Quantum walks can be connected to extended
Quantum walks can be connected to extended CMV matrices as follows. If all Verblunsky coefficients with even index vanish, then the extended CMV matrix becomes which resembles the matrix representation U.
One may notice, however, that in the extended CMV matrix , whereas in the quantum walk matrix U could be a complex number with non-zero argument, which means that these two matrices may not match. However, this can be easily resolved by conjugation with a suitable diagonal unitary, that is, there exists a diagonal unitary matrix Λ such that . For details, see  or .
From now on we consider the case of coined quantum walks on the integer lattice where the coins are distributed quasi-periodically according to the rule with analytic sampling functions and . We denote the corresponding quantum walk by . In this section, in order to distinguish the energy and the complex number , we will use E to denote the energy.
Introduction CMV reactivation is often described in patients affected by haematological malignancies and treated with alemtuzumab, a monoclonal anti-CD52 antibody (Gallamini et al., 2007).
Case report Three weeks after the first infusion the patient developed cough and fever, two days later he started antibiotic treatment with amoxicillin/clavulanate 1 g twice daily and the day after he performed X-chest ray that showed parenchymal consolidation on right lower lobe. Two more days later he was admitted to the Infectious Disease Department of University “Federico II” of Naples for persistence of fever and cough, there he switched to a parenteral antimicrobial therapy with Piperacillin/Tazobactam 4.5 g i.v. every 6 hours and Levofloxacin 750 mg i.v. once daily (OD). At admission laboratory parameters showed white blood GS967 9,040 cells/µL with prevalence of neutrophils cells (95,1%), CD4 cells 10/microL, C-Reactive Protein (CRP) 8.53 mg/dL (n.v. <5), LDH 366 U/L, Fibrinogen >1000 mg/L, IgG anti-CMV positive, negative IgM anti-CMV, CMV-DNA 500 cp/mL. Three days later he performed CT scan that showed pulmonary lower right lobe consolidation associated with concomitant interstitial involvement (Fig. 1). He quickly became apyretic and cough improved. Conversely, CRP levels, after a rapid drop, reached a plateau standing on three times upper normal value; he also repeated CMV-DNA that increased to 1,380 cp/mL five days later the first measurement. Therefore we decided to perform a Bronchoaspiratory lavage in order to exclude CMV pneumonia. CMV-DNA resulted positive (66,000 cp/mL) on BAL sample so it was started oral therapy with valgancyclovir 900 mg bis in die for 21 days followed by oral maintenance at 900 mg OD with resolution of symptoms and patient's hospital discharge. Maintenance therapy lasted two months as CD4 count rose over 100 cells/microL.
Discussion An elevated risk of CMV reactivation has been extensively reported in in patients treated with alemtuzumab for lymphoproliferative disorders. (O'Brien et al. (2006)) In this setting, current guidelines recommend serum CMV-DNA testing in all patients in case of fever occurring immediately after infusion and use of ganciclovir or valganciclovir as first line agents for the treatment of active disease (O'Brien et al., 2006). Recent reports have provided evidence of a similar increased risk of opportunistic infections, including CMV disease, in patients treated with alemtuzumab for MS (Buonomo et al., 2018). Though no specific data regarding CMV reactivation could be drawn from randomized controlled trials, at the best of our knowledge five case-reports have documented cases of CMV disease in alemtuzumab-treated patients for MS (Clerico et al., 2017, Barone et al., 2018). In detail, a single case of CMV pneumonia has been described (Yann et al., 2017). Finally, many diagnostic and clinical issues in the management of CMV reactivation in patients treated with alemtuzumab for MS still need to be addressed: