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arXiv:1912.10592 · Quantum

Complete Information Balance in Quantum Measurement

Seung-Woo Lee, Jaewan Kim, Hyunchul Nha

Published Mar 17, 2021• 17 citations

Abstract

Quantum measurement is a basic tool to manifest intrinsic quantum effects from fundamental tests to quantum information applications. While a measurement is typically performed to gain information on a quantum state, its role in quantum technology is indeed manifold. For instance, quantum measurement is a crucial process element in measurement-based quantum computation. It is also used to detect and correct errors thereby protecting quantum information in error-correcting frameworks. It is therefore important to fully characterize the roles of quantum measurement encompassing information gain, state disturbance and reversibility, together with their fundamental relations. Numerous efforts have been made to obtain the trade-off between information gain and state disturbance, which becomes a practical basis for secure information processing. However, a complete information balance is necessary to include the reversibility of quantum measurement, which constitutes an integral part of practical quantum information processing. We here establish all pairs of trade-off relations involving information gain, disturbance, and reversibility, and crucially the one among all of them together. By doing so, we show that the reversibility plays a vital role in completing the information balance. Remarkably, our result can be interpreted as an information-conservation law of quantum measurement in a nontrivial form. We completely identify the conditions for optimal measurements that satisfy the conservation for each tradeoff relation with their potential applications. Our work can provide a useful guideline for designing a quantum measurement in accordance with the aims of quantum information processors.

DOI: 10.22331/q-2021-03-17-414

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Quantum measurement unavoidably disturbs the state of a quantum system if any information about the system is extracted. Recently, the concept of reversing quantum measurement has been introduced and has attracted much attention. Numerous efforts have thus been devoted to understanding the fundamental relation of the amount of information obtained by measurement to either state disturbance or reversibility. Here, we experimentally prove the trade-off relations in quantum measurement with respect to both state disturbance and reversibility. By demonstrating the quantitative bound of the trade-off relations, we realize an optimal measurement for estimating quantum systems with minimum disturbance and maximum reversibility. Our results offer fundamental insights on quantum measurement and practical guidelines for implementing various quantum information protocols.

Quantum measurement is essentially taken to gain information encoded in quantum states, however, it also inevitably disturbs the measured state such that the postmeasurement state loses its quantum information partially or fully. Thus, it is an important issue to characterise such information gain and state disturbance associated with measurement. In this work, we study the relationships between the information gain and measurement disturbance for two-valued qubit measurements. In particular, we employ a powerful tradeoff relation between the measurement strength and reversibility to obtain, generalise, and even strengthen certain well-known tradeoff relations between information gain and measurement disturbance of qubit measurements. Moreover, it is pointed out that our results are useful to understand the measurement effects at the fundamental level as they are state-independent. Hence, our work is helpful for designing quantum information processing tasks aided by measurement.

Quantum Information Theory (QIT) has emerged as one of the most influential fields in modern science, combining the principles of quantum mechanics with classical information theory. It forms the theoretical foundation of advanced technologies such as quantum computing, quantum communication, and quantum cryptography. Despite remarkable progress in these areas, the measurement problem remains a fundamental conceptual and practical challenge. Measurement in quantum mechanics plays a dual role: it enables information extraction while simultaneously disturbing the quantum system. This paper presents a detailed study of the measurement problem within the framework of quantum information theory. It examines the role of measurement in quantum state evolution, the effects of entanglement and decoherence, and the impact on information processing tasks. The paper also discusses existing theoretical approaches and highlights the importance of resolving measurement-related issues for the future development of quantum technologies.

In the general theory of quantum measurement, one associates a positive-semidefinite operator on a

d

-dimensional Hilbert space to each of the

n

possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when

n>d

, orthogonality is restricted by positivity. This restriction allows us to more precisely state the quantum adage: information gain of a system is always accompanied by unavoidable disturbance. Specifically, we investigate three properties of a measurement with L\"uders rule updating: its disturbance, a measure of how the expected postmeasurement state deviates from the input; its measurement strength, a measure of the intrinsic information producing capacity of the measurement; and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. These quantities satisfy an information-disturbance tradeoff relation that highlights the additional role played by orthogonality. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.