我们分析并进一步发展了一种新的方法来表示一个系统的量子态的$n$量子位在相位空间网格的$N\times N$点（其中$N=2^n$）。最近由Wootters及其同事提出的方法（Gibbons { \it et al.}，quant-ph/0401155）基于使用有限域$GF(2^n)$的元素来标记相空间轴。我们提出了一个自包含的概述的方法，我们给出了新的见解，它的一些功能，我们应用它来调查量子信息理论感兴趣的问题：我们分析了稳定状态和量子纠错码的相空间表示，并提出了一个相空间解决方案，所谓的“平均王问题”。
英文摘要：
We analyze and further develop a new method to represent the quantum state of a system of $n$ qubits in a phase space grid of $N\times N$ points (where $N=2^n$). The method, which was recently proposed by Wootters and co--workers (Gibbons {\it et al.}, quant-ph/0401155), is based on the use of the elements of the finite field $GF(2^n)$ to label the phase space axes. We present a self--contained overview of the method, we give new insights on some of its features and we apply it to investigate problems which are of interest for quantum information theory: We analyze the phase space representation of stabilizer states and quantum error correction codes and present a phase space solution to the so--called ``mean king problem''.
We experimentally demonstrate the steady-state generation of propagating Wigner-negative states from a continuously driven superconducting qubit. We reconstruct the Wigner function of the radiation emitted into propagating modes defined by their temporal envelopes, using digital filtering. For an optimized temporal filter, we observe a large Wigner logarithmic negativity, in excess of 0.08, in agreement with theory. The fidelity between the theoretical predictions and the states generated experimentally is up to 99%, reaching state-of-the-art realizations in the microwave frequency domain. Our…
We present the experimental reconstruction of the Wigner function of an individual electronic spin qubit associated with a nitrogen-vacancy (NV) center in diamond at room temperature. This spherical Wigner function contains the same information as the density matrix for arbitrary spin systems. As an example, we exactly characterize the quantum state of a single qubit undergoing a nearly pure dephasing process by Wigner function. The fidelities and purities during this process are extracted from the experimental reconstructed Wigner functions, whose dynamics agree with the theoretical predictio…
In this manuscript, the behavior of the Wigner function of accelerated and non-accelerated two qubit system passing through different noisy channels is discussed. The decoherence of the initial quantum correlation due to the noisy channels and the acceleration process is investigated by means of Wigner function. The negative (positive) behavior of the Wigner function predicts the gain of the quantum (classical) correlations. Based on the upper and lower bounds of the Wigner function, the entangled initial state loses its quantum correlation due the acceleration process and the strengths of the…
Anomalous weak values and Wigner function's negativity are well known witnesses of quantum contextuality. We show that these effects occur when analyzing the energetics of a single qubit gate generated by a resonant coherent field traveling in a waveguide. The buildup of correlations between the qubit and the field is responsible for bounds on the gate fidelity, but also for a nontrivial energy balance recently observed in a superconducting setup. In the experimental scheme, the field is continuously monitored through heterodyne detection and then post-selected over the outcomes of a final qub…
We study the tomography of propagators for spin systems in the context of finite-dimensional Wigner representations, which completely characterize and visualize operators using shapes assembled from linear combinations of spherical harmonics. The Wigner representation of a propagator can be experimentally recovered by measuring expectation values of rotated axial spherical tensor operators in an augmented system with an additional ancilla qubit. The methodology is experimentally demonstrated for standard one-qubit quantum gates using nuclear magnetic resonance spectroscopy. In particular, this…
Well-known (spinless) fermionic qubits may need more subtle consideration in comparison with usual (spinful) fermions. Taking into account a model with local fermionic modes, formally only the 'occupied' states |1> could be relevant for antisymmetry with respect to particles interchange, but 'vacuum' state |0> is not. Introduction of exchange rule for such fermionic qubits indexed by some 'positions' may look questionable due to general super-selection principle. However, a consistent algebraic construction of such 'super-indexed' qubits is presented in this work. Considered method has some re…
Effects of DM and KSEA interactions on entanglement, Fisher and Wigner-Yanase information correlations of two XYZ-Heisenberg-qubit states under a magnetic field
In a recent paper, Tilma, Everitt {\it et al.} derived a generalized Wigner function that can characterize both the discrete and continuous variable states, i.e., hybrid states. As such, one can expect that the negativity of the generalized Wigner function applied to the hybrid states can reveal their nonclassicality, in analogy with the well-known Wigner function defined for the continuous variable states. In this work, we demonstrate that, indeed, the negativity volume of the generalized Wigner function of the hybrid bipartite states can be used as an entanglement witness for such states, pr…
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use t…
Quantum Wasserstein isometries of the $n$-qubit state space: a Wigner-type result
HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）；ELTE Eötvös Loránd University（罗兰大学）
It is demonstrated that exact diagonalization of the microscopic many-body Hamiltonian via systematic full configuration-interaction (FCI) calculations is able to predict the spectra as a function of detuning of three-electron hybrid qubits based on GaAs asymmetric double quantum dots. It is further shown that, as a result of strong inter-electron correlations, these spectroscopic patterns, including avoided crossings between states associated with different electron occupancies of the left and right wells, are inextricably related to the formation of Wigner molecules. These physical entities …
Quantum engineering now allows to design and construct multi-qubit states in a range of physical systems. These states are typically quite complex in nature, with disparate, but relevant properties that include both single and multi-qubit coherences and even entanglement. All these properties can be assessed by reconstructing the density matrix of those states - but the large parameter space can mean physical insight of the nature of those states and their coherence can be hard to achieve. Here we explore how the Wigner function of a multipartite system and its visualization provides rich info…
By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions, in straight connection with experiments involving, among other things, the tomogra…
Quantum simulation of chemical Hamiltonians enables the efficient calculation of chemical properties. Mapping is one of the essential steps in simulating fermionic systems on quantum computers. In this work, a unified framework of transformations mapping fermionic systems to qubit systems is presented, and many existing transformations, such as Jordan-Wigner, Bravyi-Kitaev, and parity transformations, are included in this framework. Based on this framework, the Multilayer Segmented Parity (MSP) transformation is proposed. The MSP transformation is a general mapping with an adjustable parameter…
The Wigner function's behavior of accelerated and non-accelerated Greenberger Horne Zeilinger (GHZ) state is discussed. For the non-accelerated GHZ state, the minimum/maximum peaks of the Wigner function depends on the distribution's angles, where they are displayed regularly at fixed values of the distribution's angles. We show that, for the accelerated GHZ state, the minimum bounds increases as the acceleration increases. The increasing rate depends on the number of accelerated qubits. Due to the positivity/ negativity behavior of the Wigner function, one can use it as an indicators of the p…
Quantum phase space measurement and entanglement validation made easy
最近已经表明，可以将任何系统的完整量子态表示为相空间准概率分布（维格纳函数）[Phys Rev Lett 117，180401]。这样的功能采取的形式的期望值的观察，有一个直接的类比置换宇称算子。在这项工作中，我们给出了一个程序的维格纳函数的测量，应该适用于任何量子系统。我们已经将我们的程序应用于IBM的量子体验五量子比特量子处理器，以证明我们可以测量和生成两个不同的Bell态以及五量子比特Greenberger-Horne-Zeilinger（GHZ）态的Wigner函数。由于自旋系统的维格纳函数不是唯一的，我们定义，比较和对比两个不同的例子。我们展示了如何使用这些维格纳函数导致量子态分析的最佳方法，特别是在特定的特性功能是特别感兴趣的情况下（如自旋薛定谔猫态）。此外，我们表明，这种分析导致直接的，可能是非常有效的，纠缠测试和状态表征方法。
Quantum coherence is a crucial resource for quantum information processing. By employing the language of coherence orders largely applied in NMR systems, quantum coherence has been currently addressed in terms of multiple quantum coherences (MQCs). Here we investigate the $α$-MQCs, a novel class of multiple quantum coherences which is based on $α$-relative purity, an information-theoretic quantifier analogous to quantum fidelity and closely related to Rényi relative entropy of order $α$. Our framework enables linking $α$-MQCs to Wigner-Yanase-Dyson skew information (WYDSI), an asymmetry monoto…
