Level Spacing Distribution Revisited - Theoretical Group Atomic ...

Composite systems & tensor products
Consider tensor product of random matrices V = U1 ⊗ U2
of size N, where Uj are random matrices according to the Haar measure
(CUE). Eig(V) = {ψkm} = (αk + βm)|mod2π,
where {αk} N k=1 and {βm} N m=1 denote spectra of U1 and U2.
What is the (asymptotic) spacing distribution PCUE⊗CUE(s) ?
Two qubits & random local gates
Analytical results P2⊗2(s) for CUE(2) ⊗ CUE(2),
(case by case analysis & direct integration)
P2⊗2(s) = Θ(2 − s) 1
2π (s − 2)sin2 ( πs
4 ) π(s − 2) − 2 sin( πs
2 )
+ 1
128π2
2(36 + π2 (s − 4) 2 ) + 128 cos( πs
4 )
+ (56 + π 2 (s − 4) 2 )cos( πs
2 )
K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 14 / 20