Abstract
Generic non-Markovian quantum processes have infinitely long memory, implying an exact description that grows exponentially in complexity with observation time. Here, we present a finite memory ansatz that approximates (or recovers) the true process with errors bounded by the strength of the non-Markovian memory. The introduced memory strength is an operational quantity and depends on the way the process is probed. Remarkably, the recovery error is bounded by the smallest memory strength over all possible probing methods. This allows for an unambiguous and efficient description of non-Markovian phenomena, enabling compression and recovery techniques pivotal to near-term technologies. We highlight the implications of our results by analyzing an exactly solvable model to show that memory truncation is possible even in a highly non-Markovian regime.
Highlights
Our ability to manipulate quantum systems underpins potential advantages over classical technologies[1]
We have introduced the concept of memory strength for quantum stochastic processes, which is shown to bound process recoverability
Its applicability is exemplified by the case study, where we are able to accurately and efficiently reconstruct dynamics with a memory cutoff, even in a highly non-Markovian regime
Summary
Our ability to manipulate quantum systems underpins potential advantages over classical technologies[1]. If the memory strength is small for some instrument (family), the approximate process accurately predicts expectation values for related observables, namely those in the linear span of the original instrument elements This connection is akin to that between the conditional mutual information (CMI) and the fidelity of recovery for quantum states[30,31], and involves a generalization of the measured relative entropy[32] to quantum stochastic processes. Grouping the times into three segments: the history H = {t1, ..., tk−l−1}, memory M = {tk−l, ..., tk−1} and future F = {tk, ..., tn} (see Fig. 2), Markov order l implies the conditional factorization Both the process tensor and the instrument elements are higher-order quantum maps[8,34,36,42] that can respectively be represented as quantum states Υn:[1] and fOnðx:1n:1Þg via the Choi–Jamiołkowski isomorphism. When J M is not informationally complete, i.e., does not span the full space, ΛJFMMH only approximates the original process
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