Level Spacing Distribution Revisited - Theoretical Group Atomic ...
Level Spacing Distribution Revisited - Theoretical Group Atomic ... 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
Level spacing for unitary matrices of order N = 4 Comparison of spacing distribution P(s) for a) Poisson CPE(4), b) CUE(2) ⊗ CUE(2), c) CUE(4), Poisson ensemble, β = 0, CPE(4), P (4) 0 (s) = (1 − s/4)3 , Large matrices, N → ∞ asymptotics P (∞) 0 (s) = e −s , K ˙Z (IF UJ/CFT PAN ) Level Spacing Distribution March 14, 2011 15 / 20
- Page 1 and 2: Level Spacing Distribution Revisite
- Page 3 and 4: Random Matrices & Universality Univ
- Page 5 and 6: Wigner Semicircle Law Spectral dens
- Page 7 and 8: Extremal eigenvalues & Tracy-Widom
- Page 9 and 10: Some applications of Tracy-Widom La
- Page 11 and 12: Random Domino & Arctic Circle Theor
- Page 13: Level spacing distribution P(s) Nea
- Page 17 and 18: Minimal spacing P(smin) for N = 4 u
- Page 19 and 20: Average maximal spacing 〈smax〉
- Page 21: Concluding Remarks Random Matrices:
<strong>Level</strong> spacing for unitary matrices of order N = 4<br />
Comparison of spacing distribution P(s) for<br />
a) Poisson CPE(4), b) CUE(2) ⊗ CUE(2), c) CUE(4),<br />
Poisson ensemble, β = 0, CPE(4), P (4)<br />
0 (s) = (1 − s/4)3 ,<br />
Large matrices, N → ∞ asymptotics P (∞)<br />
0 (s) = e −s ,<br />
K ˙Z (IF UJ/CFT PAN ) <strong>Level</strong> <strong>Spacing</strong> <strong>Distribution</strong> March 14, 2011 15 / 20
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