Markovian and Non-Markovian Quantum Measurements
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Markovian and Non‑Markovian Quantum Measurements Jennifer R. Glick1,2 · Christoph Adami1 Received: 25 September 2019 / Accepted: 14 July 2020 © The Author(s) 2020
Abstract Consecutive measurements performed on the same quantum system can reveal fundamental insights into quantum theory’s causal structure, and probe different aspects of the quantum measurement problem. According to the Copenhagen interpretation, measurements affect the quantum system in such a way that the quantum superposition collapses after each measurement, erasing any memory of the prior state. We show here that counter to this view, un-amplified measurements (measurements where all variables comprising a pointer are in principle controllable) have coherent ancilla density matrices that encode the memory of the entire set of (un-amplified) quantum measurements that came before, and that the chain of this entire set is therefore non-Markovian. In contrast, sequences of amplified measurements (measurements where at least one pointer variable has been lost) are equivalent to a quantum Markov chain. We argue that the non-Markovian nature of quantum measurement has empirical consequences that are incompatible with the assumption of wave function collapse, thus elevating the collapse assumption into a testable hypothesis. Finally, we find that all of the information necessary to reconstruct an arbitrary non-Markovian quantum chain of measurements is encoded on the boundary of that chain (the first and the final measurement), reminiscent of the holographic principle. Keywords Quantum measurement · Consecutive quantum measurements · Quantum eraser · Quantum Zeno effect · Double-slit experiment · Wave-function collapse
* Christoph Adami [email protected] Jennifer R. Glick [email protected] 1
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
2
Present Address: IBM T.J. Watson Research Center, Yorktown Heights, NY, USA
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Foundations of Physics
1 Introduction The physics of consecutive (sequential) measurements on the same quantum system has enjoyed increased attention as of late, as it probes the causal structure of quantum mechanics [7]. It is of interest to researchers concerned about the apparent lack of time-reversal invariance of Born’s rule [46, 53], as well as to those developing a consistent formulation of covariant quantum mechanics [45, 52], which does not allow for a time variable to define the order of (possibly non-commuting) projections [47]. Consecutive measurements on the same quantum system have also been used to test whether the statistics of the set of measurements is compatible with a macroscopic description of them [36]. Consecutive measurements can be seen to challenge our understanding of quantum theory in an altogether different manner, however. According to standard theory, a measurement causes the state of a quantum system to “collapse”, re-preparing it in an eigenstate of the measured operator so that after multiple consecutive measurements on the quantum sy
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