Manuscript Em05-B@1 · Em05-B-v1.tex · theorem · 6c050f5b

Theorem: Exact marginal diagnostics

Marginal diagnostics and Bell-input correlations · lines 534–596 · proof included or linked

Em05-B@1#thm:marginals, Em05-B@1#eq:rhoR, Em05-B@1#eq:rhoE, Em05-B@1#eq:occupation, Em05-B@1#eq:coherence, Em05-B@1#eq:purity

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In the declared bases,

\rho_R
 &=
 \begin{pmatrix}
 1-p+pt^2&0&0&\eta z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 \eta z^*&0&0&p\eta^2
 \end{pmatrix},
 \label{eq:rhoR}\\
 \rho_E
 &=
 \begin{pmatrix}
 1-p+p\eta^2&0&0&tz\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 tz^*&0&0&pt^2
 \end{pmatrix}.
 \label{eq:rhoE}

For the respective occupation operators

N_{R} = N_{E} = diag (0, 1, 1, 2),

one has

⟨ N_{R} ⟩ ∣ (ρ_{R})_{00, 11} ∣ = 2 p η, = η ∣ z ∣, ⟨ N_{E} ⟩ ∣ (ρ_{E})_{00, 11} ∣ = 2 pt, = t ∣ z ∣.

The accessible purity is

P_{R} = (1 - p + p t^{2})^{2} + p^{2} η^{4} + 2 p^{2} η^{2} t^{2} + 2 η^{2} ∣ z ∣^{2} .

The residual purity is obtained by interchanging $η$ and $t$ .

Proof. Insert (eq:input) into the Kraus expressions. Only $A_{00}$ contributes the accessible $00, 11$ coherence. Equation (eq:complement) gives the residual coherence $t z$ . The occupation formulas follow by tracing against $N_{R}, N_{E}$ , using

2 p η t + 2 p η^{2} = 2 p η, 2 p η t + 2 p t^{2} = 2 pt .

Finally, for Hermitian $ρ$ , $Tr (ρ^{2}) = \sum_{i, j} ∣ ρ_{ij} ∣^{2}$ , giving (eq:purity).

Displayed formulas in this passage

Em05-B@1#eq:rhoR · line 537

\rho_R
 &=
 \begin{pmatrix}
 1-p+pt^2&0&0&\eta z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 \eta z^*&0&0&p\eta^2
 \end{pmatrix},
 \label{eq:rhoR}\\
 \rho_E
 &=
 \begin{pmatrix}
 1-p+p\eta^2&0&0&tz\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 tz^*&0&0&pt^2
 \end{pmatrix}.
 \label{eq:rhoE}

Em05-B@1#eq:rhoE · line 537

\rho_R
 &=
 \begin{pmatrix}
 1-p+pt^2&0&0&\eta z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 \eta z^*&0&0&p\eta^2
 \end{pmatrix},
 \label{eq:rhoR}\\
 \rho_E
 &=
 \begin{pmatrix}
 1-p+p\eta^2&0&0&tz\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 tz^*&0&0&pt^2
 \end{pmatrix}.
 \label{eq:rhoE}

Em05-B@1#eq:occupation · line 562
$⟨ N_{R} ⟩ ∣ (ρ_{R})_{00, 11} ∣ = 2 p η, = η ∣ z ∣, ⟨ N_{E} ⟩ ∣ (ρ_{E})_{00, 11} ∣ = 2 pt, = t ∣ z ∣.$
Em05-B@1#eq:coherence · line 562
$⟨ N_{R} ⟩ ∣ (ρ_{R})_{00, 11} ∣ = 2 p η, = η ∣ z ∣, ⟨ N_{E} ⟩ ∣ (ρ_{E})_{00, 11} ∣ = 2 pt, = t ∣ z ∣.$
Em05-B@1#eq:purity · line 573
$P_{R} = (1 - p + p t^{2})^{2} + p^{2} η^{4} + 2 p^{2} η^{2} t^{2} + 2 η^{2} ∣ z ∣^{2} .$
unlabeled · line 558
$N_{R} = N_{E} = diag (0, 1, 1, 2),$
unlabeled · line 588
$2 p η t + 2 p η^{2} = 2 p η, 2 p η t + 2 p t^{2} = 2 pt .$

Theorem: Exact marginal diagnostics

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