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arXiv:1706.04853 · Physical Review A

Unification of nonclassicality measures in interferometry

Xiao Yuan, Hongyi Zhou, Mile Gu, Xiongfeng Ma

Published Jan 24, 2018• 14 citations
arXivHTMLPDF

Abstract

From an operational perspective, nonclassicality characterizes the exotic behavior in a physical process which cannot be explained with Newtonian physics. There are several widely used measures of nonclassicality, including coherence, discord, and entanglement, each proven to be essential resources in particular situations. There exists evidence of fundamental connections among the three measures. However, the sources of nonclassicality are still regarded differently and such connections are yet to be elucidated. Here, we introduce a general framework of defining a unified nonclassicality with an operational motivation founded on the capability of interferometry. Nonclassicality appears differently as coherence, discord, and entanglement in different scenarios with local measurement, weak basis-independent measurement, and strong basis-independent measurement, respectively. Our results elaborate how these three measures are related and how they can be transformed from each other. Experimental schemes are proposed to test the results.

DOI
10.1103/physreva.97.012331

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From an operational perspective, nonclassicality characterizes the exotic behavior in a physical process which cannot be explained with Newtonian physics. There are several widely used measures of nonclassicality, including coherence, discord, and entanglement, each proven to be essential resources in particular situations. There exists evidence of fundamental connections among the three measures. However, the sources of nonclassicality are still regarded differently and such connections are yet to be elucidated. Here, we introduce a general framework of defining a unified nonclassicality with an operational motivation founded on the capability of interferometry. Nonclassicality appears differently as coherence, discord, and entanglement in different scenarios with local measurement, weak basis-independent measurement, and strong basis-independent measurement, respectively. Our results elaborate how these three measures are related and how they can be transformed from each other. Experimental schemes are proposed to test the results.
Macroscopic behavior such as the lack of interference patterns has been attributed to ``decoherence,'' a word with several possible definitions such as (1) the loss of off-diagonal density matrix elements, (2) the flow of information to the environment, (3) the loss of complementary information, and (4) the loss of the ability to create entanglement in a measurement. In this article, we attempt to unify these distinct definitions by providing general quantitative connections between them, valid for all finite-dimensional quantum systems or quantum processes. The most important application of our results is to the understanding of quantum discord, a measure of the nonclassicality of the correlations between quantum systems. We show that some popular measures of discord measure the information missing from the purifying system and hence quantify security, which can be stated operationally in terms of distillable secure bits. The results also give some strategies for constructing discord measures.
In the first part of this dissertation a framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented. To demonstrate how these measures differ from entanglement we move to the set of Bell-diagonal states for two qubits, which can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and of the correlation measures from our framework for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that all of the proposed correlation measures are neither convex nor concave. In the second part we look at two practical interferometric setups that use nonclassical states of light to enhance their performance. First we consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer's phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cram\'er-Rao bound for such a setup. The answer is squeezed vacuum. Then we analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer's initial 50:50 beam splitter B is a product state of the two input modes. Requiring a product state is a natural restriction: if one were allowed to input an arbitrary, entangled two-mode state ∣Ξ⟩|\Xi \rangle∣Ξ⟩ to the beam splitter, one could generally just as easily input the state B∣Ξ⟩B|\Xi \rangleB∣Ξ⟩ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number. Our results indicate that entanglement is not a crucial resource for quantum-enhanced interferometry. Initially the analysis for both of these setups is performed for the idealized case of a lossless interferometer. Then the analysis is extended to the more realistic scenario where the interferometer suffers from photon losses.
Quantum resources, such as coherence, discord, and entanglement, play a key role for demonstrating an advantage in many computation and communication tasks. In order to find the nature behind these resources, tremendous efforts have been made to explore the connections between them. In this work, we extend the single party coherence resource framework to the distributed scenario and relate it to basis-dependent discord. We show the operational meaning of basis-dependent discord in quantum key distribution and propose its resource measures. With any basis-dependent discord measure, we show how to extend it to measure coherence, discord, and entanglement by considering the relation of the resource free states. Our results lead to a unification of measures of quantum resources.
In the works leading to this thesis the aim was to give a clear operational interpretation of different aspects of non-classicality. Doing so, enables to see the different properties as different aspects of the same underlying question, namely whether there is a classical explanation of the measurements at hand. If one incorporates the additional assumptions in the setting (such as locality by the separation of the two parties in entanglement theory), the question of whether some system is classical, boils down to ask whether one can simulate everything one can possibly observe by classical variables governed by a stochastic process. This new perspective allows to characterise dynamic properties of non-classicality, and also to connect them to more static types of non-classicality such as entanglement. Indeed we unify entanglement, discord and coherence from one single perspective. We do this by restricting our setting such that the operations have a clear classical interpretation and hence incorporating natural restrictions of classical systems, if the initial state of the system is not entangled does not have discord or does not have coherence. If the initial state of the system is different, we see the magic of quantum mechanics happening again. For instance we show for one specific quantum algorithm, which is thought to be better than any classical algorithm for its task, that the precision of the algorithm scales with the coherence present at its start. In other words, it is these resources (coherence, discord and entanglement) that make the magic possible and with our perspective we can interpret the role of the different resources and analyse how they are linked to each other. This does not give an explanation of the magic of quantum mechanics, but it does make the mechanisms more transparent allowing for new explanations and there are some examples already in this thesis. Finally we also look at the dual of when states are non-classical, asking, and answering, in a specific setting, what makes measurements non-classical.
Quantum theory departs from classical physics in its treatment of correlations, most prominently through the phenomena of contextuality and nonlocality. Once regarded primarily as foundational curiosities, these effects are now understood as key operational resources for quantum computation, communication, and simulation. Although traditionally investigated in distinct settings, recent theoretical and experimental advances have revealed deep conceptual, mathematical, and operational connections between them. This review presents a unified perspective on these developments based on sheaf-theoretic and graph-theoretic frameworks, which provide theory-independent characterizations of statistical correlations. These approaches clarify the structural relationship between contextuality and nonlocality, facilitate the formulation of experimentally testable inequalities, and guide implementations in realistic physical platforms, with particular emphasis on photonic systems. By bridging abstract theoretical structures and concrete experimental realizations, this review sheds light on the nonclassical foundations of quantum correlations and their emerging role in quantum technologies.
We demonstrate in theory and experiment the strict equivalence between nonclassical polarization and the entanglement of indistinguishable photons, thereby unifying these two phenomena that appear dissimilar at first sight. This allows us to analyze nonclassicality and multi-photon entanglement within the same framework. We experimentally verify this double-sided form of quantumness and its independence from the polarization basis, contrasting other notions of coherence that are highly basis-dependent. Our findings show how nonclassical polarization turns out to be equally resourceful for quantum protocols as entanglement, emphasizing its importance in practical applications.