Positive contraction mappings for classical and quantum Schrodinger systems

The classical Schrodinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior, and the law dictates a controlled path that abides by the specified marginals. Schrodinger proved that the optimal steering of the density between the two end points is effected by a multiplicative functional transformation of the prior; this transformation represents an automorphism on the space of probability measures and has since been studied by Fortet, Beurling and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins with the problem of steering a Markov chain between given marginals. Our approach is based on the Hilbert metric and leads to an alternative proof which, however, is constructive. More specifically, we show that the solution to the Schrodinger bridge is provided by the fixed point of a contractive map. We approach in a similar manner the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map, but only for the case of uniform marginals. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page

Discrete-time classical and quantum Markovian evolutions: Maximum entropy problems on path space

The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced.Comment: 34 page

The future of resilient supply chains

While supply chain resilience has been touched upon frequently, research remains (with the exception of often repeated anecdotal examples) relatively disparate on what disruptions actually are. This research aims to advance theoretical and managerial understandings around the management of supply chain disruptions. A two-stage research process is used which focuses first on polling academic experts. This stage is followed by the extraction of insights from practitioners in the automotive, electronics and food industries. Our findings coalesce around: (1) the types of disruptions that respondents are most concerned about; (2) the associated strategies suggested to cope with disruptions; and, (3) how resilience can be measured. It is apparent that there are some areas where academics and practitioners agree and others where they agree to a lesser extent. Both sets of actors tend to agree on how resilience can be quantified, with recovery time the preferred indicator. However, there is a discrepancy around how resilience is achieved within the supply chain. Academics emphasise the importance of redundancy while practitioners refer more to flexibility. Also, they disagree around what constitutes “key disruptions”: academics suggested high-profile events, while practitioners are more concerned with day-to-day problems

Rehabilitation of high upper limb amputees after Targeted Muscle Reinnervation

STUDY DESIGN: This is a Delphi study based on a scoping literature review. INTRODUCTION: Targeted muscle reinnervation (TMR) enables patients with high upper limb amputations to intuitively control a prosthetic arm with up to six independent control signals. Although there is a broad agreement regarding the importance of structured motor learning and prosthetic training after such nerve transfers, to date, no evidence-based protocol for rehabilitation after TMR exists. PURPOSE OF THE STUDY: We aimed at developing a structured rehabilitation protocol after TMR surgery after major upper limb amputation. The purpose of the protocol is to guide clinicians through the full rehabilitation process, from presurgical patient education to functional prosthetic training. METHODS: European clinicians and researchers working in upper limb prosthetic rehabilitation were invited to contribute to a web-based Delphi study. Within the first round, clinical experts were presented a summary of recent literature and were asked to describe the rehabilitation steps based on their own experience and scientific evidence. The second round was used to refine these steps, while the importance of each step was rated within the third round. RESULTS: Experts agreed on a rehabilitation protocol that consists of 16 steps and starts before surgery. It is based on two overarching principles, namely the necessity of multiprofessional teamwork and a careful selection and education of patients within the rehabilitation team. Among the different steps in therapy, experts rated the training with electromyographic biofeedback as the most important one. DISCUSSION: Within this study, a first rehabilitation protocol for TMR patients based on a broad experts' consensus and relevant literature could be developed. The detailed steps for rehabilitation start well before surgery and prosthetic fitting, and include relatively novel interventions as motor imagery and biofeedback. Future studies need to further investigate the clinical outcomes and thereby improve therapists' practice. CONCLUSION: Graded rehabilitation offered by a multiprofessional team is needed to enable individuals with upper limb amputations and TMR to fully benefit from prosthetic reconstruction. LEVEL OF EVIDENCE: Low