Visualization of correlations in hybrid discrete—continuous variable quantum systems

Abstract

In this work we construct Wigner functions for hybrid continuous and discrete variable quantum systems. We demonstrate new capabilities in the visualization of the interactions and correlations between discrete and continuous variable quantum systems, where visualizing the full phase space has proven difficult in the past due to the high number of degrees of freedom. Specifically, we show how to clearly distinguish signatures that arise due to quantum and classical correlations in an entangled Bellcat state. We further show how correlations are manifested in different types of interaction, leading to a deeper understanding of how quantum information is shared between two subsystems. Understanding the nature of the correlations between systems is central to harnessing quantum effects for information processing; the methods presented here reveal the nature of these correlations, allowing a clear visualization of the quantum information present in these hybrid discrete-continuous variable quantum systems. The methods presented here could be viewed as a form of quantum state spectroscopy.Bloch sphere [20][21][22][23][24][25][26]. For example, there have been various proposals put forward that use a continuous Wigner function to reveal correlations between DV systems [26][27][28]. These methods have further been validated through the direct measurement of phase-space to reveal quantum correlations [28][29][30][31]. Recently this has been extended to experiments validating atomic Schrödinger cat states of up to 20 superconducting qubits [32].A case that has not been explored in much detail is the phase-space representation of CV-DV hybridization. This hybridisation is seen in many applications of quantum technologies, including simple gate models for quantum computers, such as hybrid two-qubit gates [33,34], and CV microwave pulse control of DV qubits [35]. The generation of hybrid quantum correlations within CV-DV hybrid 5 systems commonly takes place within the framework of cavity quantum electrodynamics, that describes the interaction between a two-level quantum system and a single mode of a microwave field. These models can be further used to describe the effect of circuit quantum electrodynamics, and to consider the interaction of the microwave field with an artificial atom. Analyzing these interactions within the framework of the Jaynes-Cummings model [36] allows us to display how quantum information is shared between the CV and DV systems.A number of papers [23, 24, 37] have shown the mathematical construction of hybrid states within the phase space, these have been constructed without giving a way to visually display the degrees of freedom of such composite systems. A method for displaying states with heterogeneous degrees of freedom, using the Wigner function, came from the application of composite phase-space methods to quantum chemistry [38]. The technique presented here is based on this approach, however in [38], reduced Wigner functions are used and an envelope is further applied, potentially losing many o...

Universal non-Markovianity detection in hybrid open quantum systems

“…Moreover, the experimental advances 57 allow us to expect that the experimental measurements of the GWF will become a standard tool in characterizing complex composite quantum systems and their dynamics in the near future. Additionally, the knowledge of the GWF offers the opportunity to perform not only quantitative but also qualitative analysis of the time evolution of any quantum hybrid systems 32 , 33 .…”

“… 30 , 31 the unifying approach based on the transformation kernels for each subsystem has been presented, which permits to define a generalized Wigner function (GWF) for an arbitrary complex quantum system, consisting of any DV or CV subsystems, so-called “hybrid” systems. The proposed framework not only unifies different variable domains, but also enables to visualize the quantum correlations in such systems 32 , 33 .…”

Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

“…Finally in (v) we show a hybrid representation developed in refs. [148, 149] which have colorbar values .…”

“…However here we want to consider hybridizing the Stratonovich–Kernel Wigner function with the the HW Wigner function. [ 148,149 ] At each point we can plot the Wigner function for the qubit, producing a lattice of Wigner functions where We can then add transparency to each of the spheres according to the maximum value of the Wigner function at that point , , resulting in an envelope over the distribution. Examples of this are given in Figure 5a–c(v), where for the separable states we can see the envelope of the vacuum state.…”

An Informationally Complete Wigner Function For The Tavis-Cummings Model: Visualization of Cat Swapping and Quantum Correlations in Field-Many Atom Systems

“…We can calculate the Wigner function [7,8] of a system of arbitrary components by taking the expectation value of a suitable displaced parity operator over all its possible configurations -the phase space [9][10][11][12]. The total displaced parity operator for the composite system simply comprises the tensor product of the displaced parity operators for each element of the system.…”