So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. Under these axioms, we identify the largest and the smallest metrics. Also, we studied asymptotic theory of metric in parallel and adaptive repetition settings, and applied them to the study of channel estimation. First we express the achievable lower bound of the mean square error (MSE) of an estimate by a monotone channel metric, and show this equals O(1/n) for noisy channels, where n is the number of times of channel use. This result shows Heisenberg rate, or O(1/n^2)-rate of the MSE observed in case of estimation of unitary, collapses with very small arbitrary noise.
So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. Under these axioms, we identify the largest and the smallest metrics. Also, we studied asymptotic theory of metric in parallel and adaptive repetition settings, and applied them to the study of channel estimation. First we express the achievable lower bound of the mean square error (MSE) of an estimate by a monotone channel metric, and show this equals O(1/n) for noisy channels, where n is the number of times of channel use. This result shows Heisenberg rate, or O(1/n^2)-rate of the MSE observed in case of estimation of unitary, collapses with very small arbitrary noise.
The aim of the manuscript is to characterize monotone `metric' in the space of Markov map. Here, `metric' means the square of the norm defined on the tangent space, and not necessarily induced from an inner product (this property hereafter will be called inner-product-assumption), different from usual metric used in differential geometry. As for metrics in So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov proved the monotone metric in probability distribution space is unique up to constant multiple, and identical to Fisher information metric. Petz characterized all the monotone metrics in the quantum state space using operator mean. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. (Notably, we do not use the inner-product-assumption.) To obtain this result, `resource conversion' technique, which is widely used in quantum information, is used. We consider distillation from and formation to a family of channels. Under these axioms, we identify the largest and the smallest `metrics'. Interestingly, they are not induced from any inner product, i.e., not a metric. Indeed, one can prove that any `metric' satisfying our axioms can not be a metric. This result has some impact on the axiomatic study of the monotone metric in the space of classical and quantum states, since both conventional theory relies on the inner-product-assumption. Also, we compute the lower and the upper bound for some concrete examples.
The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.
The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.
The aim of the manuscript is to characterize monotone metric in the space of Markov map. Here, metric may not be Riemanian, or equivalently, may not be induced from an inner product. So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov and Petz characterized all the monotone metrics in the classical and quantum state space. As for channels, however, only a little is known about its geometrical structures. In that author's previous manuscript, the upper and the lower bound of monotone channel metric was derived using resource conversion theory, and it is proved that any monotone metric cannot be Riemanian. . Due to the latter result, we cannot rely on Cencov's theory, to build a geometric theory consistent across probability distributions and channels. To dispense with the assumption that a metric is Riemanian, we introduce some assumptions on asymptotic behavior;weak asymptotic additivity and lower asymptotic continuity. The proof utilizes resource conversion technique. In the end of the paper, an implication on quantum state metrics is discussed.
The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field. Joint work with Zixin Huang, Johannes Jakob Meyer, and Theshani Nuradha.
The most general strategy for estimating parameters of a quantum channel is to probe the channel many times sequentially with the help of coherent ancillary systems. However, finding the strategy that minimizes the mean-squared error (MSE) is a notoriously hard problem, requiring optimization over the initial probe state, intermediate evolutions, final measurement, and classical postprocessing. In this work, we study how the minimum attainable MSE can be bounded from below in this setting. We propose a lower bound for the MSE of any physically implementable sequential sensing strategy that applies to estimating parameters of any physical quantum evolution, including quantum channels probed over multiple time steps under temporally correlated noise and general temporally correlated quantum processes. We show that this lower bound can be formulated as a semidefinite program (SDP). Our bound generalizes the Nagaoka-Hayashi lower bound (that applies to parameter estimation tasks with separable measurements) to the regime where the quantum channel can be accessed sequentially multiple times. For regular finite-dimensional single-parameter sequential estimation tasks admitting an optimal strategy, we show that our SDP bound is tight and provide a procedure to obtain a locally optimal strategy from the SDP optimizer. Using our optimization bound, we compute the MSE lower bound and derive new protocols for several other physically motivated problems including error-corrected quantum sensing under an unknown correlated noise model, multiparameter estimation, and estimation with qudit systems.