Recurrence for Discrete Time Unitary Evolutions
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
Recurrence for Discrete Time Unitary Evolutions. / Grünbaum, F. A.; Velázquez, L.; Werner, A. H.; Werner, R. F.
I: Communications in Mathematical Physics, Bind 320, Nr. 2, 01.06.2013, s. 543-569.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
TY - JOUR
T1 - Recurrence for Discrete Time Unitary Evolutions
AU - Grünbaum, F. A.
AU - Velázquez, L.
AU - Werner, A. H.
AU - Werner, R. F.
PY - 2013/6/1
Y1 - 2013/6/1
N2 - We consider quantum dynamical systems specified by a unitary operator U and an initial state vector φ. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to φ. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.
AB - We consider quantum dynamical systems specified by a unitary operator U and an initial state vector φ. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to φ. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.
UR - http://www.scopus.com/inward/record.url?scp=84877617738&partnerID=8YFLogxK
U2 - 10.1007/s00220-012-1645-2
DO - 10.1007/s00220-012-1645-2
M3 - Journal article
AN - SCOPUS:84877617738
VL - 320
SP - 543
EP - 569
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
IS - 2
ER -
ID: 236787382