A Pure Bell State with a Mixed Marginal

Abstract

The pure Bell density has the maximally mixed one-qubit marginal.

More generally, a positive joint density with a normalized pure first marginal is the tensor product of that pure density and a second density. Let P project onto the pure vector tensored with the full second interface. The marginal identity gives full trace on P. Factoring the joint density as R-adjoint times R shows that R times the complementary projection has zero squared norm, hence the joint density equals P times itself times P. Contracting with the pure vector on both sides supplies the positive trace-one second factor.

Two Bell densities on overlapping pairs have the same middle-qubit marginal, one half of the identity, but admit no common three-qubit positive state. The first Bell marginal has zero diagonal entries at the unequal computational-basis addresses. Positivity makes every joint diagonal summand at either such address zero. A zero diagonal entry of a positive matrix forces its entire row and column to vanish.

The off-diagonal Bell coherence required by the second pair is a sum of two joint entries. In the first entry the column has an unequal first-pair address; in the second the row has one. Both vanish, contradicting the nonzero Bell coherence. This obstruction concerns a common positive realization of both prescribed marginals; equality of their overlap does not supply such a realization.

Theorem 1.1 (The Bell pure state reduces to one half of the identity).

$$\begin{array}{c}⟨\mathrm{bellVector}\mid \mathrm{bellVector}⟩=1\wedge \\ \mathrm{bellDensity}\ge 0\wedge \\ \mathrm{Tr}(\mathrm{bellDensity})=1\wedge \\ \mathrm{rank}(\mathrm{bellDensity})=1\wedge \\ {\mathrm{bellDensity}}^2=\mathrm{bellDensity}\wedge \\ \mathrm{traceEnvironment}(\mathrm{bellDensity})=\frac{1}{2}I\wedge \\ \mathrm{traceEnvironment}(\mathrm{bellDensity}{)}^2\ne \mathrm{traceEnvironment}(\mathrm{bellDensity}).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/BellPureStateMixedMarginal.bell_pure_state_has_maximally_mixed_marginal (✓ std3). ∎

Source. Repository-derived.

Commentary.

The amplitude is the canonical normalized Bell vector obtained from the standard coefficients for the two computational-basis terms. Its outer product is the canonical Bell density matrix.

Normalization, positivity, trace one, rank one, and idempotence are all public clauses. Together they certify that the joint two-qubit density is pure rather than merely naming it as a pure state.

The partial trace is constructed by summing the equal environment indices. It is exactly one half of the qubit identity, and its failure of idempotence is public evidence that the marginal is mixed.

The proof applies the existing rank-one handshake and Bell-state certificate, then evaluates the four finite matrix entries. No repository theorem already packaged the Bell partial trace.

References

• Truth anchor: D5/S3/Quantum/Entanglement/BellPureStateMixedMarginal.bell_pure_state_has_maximally_mixed_marginal
• Dependency: D5/S3/Quantum/EnvironmentRecords
• Dependency: D5/S3/Quantum/PureState/PureStateHandshake
• Dependency: D5/S3/QuantumBounds/CHSHWitness