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Information Gain in Repeated Quantum Measurements

Erkka Haapasalo, Teiko Heinosaari, Yui Kuramochi

Published Mar 14, 2016• 0 citations
OpenAlex

Abstract

We study information gain in a sequential measurement scenario where the system is repeatedly subjected to the same measurement process. We first provide examples of such repeated measurements where all the obtainable information is reached after a finite number of repetitions. We also prove, however, that repeating the L\uders measurement of an unsharp non-trivial two-outcome observable yields novel information at each step, and we characterize the information obtained in the limit of infinitely many repetitions. Our result implies that a repeated measurement can be used to correct the inherent noise of an unsharp observable.

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We study information gain in a sequential measurement scenario where the system is repeatedly subjected to the same measurement process. We first provide examples of such repeated measurements where all the obtainable information is reached after a finite number of repetitions. We also prove, however, that repeating the L\uders measurement of an unsharp non-trivial two-outcome observable yields novel information at each step, and we characterize the information obtained in the limit of infinitely many repetitions. Our result implies that a repeated measurement can be used to correct the inherent noise of an unsharp observable.
Abstract We study sequential measurement scenarios where the system is repeatedly subjected to the same measurement process. We first provide examples of such repeated measurements where further repetitions of the measurement do not increase our knowledge on the system after some finite number of measurement steps. We also prove, however, that repeating the Lüders measurement of an unsharp two-outcome observable never saturates in this sense, and we characterize the observable measured in the limit of infinitely many repetitions. Our result implies that a repeated measurement can be used to correct the inherent noise of an unsharp observable.
How much information about the original state preparation can be extracted from a quantum system which has already been measured? That is, how many independent (noncommunicating) observers can measure the quantum system sequentially and give a nontrivial estimation of the original unknown state? We investigate these questions, and show from a simple example that information about the original preparation is not entirely lost as a result of the measurement-induced collapse of the quantum state, and that an infinite number of independent observers who have no prior knowledge about the initial state can gain partial information about the original preparation of the quantum system.
Performing a quantum measurement yields two different results: a classical outcome drawn from a probability distribution, according to Born's rule, and a quantum outcome corresponding to the post-measurement state. Quantum devices that provide both outcomes can be described through quantum instruments. In a realistic scenario, one can expect that the observer's obtained classical and quantum outcomes are non-ideal: this can be due to experimental limitations, but could also be explained by adversarial interference, that is, a second party that disturbs the device through a concealed measurement to obtain information. The second scenario can be interpreted through quantum compatibility, as it implies that both the observer's instrument and the adversary's measurement can be performed simultaneously. In this work, we show how the noise of the observer's device relates to the amount of information that the adversary can obtain. We study scenarios in which the adversary aims to acquire information on the same basis as the observer's measurement, or on a mutually unbiased basis with respect to the observer's basis. In both cases, we derive necessary and sufficient conditions for the compatibility of a single qubit non-ideal quantum instrument and a noisy meter, from which we obtain the maximum amount of information that the adversary can extract in terms of the noise parameters of the observer's instrument. Finally, we provide the device implementation from the adversary's point of view for the same basis scenario.
Unsharp measurements are increasingly important for foundational insights in quantum theory and quantum information applications. Here, we report an experimental implementation of unsharp qubit measurements in a sequential communication protocol, based on a quantum random access code. The protocol involves three parties; the first party prepares a qubit system, the second party performs operations that return both a classical and quantum outcome, and the latter is measured by the third party. We demonstrate a nearly optimal sequential quantum random access code that outperforms both the best possible classical protocol and any quantum protocol that utilizes only projective measurements. Furthermore, while only assuming that the involved devices operate on qubits and that detected events constitute a fair sample, we demonstrate the noise-robust characterization of unsharp measurements based on the sequential quantum random access code. We apply this characterization towards quantifying the degree of incompatibility of two sequential pairs of quantum measurements.
We show that there are informationally complete joint measurements of two conjugated observables on a finite quantum system, meaning that they enable the identification of all quantum states from their measurement outcome statistics. We further demonstrate that it is possible to implement a joint observable as a sequential measurement. If we require minimal noise in the joint measurement, then the joint observable is unique. If ddd is odd, then this observable is informationally complete. But if ddd is even, then the joint observable is not informationally complete, and one has to allow more noise in order to obtain informational completeness.
One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown by two of the authors of the present article [T. Heinosaari and T. Miyadera, Phys. Rev. A 91, 022110 (2015)] that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement. In this work, we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme.