trureturingGitHub MATHEMATICAL ATLAS

Quantum / CONCEPT

Partial Trace and Quantum Mutual Information

Partial traces give density-state marginals, and independent product states have zero quantum mutual information.

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DIRECT PREREQUISITES1
DIRECT CONSEQUENCES61
PROOF DEPTH5
DOCUMENT LINKS0
THEOREM

Density states are positive in matrix coordinates

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Von Neumann Entropy Under Basis PinchingVon Neumann Entropy Under BasisPinchingPartial Trace and Quantum Mutual InformationPartial Trace and QuantumMutual InformationA triangle inequality fails for unified-entropy correlationsA triangle inequality fails forunified-entropy correlationsFibonacci carry output algebraFibonacci carry output algebraSteering does not force atemporalitySteering does not forceatemporalityCanonical Product RecoveryCanonical Product RecoveryThe three-state root-fidelity Holevo boundThe three-state root-fidelityHolevo boundRaw Clause Hamiltonian and Full TraceRaw Clause Hamiltonian and FullTraceAll gate orderings of an open-boundary transfer-matrix circuitAll gate orderings of an open-boundary transfer-matrixcircuitSubsystem Lanczos positivity failsSubsystem Lanczos positivityfailsThe Gurvits--Barnum separable ballThe Gurvits--Barnum separableballSymmetric qudit states with an absolutely PPT spectrumSymmetric qudit states with anabsolutely PPT spectrumFinite product-vector averagesFinite product-vector averagesTwo reduced states of one three-qutrit state both violate the CGLMP inequalityTwo reduced states of onethree-qutrit state both violatethe CGLMP inequalityFinite Additive Readout BlocksFinite Additive Readout BlocksFinite Sector Channel OptimalityFinite Sector ChannelOptimalityFlat Sector Rank FeasibilityFlat Sector Rank FeasibilityPhysical Sector Channel ConstructionPhysical Sector ChannelConstructionFormationBoundsFormationBoundsTotal spectral rows and direct concatenationTotal spectral rows and directconcatenationSelective LOCC can increase the Moreau-Yosida entanglement of formationSelective LOCC can increase theMoreau-Yosida entanglement offormationThe minimum of purity plus time-reversal overlapThe minimum of purity plustime-reversal overlapThe fully entangled fraction and the steering-ellipsoid centreThe fully entangled fractionand the steering-ellipsoidcentreDicke diagonal extension conesDicke diagonal extension conesRefutation of the geometric tangle ansatzRefutation of the geometrictangle ansatzThe kernel and trace criteria are not equivalentThe kernel and trace criteriaare not equivalentReal Copy Statistics and ConjugationReal Copy Statistics andConjugationFinite Density PurificationFinite Density PurificationMaximum marginal entropy can increase under a local measurementMaximum marginal entropy canincrease under a localmeasurementThe minimum of the long-time algebra OTOCThe minimum of the long-timealgebra OTOCConditional 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 contractionconjectureCoherent Premeasurement and Correlation TaxCoherent Premeasurement andCorrelation TaxTwo-qubit conditional domains need not be subalgebrasTwo-qubit conditional domainsneed not be subalgebrasEnergy and Information with Initial CorrelationsEnergy and Information withInitial CorrelationsFisher discord conventionsFisher discord conventionsInput Information BalanceInput Information BalancePurify-and-clone is not optimal for one-site worst-case riskPurify-and-clone is not optimalfor one-site worst-case riskOrthogonal Records and Pointer InformationOrthogonal Records and PointerInformationQubitNormalFormStateQubitNormalFormStateRényi CMI local-channel monotonicityRényi CMI local-channelmonotonicityOrder-three Renyi information combiningOrder-three Renyi informationcombiningSeparable states outside every local-unitary stabilizer frameSeparable states outside everylocal-unitary stabilizer frameA mixed-state refutation of skew-coherence polygamyA mixed-state refutation ofskew-coherence polygamyStationarity does not imply tabletop time reversibilityStationarity does not implytabletop time reversibilityA counterexample to Zhang–Wu–Fei superadditivityA counterexample toZhang–Wu–Fei superadditivityAn identity ancilla defeats maximal measurement coherence monotonicityAn identity ancilla defeatsmaximal measurement coherencemonotonicityOccupationPartialTransposeSpectralOccupationPartialTransposeSpectralHardware Circuit FactorsHardware Circuit FactorsAmplitude-damping channels resist perfect measure-and-prepare cloningAmplitude-damping channelsresist perfect measure-and-prepare cloningWassersteinComplexityNonconvexWassersteinComplexityNonconvexConcatenation can increase the cHaar moment normConcatenation can increase thecHaar moment normAn explicit source amplitudeAn explicit source amplitudeCanonical purification with the correct complex-state marginalCanonical purification with thecorrect complex-state marginalDeterminant-integral subentropyDeterminant-integral subentropyA strong-subadditivity counterexample for subentropyA strong-subadditivitycounterexample for subentropyThe exact quantum bound for nonlocality depthThe exact quantum bound fornonlocality depthA cooling transition with no ground-space thermal realizationA cooling transition with noground-space thermalrealizationA non-extreme minimum-rank superchannel extensionA non-extreme minimum-ranksuperchannel extensionCompatibility need not imply Jordan compatibilityCompatibility need not implyJordan compatibilityPassive memory with a nonclassical Gram matrixPassive memory with anonclassical Gram matrixOne measurement does not force noncontextualityOne measurement does not forcenoncontextuality
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Lean sourceResearch workspace
Status
Closed
Depth
5
Layer
D5/S3
Domain
Quantum
Node ID
D5/S3/Quantum/Information/PartialTraceMutualInformation
Repository path
D5/S3/Quantum/Information/PartialTraceMutualInformation.lean
Truth release
sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608
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b52262a023c4db9bb30a128ebfd80ea4ef4c1a53
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37a24ddf3b9983bdabf0acbfd9b32b06c35c28d3

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