One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigner function description. Conversely, it was known that the Wigner representation provides a non-negative non-contextual description of Gaussian quantum optics. It follows that contextuality and Wigner negativity are equivalent notions of non-classicality and equivalent resources for this quantum subtheory. In particular, both contextuality and Wigner negativity are necessary for a computational speed-up of quantum Gaussian optics. At the technical level, our result holds true for any subfamily of Gaussian measurements that include homodyne measurements, i.e., measurements of standard quadrature observables.
One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigner function description. Conversely, it was known that the Wigner representation provides a non-negative non-contextual description of Gaussian quantum optics. It follows that contextuality and Wigner negativity are equivalent notions of non-classicality and equivalent resources for this quantum subtheory. In particular, both contextuality and Wigner negativity are necessary for a computational speed-up of quantum Gaussian optics. At the technical level, our result holds true for any subfamily of Gaussian measurements that include homodyne measurements, i.e., measurements of standard quadrature observables.
Understanding what distinguishes quantum mechanics from classical mechanics is crucial for quantum information processing applications. In this work, we consider two notions of non-classicality for quantum systems, negativity of the Wigner function and contextuality for Pauli measurements. We prove that these two notions are equivalent for multi-qudit systems with odd local dimension. For a single qudit, the equivalence breaks down. We show that there exist single qudit states that admit a non-contextual hidden variable model description and whose Wigner functions are negative.
The use of quantum information in technology promises to supersede the so-called classical devices used nowadays. Understanding what features are inherently non-classical is crucial for reaching better-than-classical performance. This thesis focuses on two nonclassical behaviours: quantum contextuality and Wigner negativity. The former is a notion superseding nonlocality that can be exhibited by quantum systems. To date, it has mostly been studied in discrete-variable scenarios. In those scenarios, contextuality has been shown to be necessary and sufficient for advantages in some cases. On the other hand, negativity of the Wigner function is another unsettling non-classical feature of quantum states that originates from phase-space formulation in continuous-variable quantum optics. Continuous-variable scenarios offer promising candidates for implementing quantum computations. Wigner negativity is known to be a necessary resource for quantum speedup with continuous variables. However contextuality has been little understood and studied in continuous-variable scenarios. We first set out a robust framework for properly treating contextuality in continuous variables. We also quantify contextuality in such scenarios by using tools from infinite-dimensional optimisation theory. Building upon this, we show that Wigner negativity is equivalent to contextuality in continuous variables with respect to Pauli measurements thus establishing a continuous-variable analogue of a celebrated result by Howard et al. We then introduce experimentally-friendly witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, using again tools from infinite-dimensional optimisation theory. We further extend the range of previously known discrete-variable results linking contextuality and advantage into a new territory of information retrieval.
Quantum computers promise considerable speedups with respect to their classical counterparts. However, the identification of the innately quantum features that enable these speedups is challenging. In the continuous-variable setting-a promising paradigm for the realization of universal, scalable, and fault-tolerant quantum computing-contextuality and Wigner negativity have been perceived as two such distinct resources. Here we show that they are in fact equivalent for the standard models of continuous-variable quantum computing. While our results provide a unifying picture of continuous-variable resources for quantum speedup, they also pave the way toward practical demonstrations of continuous-variable contextuality and shed light on the significance of negative probabilities in phase-space descriptions of quantum mechanics.
Gaussian states have played an important role in the physics of continuous-variable quantum systems.They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description.Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements.In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states.We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems.We then use this framework in various ways to explore the structure of the state space.We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank.For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations.To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state.Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states.On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations.Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage.At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.
Since being defined by Kochen and Specker, and separately by Bell, contextuality has been proposed as one of the key phenomena that distinguishing quantum theory from classical theories. However, with the rise of quantum information contextuality's position as the leading definition of the quantum/classical boundary has been called into question. This is due to the fact that a contextual explanation is required by subtheories that offer no exponential quantum computational advantage over classical computation. In this thesis, submitted in requirement for a PhD in physics with quantum information, we shall explore this unwanted contextuality, and show that generalized contextuality is more prevalent than was previously thought. \n \nFirst we will show that the single-qubit stabilizer subtheory, which was previously thought of as a non-contextual subtheory, requires a generalized contextual ontological model, when transformations are included in the operational description. In addition to this we show that the even smaller single-rebit stabilizer subtheory, a strict subtheory of the single-qubit stabilizer subtheory, must also admit a generalized contextual ontological model, again when transformations are included in the operational description. We then show that both these subtheories require negatively represented quasi-probability representations, re-establishing the link between contextuality and negativity for transformations. We also investigate the representation of transformations in generic quasi-probability representations, showing that under a reasonable assumption almost-all unitaries must be negatively represented by a finite quasi-probability representation. \n \nSecond we will investigate the efficiently simulable n-qubit stabilizer subtheory, which exhibits all forms of contextuality and thus represents the main obstacle to identifying contextuality as a resource for quantum computation. To this end we present an attempt at constructing a model based on a frame-switching Wigner function. This leads us to constructing a contextual ψ-epistemic ontological model of the n-qubit stabilizer formalism. We shall see that this model is outcome deterministic, which is one of the core assumptions in the definition of traditional non-contextuality. This model then will lead us to a result that proves that any ontological model of the n-qubit stabilizer formalism requires at least n-1 generators to be encoded in the ontology of the model. As n-1 generators represents almost full knowledge about the stabilizer state, we therefore posit that ψ-onicity, a more controversial notion of non-classicality, is actually the resource for quantum computation.
Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.