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Physical Review Letters

Demonstration of Complete Information Trade-Off in Quantum Measurement

Seongjin Hong, Yong‐Su Kim, Young‐Wook Cho, Jaewan Kim, Seung-Woo Lee, Hyang‐Tag Lim

Published Feb 3, 2022• 20 citations
Publisher

Abstract

While an information-disturbance trade-off in quantum measurement has been at the core of foundational quantum physics and constitutes a basis of secure quantum information processing, recently verified reversibility of a quantum measurement requires to refine it toward a complete version of information trade-off in quantum measurement. Here we experimentally demonstrate a trade-off relation among all information contents, i.e., information gain, disturbance, and reversibility in quantum measurement. By exploring quantum measurements applied on a photonic qutrit, we observe that the information of a quantum state is split into three distinct parts accounting for the extracted, disturbed, and reversible information. We verify that such different parts of information are in trade-off relations not only pairwise but also triplewise all at once, and find that the triplewise relation is tighter than any of the pairwise relations. Finally, we realize optimal quantum measurements that inherently preserve quantum information without loss of information, which offer wider applications in measurement-based quantum information processing.

DOI
10.1103/physrevlett.128.050401

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While an information-disturbance trade-off in quantum measurement has been at the core of foundational quantum physics and constitutes a basis of secure quantum information processing, recently verified reversibility of a quantum measurement requires to refine it toward a complete version of information trade-off in quantum measurement. Here we experimentally demonstrate a trade-off relation among all information contents, i.e., information gain, disturbance, and reversibility in quantum measurement. By exploring quantum measurements applied on a photonic qutrit, we observe that the information of a quantum state is split into three distinct parts accounting for the extracted, disturbed, and reversible information. We verify that such different parts of information are in trade-off relations not only pairwise but also triplewise all at once, and find that the triplewise relation is tighter than any of the pairwise relations. Finally, we realize optimal quantum measurements that inherently preserve quantum information without loss of information, which offer wider applications in measurement-based quantum information processing.
State disturbance by a quantum measurement is at the core of foundational quantum physics and constitutes a fundamental basis of secure quantum information processing. While quantifying an information-disturbance relation has been a long-standing problem, recently verified reversibility of a quantum measurement requires to refine such a conventional information trade-off toward a complete picture of information conservation in quantum measurement. Here we experimentally demonstrate complete trade-off relations among all information contents, i.e., information gain, disturbance and reversibility in quantum measurement. By exploring various quantum measurements applied on a photonic qutrit, we observe that the information of a quantum state is split into three distinct parts accounting for the extracted, disturbed, and reversible information. We verify that such different parts of information are in trade-off relations not only pairwise but also globally all-at-once, and find that the global trade-off relation is tighter than any of the pairwise relations. Finally, we realize optimal quantum measurements that inherently preserve quantum information without loss of information, which offer wider applications in measurement-based quantum information processing.
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.
The measurement process necessarily disturbs the state of a quantum system, unless the state is an eigenstate of the observable being measured. Once we perform a complete measurement in a given basis, the system undergoes decoherence and loses its coherence. If there is no disturbance, the state retains all of its coherence. It is therefore natural to ask if there is a trade-off between disturbance caused to a state and its coherence. We present coherence disturbance trade-off relations using the relative entropy of coherence for measurement channel as well as for general channels (completely positive trace-preserving maps). For bipartite states we prove a trade-off relation between the quantum coherence, entanglement, and disturbance. Similar relation also holds for quantum coherence, quantum discord, and disturbance for any bipartite state. We illustrate the trade-off between the coherence and the disturbance for single-qubit and -qutrit states subject to various quantum channels.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when n>dn>dn>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. 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.
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.
In the general theory of quantum measurement, one associates a positive-semidefinite operator on a ddd-dimensional Hilbert space to each of the nnn possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert-Schmidt orthogonal, but when n>dn>dn>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.