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arXiv:1708.03090 · Physical Review A

Trade-off relation for coherence and disturbance

Gautam Sharma, Arun Kumar Pati

Published Jun 6, 2018• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.97.062308

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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.
Abstract When a quantum system is sent through a noisy channel, it is usually disturbed. At the same time, the system undergoes decoherence and tends to lose some delicate quantum features. For a particular basis, the coherence of the state changes. Otherwise, if the system is not disturbed, its state might retain all of coherence. As quantum noisy channels lead to both disturbance and decoherence, it is natural to ask about the relation between disturbance and decoherence. Recently, a trade-off relation for coherence and disturbance has been presented by Sharma and Pati (2018 Phys. Rev. A 97 062308). In this paper, with entangled photon pairs and linear optics, we experimentally verify this trade-off relation for a single-qubit system undergoing various noisy channels. Our experimental results agree with the theoretical predictions and provide a quantitative understanding of the relation between quantum channels and resources.
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.
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.
Entanglement and coherence are two essential quantum resources for quantum information processing. A natural question arises of whether there are direct link between them. And by thinking about this question, we propose a new measure for quantum state that contains concurrence and is called intrinsic concurrence. Interestingly, we discover that the intrinsic concurrence is always complementary to coherence. Note that the intrinsic concurrence is related to the concurrence of a special pure state ensemble. In order to explain the trade-off relation more intuitively, we apply it in some composite systems composed by a single-qubit state coupling four typical noise channels with the aim at illustrating their mutual transformation relationship between their coherence and intrinsic concurrence. This unified trade-off relation will provide more flexibility in exploiting one resource to perform quantum tasks and also provide credible theoretical basis for the interconversion of the two important quantum resources.
Quantum coherence, a fundamental resource in quantum computing and quantum information, often competes with localization effects that affect quantum states in disordered systems. In this work, we prove exact trade-off relations between quantum coherence and a measure of localization and many-body localization, namely, the inverse participation ratio (IPR). We prove that for a pure quantum state, the l1{l}_{1}l1​ norm of quantum coherence and the relative entropy of coherence satisfy complementarity relations with IPR. For a mixed state, IPR and the l2{l}_{2}l2​ norm of quantum coherence as well as relative entropy of coherence satisfy trade-off inequalities. We further explore the applicability of these general relations between quantum coherence and measure of localization in the context of delocalization to many-body localization transition in a disordered interacting quantum many-body system and demonstrate that quantum coherence is also an ideal characterization of the delocalization to many-body localization transition. Our numerical analysis shows that though the full system loses coherence in the many-body localization phase, coherence of a subsystem can increase with the combined effect of disorder and interactions. We believe that these trade-off relations can help in better understanding of how coherence can be preserved or lost in realistic many-body quantum systems, which is vital for developing robust quantum technologies and uncovering new phases of quantum matter.