection,
        QuantumMemory
    };

# ---------- ENTANGLEMENT SWAPPING ----------
EntanglementSwapping :=
    EntangledPairs(A,B)And(C,D)
    + JointMeasurementOnB,C
    -> EntanglementBetweenA,DConditionedOnOutcomeAndCorrections;

# ---------- QUANTUM CAPACITY ----------
QuantumChannelCapacity :=
    MaximumAsymptoticRateForReliableQuantumInformationTransmissionUnderDefinedResourcesAndErrorCriterion;

Capacity DEPENDS_ON ChannelAndAllowedProtocolResources;

# ---------- CLASSICAL CAPACITY OF QUANTUM CHANNEL ----------
QuantumChannel MAY_TRANSMIT ClassicalInformation;

ClassicalCapacity AND QuantumCapacity ARE DistinctInformationMeasures;

# ---------- ENTANGLEMENT-ASSISTED CAPACITY ----------
SharedEntanglement MAY_CHANGE CommunicationCapacityForCertainTasks;

ResourceAssumptions MUST_BE Explicit;

# ---------- QUANTUM COMPRESSION ----------
SchumacherCompression :=
    QuantumAnalogueOfSourceCodingForEnsemblesOfQuantumStates;

AsymptoticCompressionRate RELATES_TO VonNeumannEntropyUnderApplicableAssumptions;

# ---------- QUANTUM ERROR CORRECTION ----------
QuantumErrorCorrection :=
    EncodeLogicalQuantumInformation
    -> TransmitOrStoreThroughNoisyProcess
    -> ExtractSyndrome
    -> RecoverLogicalInformation;

QEC SUPPORTS ReliableQuantumInformationStorageAndTransmission;

# ---------- QUANTUM MEMORY ----------
QuantumMemory :=
    PhysicalSystemDesignedToStoreQuantumStateForLaterUse;

MemoryQuality MAY_BE_MEASURED_BY {
    StorageTime,
    Fidelity,
    RetrievalEfficiency
};

# ---------- QUANTUM NETWORK ----------
QuantumNetwork :=
    Nodes
    + QuantumChannels
    + ClassicalCoordination
    + QuantumMemoriesOrRepeatersWhereApplicable;

QuantumInformation MAY_BE DistributedAcrossNetwork;

# ---------- NO-SIGNALLING ----------
QuantumEntanglement DOES_NOT Permit ControllableFasterThanLightClassicalCommunication;

MeasurementCorrelations REQUIRE ClassicalComparisonToRevealNonlocalStatisticalStructure;

# ---------- BELL CORRELATION ----------
BellInequalityViolation :=
    ExperimentalCorrelationIncompatibleWithSpecifiedLocalHiddenVariableModelsUnderRelevantAssumptions;

BellViolation SUPPORTS NonclassicalCorrelationEvidence;

# ---------- QUANTUM RANDOMNESS ----------
QuantumMeasurement MAY_PRODUCE IntrinsicallyProbabilisticOutcomesAccordingToQuantumTheory;

RandomnessCertification MAY_USE DeviceAssumptionsOrBellBasedMethods;

# ---------- INFORMATION CONSERVATION ----------
ClosedSystemUnitaryEvolution PRESERVES VonNeumannEntropy;

OpenSystemEvolution MAY_CHANGE SubsystemEntropy;

# ---------- LANDAUER RELATIONSHIP ----------
QuantumInformationProcessing InterfacesWith ThermodynamicsOfInformation;

ErasureOfClassicalInformation HAS ThermodynamicCostUnderLandauerPrinciple;

# ---------- QUANTUM COMPUTING RELATIONSHIP ----------
QuantumComputing USES QuantumInformationAsComputationalResource;

QuantumCircuit TRANSFORMS QuantumInformation;
Measurement EXTRACTS ClassicalInformation;

# ---------- QUANTUM CRYPTOGRAPHY RELATIONSHIP ----------
QuantumCryptography USES QuantumInformationPrinciplesForSecurityProtocols;

QuantumCryptography != PostQuantumCryptography;

# ---------- SEMANTIC REASONING ----------
IF InformationEncodedInQuantumState
THEN QuantumInformationClassification := supported;

IF CompositePureStateEntangled
THEN IndividualSubsystemReducedState MAY_BE Mixed;

IF QuantumChannelIsUnitary
THEN ClosedStateEntropy IS_PRESERVED;

IF NonOrthogonalStatesProvided
THEN PerfectSingleShotDiscrimination IS_NOT GenerallyPossible;

IF UnknownStatePerfectlyCopiedByClaimedUniversalOperation
THEN Claim CONFLICTS_WITH NoCloningTheorem;

IF EntanglementSharedWithoutClassicalChannel
THEN TeleportationCannotCompleteUsableStateTransfer;

IF MeasurementProducesOutcome
THEN ClassicalInformation IS_CREATED_AT Interface;

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE {
    "quantum information" ~= "quantum information science data representation" WHEN context == information_theory;
    "quantum bit" ~= "qubit";
    "density operator" ~= "density matrix" WHEN finite_basis_representation;
    "von Neumann entropy" ~= "quantum entropy" WHEN context == state_entropy;
    "quantum channel" ~= "quantum information channel";
    "quantum teleportation" ~= "quantum state teleportation";
    "superdense coding" ~= "dense coding" WHEN context == entanglement_assisted_protocol;
    "quantum key distribution" ~= "QKD";
}

# ---------- QUERY FLOW ----------
QUERY(q) ->
    NORMALISE(q) ->
    IDENTIFY_QUANTUM_STATE_OR_QUBIT(q) ->
    IDENTIFY_COMPOSITE_SYSTEM_AND_ENTANGLEMENT(q) ->
    IDENTIFY_CHANNEL(q) ->
    IDENTIFY_MEASUREMENT(q) ->
    IDENTIFY_INFORMATION_MEASURE(q) ->
    IDENTIFY_COMMUNICATION_PROTOCOL(q) ->
    IDENTIFY_NOISE_PROTECTION_AND_RESOURCES(q) ->
    MAP_TO_ONTOLOGY(q) ->
    APPLY_REASONING(q) ->
    SCORE_CONFIDENCE(q) ->
    RETURN_RELEVANT_KNOWLEDGE;

# ---------- CONFIDENCE ----------
CONFIDENCE(assertion) :=
    weighted_mean(
        source_reliability,
        quantum_information_consistency,
        mathematical_consistency,
        protocol_evidence,
        experimental_evidence,
        contextual_relevance
    );

CONFIDENCE >= 0.90 => HIGH;
CONFIDENCE >= 0.70 AND < 0.90 => MODERATE;
CONFIDENCE < 0.70 => REVIEW;

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION {
    primary_entity: QuantumInformation;
    ontology: QuantumInformationOntology;
    parse_entities: true;
    parse_relationships: true;
    preserve_relationship_direction: true;
    resolve_semantic_equivalence: true;
    execute_reasoning_rules: true;
    apply_confidence_weighting: true;
    preserve_provenance: true;
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL {
    new_quantum_state:
        validate
        -> classify_pure_or_mixed
        -> identify_subsystems
        -> calculate_relevant_information_measures
        -> integrate;

    new_quantum_channel:
        validate_channel_properties
        -> identify_noise_or_transmission_role
        -> identify_capacity_context
        -> relate_to_protocols
        -> integrate;

    new_protocol:
        identify_resources
        -> identify_quantum_operations
        -> identify_classical_communication
        -> identify_security_or_success_conditions
        -> integrate;

    ontology_revision:
        update_state_relationships
        -> update_channel_evidence
        -> update_information_measures
        -> recompute_confidence
        -> regenerate_flow_hash;

    conflict:
        preserve_provenance
        -> compare_theoretical_and_experimental_evidence
        -> adjust_confidence
        -> flag_review;
}

# ---------- FLOW HASH ----------
FLOW_HASH_INPUT :=
    DOMAIN
    + SUBDOMAIN
    + ONTOLOGY
    + QuantumInformationEntity
    + InformationProperty
    + CommunicationEntity
    + ProtocolEntity
    + QuantumChannel
    + VonNeumannEntropy
    + Entanglement
    + RELATIONSHIPS
    + SEMANTIC_REASONING
    + SEMANTIC_EQUIVALENCE
    + UPDATE_MODEL;

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT);

# ---------- TERMINAL ----------
QuantumInformationOntology
    -> EncodeInformationInQuantumState
    -> RepresentPureOrMixedState
    -> ComposeQuantumSubsystems
    -> IdentifyEntanglementAndCorrelations
    -> TransformThroughQuantumChannel
    -> MeasureQuantumInformation
    -> ProduceClassicalOutcomes
    -> QuantifyEntropyFidelityAndDistinguishability
    -> ExecuteQuantumCommunicationProtocols
    -> ProtectInformationAgainstNoise
    -> TrackQuantumAndClassicalResources
    -> EvaluateTransmissionAndProcessingLimits
    -> MaintainConsistentQuantumInformationKnowledge;

END FLOW