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arXiv:1803.07580 · Physical Review A

Resource theory of non-Gaussian operations

Quntao Zhuang, Peter W. Shor, Jeffrey H. Shapiro

Published May 14, 2018• 88 citations
arXivHTMLPDF

Abstract

Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is nonincreasing under the set of free superoperations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (i) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition, and all Gaussian-dilatable non-Gaussian channels; and (ii) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian dilatable. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.

DOI
10.1103/physreva.97.052317

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Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is nonincreasing under the set of free superoperations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (i) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition, and all Gaussian-dilatable non-Gaussian channels; and (ii) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian dilatable. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.
We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set---i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone---the Wigner logarithmic negativity. We use the latter to assess the resource content of experimentally relevant states---e.g., photon-added, photon-subtracted, cubic-phase, and cat states---and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to subuniversal and universal quantum information processing over continuous variables.
We present a framework for studying linear bosonic non-Gaussian channels. Our emphasis is on a class of channels that we call photon-added Gaussian channels and these are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian regime for numerous tasks in continuous-variable quantum information processing including universal quantum computation. These channels are obtained by extending Gaussian channels with photon addition applied to the ancilla ports giving rise to a one-parameter family of non-Gaussian channels indexed by photon number n≥1n \geq 1n≥1. We derive its operator-sum representation which is indispensable in the present context. We observe that these channels are Fock-preserving (or coherence non-generating in the Fock basis). Further, noisy Gaussian channels can be expressed as a convex mixture of these non-Gaussian channels. We then report examples of activation of nonclassicality, using this method of photon-addition, at outputs of channels that would otherwise output only classical states. We also derive many structure theorems and information theoretic properties of these channels.
In the context of quantum technologies over continuous variables, Gaussian states and operations are typically regarded as freely available, as they are relatively easily accessible experimentally. In contrast, the generation of non-Gaussian states, as well as the implementation of non-Gaussian operations, pose significant challenges. This divide has motivated the introduction of resource theories of non-Gaussianity. As for any resource theory, it is of practical relevance to identify free conversion protocols between resources, namely, Gaussian conversion protocols between non-Gaussian states. Via systematic numerical investigations, we address the approximate conversion between experimentally relevant single-mode non-Gaussian states via arbitrary deterministic one-to-one mode Gaussian maps. First we show that cat and binomial states are approximately equivalent for finite energy, while this equivalence was previously known only in the infinite-energy limit. Then we consider the generation of cat states from photon-added and photon-subtracted squeezed states, improving over known schemes by introducing additional squeezing operations. The numerical tools that we develop also allow one to devise conversions of trisqueezed into cubic-phase states beyond previously reported performances. Finally, we identify various other conversions which instead are not viable.
Abstract Which quantum phenomena are advantageous for information processing tasks? By classifying quantum states and operations as resourceful versus non-resourceful, or free, the mathematical formalism of quantum resource theories helps to address such questions. For the task of discriminating channels applied to a probe state, it has been shown that under certain conditions—namely, the set of free states being convex in any dimension or possibly non-convex but finite dimensional—every resourceful probe state can provide an advantage, quantified by a resource monotone known as generalized robustness. In this work, bypassing the limitations of previous studies, we define the generalized robustness for an arbitrary resource theory and show that it admits two operational interpretations. Firstly, it provides an upper bound on the maximal advantage in channel discrimination tasks implemented on multiple copies of the probe states. Secondly, in many physically relevant theories, it quantifies the advantage in single-copy channel discrimination tasks in a worst case scenario. We further present a general construction of multi-copy resource witnesses and provide practical methods to bound the generalized robustness in experiments. Finally, we apply our results to a key case study in continuous variable quantum information: the resource theory of non-Gaussianity. This theory is naturally defined by a set of free states (Gaussian states) that is non-convex and infinite-dimensional. Our work then shows conclusively that all non-Gaussian states can provide an advantage in some channel discrimination task, even those that are simply mixtures of Gaussian states and are typically disregarded for other quantum information tasks. To illustrate our findings, we provide exact formulas for the robustness of non-Gaussianity of Fock states, along with an analysis of the robustness for a family of non-Gaussian states within the convex hull of Gaussian states.
Continuous-variable systems realized in quantum optics play a major role in quantum information processing, and it is also one of the promising candidates for a scalable quantum computer. We introduce a resource theory for continuous-variable systems relevant to universal quantum computation. In our theory, easily implementable operations---Gaussian operations combined with feed-forward---are chosen to be the free operations, making the convex hull of the Gaussian states the natural free states. Since our free operations and free states cannot perform universal quantum computation, genuine non-Gaussian states---states not in the convex hull of Gaussian states---are the necessary resource states for universal quantum computation together with free operations. We introduce a monotone to quantify the genuine non-Gaussianity of resource states, in analogy to the stabilizer theory. A direct application of our resource theory is to bound the conversion rate between genuine non-Gaussian states. Finally, we give a protocol that probabilistically distills genuine non-Gaussianity---increases the genuine non-Gaussianity of resource states---only using free operations and postselection on Gaussian measurements, where our theory gives an upper bound for the distillation rate. In particular, the same protocol allows the distillation of cubic phase states, which enable universal quantum computation when combined with free operations.
Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.