bibkey: jacqlardizabal2017oqwhalfline
authors: Thomas S. Jacq; Carlos F. Lardizabal
year: 2017
title: "Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster's Theorem"
doi: 10.1007/s10955-017-1876-x
url: "https://arxiv.org/abs/1701.08400v3"
claim: "For the gambler's-ruin open quantum random walk on {0,...,M} induced by matrices L, R with L*L+R*R=I, the paper computes absorption probabilities and mean hitting times through first-visit generating functions, and conjectures that for every such pair the mean hitting time has the form 1+h(M,M-k)ρ11+h(M,k)ρ22+j(M,k)Re(ρ12) with j(M,k)=-j(M,M-k)."
strata_touched:
  - D5/S3/Quantum/Walks/OpenQuantumWalkGamblerHittingTimeRefutation
license: citation-only
triage: anchor

Jacq and Lardizabal, Open quantum random walks on the half-line

T. S. Jacq and C. F. Lardizabal, J. Stat. Phys. 169(3), 547–594 (2017); arXiv:1701.08400v3. Quotations are from the v3 TeX source.

An open quantum random walk on the half-line has transition matrices $B_i^j$ with ${\sum }_j{B}_i^{j\ast }{B}_i^j=I$; from ${\rho }_i\otimes |i⟩⟨i|$ it moves to $|j⟩$ with probability $\mathrm{Tr}(B_i^j{\rho }_i{B}_i^{j\ast })$, and the state becomes $B_i^j{\rho }_i{B}_i^{j\ast }/\mathrm{Tr}(B_i^j{\rho }_i{B}_i^{j\ast })\otimes |j⟩⟨j|$. The nearest-neighbour walk of a pair $L,R$ with $L^{\ast }L+R^{\ast }R=I$ has $B_i^{i+1}=R$ and $B_i^{i-1}=L$. On the vertices $\{0,\dots ,M\}$, starting from $\rho \otimes |k⟩⟨k|$, $\tau$ is the time to reach $0$ or $M$ and $E_{k,\rho }(\tau )$ its expectation. After examples, Remark 5.3 states:

Numerical experiments allows us to recognize a behavior that such OQW has in common with Example 5.1, namely, that the mean hitting time has the form $$E_{k,\rho }(\tau )=1+h(M,M-k){\rho }_{11}+h(M,k){\rho }_{22}+j(M,k)Re({\rho }_{12}),$$ where $h,j$ are functions of $M$ and $k=1,\dots ,M-1$, with $j$ being a function satisfying $j(M,k)=-j(M,M-k)$. We conjecture that this holds for the gambler’s ruin problem associated to every pair or matrices $L$, $R$ with $L^{\ast }L+R^{\ast }R=I$ inducing an OQW on vertices $\{0,\dots ,M\}$. A proof of this statement is, up to our knowledge, unknown.

Verified locator

Source: https://arxiv.org/abs/1701.08400v3

• DOI: https://doi.org/10.1007/s10955-017-1876-x (Journal of Statistical Physics 169(3), 547–594; title, authors, journal, volume, issue and pages from Crossref).
• URL: https://arxiv.org/abs/1701.08400v3. The e-print member JL_rev2.tex has 2213 lines and md5 14ebf963591a84376ea4c598ae5fc1fc. Locations:
  • the open quantum random walk and its transition rule, l. 77–83; the nearest-neighbour pair $B_i^{i+1}=R$, $B_i^{i-1}=L$, Figure 1 caption, l. 120;
  • the gambler’s ruin and the mean hitting time, Theorem 1.7, l. 304–318, and the first-visit functions, l. 1081–1097;
  • the conjecture, Remark 5.3, l. 1291–1293.