Elsevier

Physics Letters A

Volume 397, 6 May 2021, 127264
Physics Letters A

Localizable quantum coherence

https://doi.org/10.1016/j.physleta.2021.127264Get rights and content

Highlights

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    quantum coherence can be localized in subsystems of a composite system.

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    measurement aided localization is more efficient than tracing out.

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    by localizable coherence we can distinguish topological states or characterize localization transitions.

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    average localizable coherence in the Hilbert space is studied.

Abstract

Coherence is a fundamental notion in quantum mechanics, defined relative to a reference basis. As such, it does not necessarily reveal the locality of interactions nor takes into account the accessible operations in a composite quantum system. In this paper, we put forward a notion of localizable coherence as the coherence that can be stored in a particular subsystem, either by measuring or just by disregarding the rest. We examine its spreading, its average properties in the Hilbert space and show that it can be applied to reveal the real-space structure of states of interest in quantum many-body theory, for example, localized or topological states.

Introduction

One of the most striking properties of quantum mechanics is the fact that the state of a quantum system can be expressed as a coherent superposition of different physical states, that is, the eigenstates corresponding to actual measurable values of some observable. Since these eigenstates constitute a basis of perfectly distinguishable states, the coefficients of this linear expansion also depend on the basis. All the purely quantum features are closely related to the presence of quantum coherence, which experimentally manifests itself in interference and quantum fluctuations [1]. The passage from classical to quantum world is indeed believed to be due to decoherence [2]. Preserving quantum coherence, and thus fighting decoherence, is one of the most fundamental challenges [3], [4], [5] for protocols of quantum information processing [6].

The quantitative theory of coherence has witnessed several advances in recent years [7], [8], [9] together with its application to the fields of quantum metrology [10], [11], quantum foundations [12], [13], quantum biology [14] and quantum thermodynamics [15], [16]. This approach has also motivated various efforts to extend the quantification of coherence from quantum states to quantum operations [17], [18], [19], [20], [21]. In particular, one notion that has surfaced is that of coherence-generating power for a quantum map [22], [23], [24], [25], namely how much coherence can be on average be obtained by a given class of quantum operations.

The notion of coherence per se makes no reference to the locality of a quantum system [8]. In other words, the basis with respect to which coherence is defined does not necessarily require any underlying tensor product structure of the Hilbert space, as is the case, e.g., for entanglement. On the other hand, every realistic quantum operation is local because of the observables one has access to [26]. To that end, a few approaches towards taking into account the subsystem structure have been proposed [27], [28], [29], [30], [31]. One of the basic ideas utilized is to consider incoherent states and operations that, at the same time, respect the underlying local structure of the Hilbert space, obtaining various hybrids between coherence and entanglement.

In this paper, we put forward a notion of localizable coherence, that is, the coherence that can be stored in a particular subsystem of a quantum system with a given tensor product structure. We investigate different protocols, that involve either disregarding or actively measuring a part of the system, so as to localize quantum coherence in the rest of it. We compute average properties of the introduced quantities in the Hilbert space and investigate the role that measurements, with or without post-selection, have in localizing coherence. Once one has introduced a notion of locality, we use this quantity to characterize the coherence of states that have a particular real space structure, e.g., localized or topological states.

Section snippets

Localizing coherence by tracing out

Consider a (finite dimensional) Hilbert space H=HS⊗HA. We see HS as the subsystem in which we want to store coherence, and HA as an environment or an ancillary system. Let dim⁡(H)=d=dSdA. Given a quantum state ρ∈B(H), a natural way of obtaining a quantum state over HS would be to just trace out the ancillary part and obtain ρS=TrA(ρ); then, picking a preferential basis BS on HS, one could simply consider the coherence of the state ρS in that basis.

However, it appears immediately that this

Spreading of localizable coherence

Consider a local quantum system HΛ=⊗x∈ΛHx on a lattice Λ endowed with graph distance d(x,y) and with each local system a d-level system Hx≃Cd. We will assume that the dynamics is described by a local Hamiltonian, that is, a Hamiltonian sum of local operators H=∑XΦX where X⊂Λ and the operators ΦX are bounded hermitian operators on HX=⊗x∈XHx. The map Φ:X↦ΦX is the interaction map that specifies the physical interactions between the particles in the system (including one-body terms). The locality

Average localizable coherence

As we have seen in section 2.1, if we obtain a reduced state to the system S by tracing out the ancillary part A, it is expected that this reduced state will not have much coherence in the large Hilbert space dimension limit; typically states are maximally entangled [32]. In this section, we investigate the average value of localizable coherence in the Hilbert space by means of measurement, using the definitions Eqs. (15), (16). These results will prove useful to understand the local coherence

Applications to quantum many-body systems

In this section, we apply some of the ideas and results introduced so far to the description of notable quantum many-body states from the coherence point of view. We are interested in states that can be representative of the ergodic phase (as described by the Eigenstate Thermalization Hypothesis [44], [45], [46] (ETH)), of the MBL phase, and of the topologically ordered phases. We model the ETH state simply like a Haar-random state in the Hilbert space. These states do indeed obey a volume law

Conclusions and outlook

In this paper, we have addressed the question of quantifying coherence in a composite quantum system where a notion of locality is imposed by a tensor product structure. We have put forward a notion of localizable coherence as the coherence that is obtainable in a subsystem of a composite quantum system after either disregarding or by measurement on the rest of the system, that serves as an ancilla. We have computed the average localizable coherence over the Hilbert space, including over

CRediT authorship contribution statement

All authors have contributed equally to the research and preparation of this manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

P.Z. acknowledges partial support from the NSF award PHY-1819189. Research was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC-2111 – 390814868. This research was (partially) sponsored by the Army Research Office and was accomplished under Grant Number W911NF-20-1-0075. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either

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