Measurement incompatibility and the negativity of quasiprobability distribution functions are well-known non-classical aspects of quantum systems. Both of them are widely accepted resources in quantum information processing. We acquaint an approach to establish a connection between the negativity of the Wigner function, a well-known phase-space quasiprobability distribution, of finite-dimensional Hermitian operators and incompatibility among them. We calculate the negativity of the Wigner distribution function for noisy eigenprojectors of qubit Pauli operators as a function of the noise and observe that the amount of negativity increases with the decrease in noise vis-à-vis the increase in the incompatibility. It becomes maximum for the set of maximally unbiased operators. Our results, although qualitatively, provide a direct comparison between relative degrees of incompatibility among a set of operators for different amounts of noise. We generalize our treatment for higher dimensional qudits for specific finite-dimensional Gell-Mann operators to observe that with an increase in the dimension of the operators, the negativity of their Wigner distribution, and hence incompatibility, decreases.
Measurement incompatibility and the negativity of quasiprobability distribution functions are well-known non-classical aspects of quantum systems. Both of them are widely accepted resources in quantum information processing. We acquaint an approach to establish a connection between the negativity of the Wigner function, a well-known phase-space quasiprobability distribution, of finite-dimensional Hermitian operators and incompatibility among them. We calculate the negativity of the Wigner distribution function for noisy eigenprojectors of qubit Pauli operators as a function of the noise and observe that the amount of negativity increases with the decrease in noise vis-à-vis the increase in the incompatibility. It becomes maximum for the set of maximally unbiased operators. Our results, although qualitatively, provide a direct comparison between relative degrees of incompatibility among a set of operators for different amounts of noise. We generalize our treatment for higher dimensional qudits for specific finite-dimensional Gell-Mann operators to observe that with an increase in the dimension of the operators, the negativity of their Wigner distribution, and hence incompatibility, decreases.
Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results, therefore, establish a connection between two notions of nonclassicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions for bosonic systems and Gaussian channels. In particular, these conditions provide useful tools for investigating the effects of errors and imperfections on the incompatibility of measurements in practice. To illustrate our method, we consider all classes of single-mode Gaussian channels. We show that pure lossy channels with 50% or more losses break the incompatibility of all measurements that can be represented by non-negative Wigner functions, which includes the set of Gaussian measurements.
In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
A fundamental feature of quantum mechanics is the incompatibility between quantum measurements, which leads to the Heisenberg uncertainty relations and is intrinsically related to detection of many other quantum features such as nonlocality and steering. Given the significance and ubiquity of incompatibility, it is desirable to quantify the degree of incompatibility between two quantum measurements which are mathematically described by positive operator valued measures. The commutator (Lie product) between operators has widely been used in quantifying incompatibility (noncommutativity) and plays a crucial role in quantum mechanics. In contrast, the anticommutator (Jordan product) between operators has been relatively less employed in the study of quantum mechanics. In this work, we explore the role of the Jordan product in characterizing measurement incompatibility. The key observation lies in that while the product of two non-negative classical observables is automatically non-negative, this is not the case for quantum observables. Indeed the Jordan product of two non-negative observables is not necessarily non-negative. This negativity of the Jordan product is a purely quantum phenomenon and is here exploited to quantify a kind of measurement incompatibility, which interpolates between noncommutativity and joint measurability. We reveal basic features of this approach, make some comparisons with existing quantifiers of incompatibility and suggest the issue of investigating its applications and implications in quantum information tasks.
Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as proof of nonclassicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feature comes from the discrete nature of Stokes variables; however, it was not observed in previous experiments, because they were performed with photon-number averaging detectors. Here we reconstruct the polarization quasiprobability distribution of a coherent state with photon-number resolving detectors, which allows us to directly observe for the first time its negativity. Furthermore we derive a theoretical polarization quasiprobability distribution for any linearly polarized quantum state.