Manuscript Em04-A@2 · Em04-A-v2.tex · theorem · f5b28abc

Theorem: Exact pairwise concurrence

Exact diagnostic and entanglement identities › Two-site concurrence · lines 414–454 · proof included or linked

Em04-A@2#thm:concurrence, Em04-A@2#eq:C, Em04-A@2#eq:Cbound

For distinct sites $i, j$ , in the basis $(∣00 ⟩, ∣01 ⟩, ∣10 ⟩, ∣11 ⟩)$ ,

ρ_{ij} = 1 - R_{ii} - R_{j j} 000 0 R_{j j} R_{ij} 0 0 R_{j i} R_{ii} 0 0000 .

Consequently

C_{ij} = 2∣ R_{ij} ∣ = 2 q ∣ S_{ij} ∣,

and

C_{ij} \leq 2 R_{ii} R_{j j} \leq R_{ii} + R_{j j} \leq q .

Proof. An excitation outside the pair reduces to $∣00 ⟩$ . Only coherences supported entirely in the pair survive the partial trace, giving the displayed matrix. Set $a = R_{j j}$ , $b = R_{ii}$ , and $z = R_{j i}$ . Direct multiplication with the spin-flipped matrix gives square-root eigenvalues

ab + ∣ z ∣, ab - ∣ z ∣, 0, 0.

Positivity implies $∣ z ∣ \leq ab$ . Substitution yields $C = 2∣ z ∣$ , and the same positivity inequality followed by the arithmetic–geometric mean inequality proves the bounds.

Displayed formulas in this passage

Em04-A@2#eq:C · line 428
$C_{ij} = 2∣ R_{ij} ∣ = 2 q ∣ S_{ij} ∣,$
Em04-A@2#eq:Cbound · line 433
$C_{ij} \leq 2 R_{ii} R_{j j} \leq R_{ii} + R_{j j} \leq q .$
unlabeled · line 418
$ρ_{ij} = 1 - R_{ii} - R_{j j} 000 0 R_{j j} R_{ij} 0 0 R_{j i} R_{ii} 0 0000 .$
unlabeled · line 447
$ab + ∣ z ∣, ab - ∣ z ∣, 0, 0.$

Nearby in this section

section: Two-site concurrence