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Physical Review A

Trade-off between information gain and fidelity under weak measurements

Longfei Fan, Wenchao Ge, Hyunchul Nha, Muhammad Suhail Zubairy

Published Aug 17, 2015• 11 citations
Publisher

Abstract

It is of general interest how a quantum measurement may disturb a quantum system while it gives information on the state of the system. We study a trade-off relation between the information gain and the output fidelity for a quantum nondemolition measurement scheme for photon numbers. Toward this aim, we obtain general expressions for the information gain and the output fidelity for an arbitrary initial state. We particularly investigate how these two quantities vary with measurement strength for some specific classes of states, through a single measurement or successive measurements.

DOI
10.1103/physreva.92.022114

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It is of general interest how a quantum measurement may disturb a quantum system while it gives information on the state of the system. We study a trade-off relation between the information gain and the output fidelity for a quantum nondemolition measurement scheme for photon numbers. Toward this aim, we obtain general expressions for the information gain and the output fidelity for an arbitrary initial state. We particularly investigate how these two quantities vary with measurement strength for some specific classes of states, through a single measurement or successive measurements.
It is of general interest how a quantum measurement may disturb a quantum system while it gives information on the state of the system. We study a trade-off relation between the information gain and the output fidelity for a quantum nondemolition measurement scheme for photon numbers. Toward this aim, we obtain general expressions for the information gain and the output fidelity for an arbitrary initial state. We particularly investigate how these two quantities vary with measurement strength for some specific classes of states, through a single measurement or successive measurements.
We present the amount of information, fidelity, and reversibility obtained by arbitrary quantum measurements on completely unknown states. These quantities are expressed as functions of the singular values of a measurement operator corresponding to the obtained outcome. As an example, we consider a class of quantum measurements with highly degenerate singular values to discuss trade-offs among information, fidelity, and reversibility. The trade-offs are at the level of a single outcome, in the sense that the quantities pertain to each single outcome rather than the average over all possible outcomes.
The trade-off between the information gain and the state disturbance is derived for quantum operations on a single qubit prepared in a uniformly distributed pure state. The derivation is valid for a class of measures quantifying the state disturbance and the information gain which satisfy certain invariance conditions. This class includes in particular the Shannon entropy versus the operation fidelity. The central role in the derivation is played by efficient quantum operations, which leave the system in a pure output state for any measurement outcome. It is pointed out that the optimality of efficient quantum operations among those inducing a given operator-valued measure is related to Davies' characterization of convex invariant functions on hermitian operators.
We study trade-off relations in information extraction from quantum systems subject to null-result weak measurements, where the absence of a detected photon continuously updates the system state. We present a detailed analysis of qubit and qutrit systems and investigate a general framework for a multilevel quantum system. We develop a dynamical characterization of null-result weak measurements that quantifies the information extracted over time, revealing the amount of the obtained information and also the rate of the information accumulation. The characterizations are obtained by examining the time-dependent evolution of the information theoretic quantities. More specifically, we consider Shannon entropy, mutual information, fidelity, and relative entropy to characterize the weak measurement dynamics. Our results provide an information theoretic analysis of the weak measurement process and highlight the dynamical nature of information extraction and reversibility in the weak measurement processes.
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also investigated, and the optimality is analyzed by explicit evaluation of fidelities. We found that the estimation fidelity does not depend on the number of users, neither for single-measure inference nor for collective one, whereas the transmission fidelity decreases with N . The resulting trade-off is no longer optimal and degrades with increasing N . We found that optimality can be restored by an effective preparation of the probe states and present explicitly calculations for the 2-user case.
We address the trade-off between information gain and state disturbance in measurement performed on qudit systems and devise a class of optimal measurement schemes that saturate the ultimate bound imposed by quantum mechanics to estimation and transmission fidelities. The schemes are minimal, i.e. they involve a single additional probe qudit, and optimal, i.e. they provide the maximum amount of information compatible with a given level of disturbance. The performances of optimal single-user schemes in extracting information by sequential measurements in a N -user transmission line are also investigated, and the optimality is analyzed by explicit evaluation of fidelities. We found that the estimation fidelity does not depend on the number of users, neither for single-measure inference nor for collective one, whereas the transmission fidelity decreases with N . The resulting trade-off is no longer optimal and degrades with increasing N . We found that optimality can be restored by an effective preparation of the probe states and present explicitly calculations for the 2-user case.