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arXiv:2409.10449 · Physical review. B./Physical review. B

Trade-off relations between quantum coherence and measure of many-body localization

Arti Garg, Arun Kumar Pati

Published Feb 4, 2025• 2 citations
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Abstract

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 ${l}_{1}$ norm of quantum coherence and the relative entropy of coherence satisfy complementarity relations with IPR. For a mixed state, IPR and the ${l}_{2}$ 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.

DOI
10.1103/physrevb.111.054202

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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.
Abstract Quantum coherence is a relevant resource for various quantum information processing tasks, but it is fragile since it is generally affected by environmental noise. This is reflected in the loss of purity of the system, which in turn limits the amount of quantum coherence of it. As a consequence, a complementarity relation between coherence and mixedness arises. Previous works characterize this complementarity through inequalities between the ℓ 1-norm of coherence and linear entropy, and between the relative entropy of coherence and von Neumann entropy. However, coherence–mixedness complementarity is expected to be a general feature of quantum systems, regardless of the measures used. Here, an alternative approach to coherence–mixedness complementarity, based on majorization theory, is proposed. Vectorial quantifiers of coherence and mixedness, namely the coherence vector and the spectrum, respectively, are used, instead of scalar measures. A majorization relation for the tensor product of both vectorial quantifiers is obtained, capturing general aspects of the trade-off between coherence and mixedness. The optimal bound for qubit systems and numerical bounds for qutrit systems are analyzed. Finally, coherence–mixedness complementarity relations are derived for a family of symmetric, concave and additive functions. These results provide a deeper insight into the relation between quantum coherence and mixedness.
Abstract One of the fundamental differences between classical and quantum mechanics is in the ways correlations can be distributed among the many parties that compose a system. While classical correlations can be shared among many subsystems, in general quantum correlations cannot be freely shared. This unique property is known as monogamy of quantum correlations. In this work, we study the monogamy properties of the correlated coherence for the l 1 -norm and relative entropy measures of coherence. For the l 1 -norm the correlated coherence is monogamous for a particular class of quantum states. For the relative entropy of coherence, and using maximally mixed state as the reference incoherent state, we show that the correlated coherence is monogamous for tripartite pure quantum systems.
Quantum coherence quantifies the amount of superposition a quantum state can have in a given basis. Since there is a difference in the structure of eigenstates of the ergodic and many-body localized systems, we expect them also to differ in terms of their coherences in a given basis. Here, we numerically calculate different measures of quantum coherence in the excited eigenstates of an interacting disordered Hamiltonian as a function of the disorder. We show that quantum coherence can be used as an order parameter to detect the well-studied ergodic to many-body localized phase transition. We also perform quantum quench studies to distinguish the behavior of coherence in thermalized and localized phases. We then present a protocol to calculate measurement-based localizable coherence to investigate the thermal and many-body localized phases. The protocol allows one to investigate quantum correlations experimentally in a nondestructive way, in contrast to measures that require tracing out a subsystem, which always destroys coherence and correlation.
Quantum coherence is a key element in topical research on quantum resource theories and a primary facilitator for design and implementation of quantum technologies. However, the resourcefulness of quantum coherence is severely restricted by environmental noise, which is indicated by the loss of information in a quantum system, measured in terms of its purity. In this work, we derive the limits imposed by the mixedness of a quantum system on the amount of quantum coherence that it can possess. We obtain an analytical trade-off between the two quantities that upperbound the maximum quantum coherence for fixed mixedness in a system. This gives rise to a class of quantum states, ``maximally coherent mixed states,'' whose coherence cannot be increased further under any purity-preserving operation. For the above class of states, quantum coherence and mixedness satisfy a complementarity relation, which is crucial to understand the interplay between a resource and noise in open quantum systems.
Identifying phase transition points is a fundamental challenge in condensed-matter physics, particularly for transitions driven by quantum interference effects such as Anderson and many-body localization. Recent studies have demonstrated that quantum coherence provides an effective means of detecting localization transitions, offering a practical alternative to full quantum state tomography and related approaches. Building on this idea, we investigate localization transitions through complementarity relations that connect local predictability, local coherence, and entanglement in bipartite pure states. Our results show that predictability serves as a robust and efficient marker for localization transitions. Crucially, its experimental determination requires exponentially fewer measurements than coherence or entanglement, making it a powerful tool for probing localization phase transitions.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.