Visualization of correlations in hybrid discrete—continuous variable quantum systems

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 Bell-cat 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.


Introduction
Quantum correlations have become central to the design and manufacture of various quantum technologies [1][2][3][4]. Whether these quantum correlations are found between macroscopically distinct superpositions of states, also known as Schrödinger cat states, or in the entanglement between multiple systems. Currently, such technologies can be broadly categorized as being based on either continuous-variable (CV) or discrete-variable (DV) quantum systems.
For CV systems, the primary focus has been on quantum optical systems; manipulating coherent states of light for various quantum information processing applications [5][6][7][8]. In such systems, the Wigner function [9,10] is commonly used due to its ability to display an intuitive representation of a quantum state. Furthermore, the Wigner function is particularly good at revealing coherences and correlations, such as squeezing and superposition [11]. For these reasons, it has become a fundamental tool in the 'search' for Schrödingers cats [12], readily identified by the iconic interference patterns arising from its quantum correlations.
By contrast the focus for DV systems has been on exploiting two-level quantum systems-qubits-in order to generate a quantum analogue of the classical bit [2,13,14]. Here, the Wigner function has received little attention as a means of visualization. Unlike the case of CV systems, there are two common approaches for generating informationally complete DV Wigner functions, both of which have found application. The approach developed in [15,16] uses discrete degrees of freedom and has proven useful for quantum information purposes, particularly in the case of contextuality and Wigner function negativity [17][18][19]. The second approach (and the one used in this work) uses a DV Wigner function with continuous degrees of freedom, similar to the

The Wigner function
The Wigner function is traditionally introduced as the Fourier transform of an autocorrelation function [9,55].
Here it is more suitable to consider a general Wigner function of some arbitrary operator Â, defined as [56] where P Ŵ ( )is the displaced parity operator for some parameterization of phase space Ω. The displaced parity operator is defined through displacing a generalized parity operator [24], and for the CV Wigner function is [57]  is the standard CV displacement operator written using the annihilation and creation operators, â and â † , respectively. Note that we have introduced the subscript f, for 'field', to indicate CV systems. The displacement operator can be used to define a coherent state [57] b as the displacement of the vacuum state, ñ 0 f | , generating a new coherent state bñ f | . As shown in [23,24], a similar approach to(2) can be used to generate Wigner functions for arbitrary quantum systems. For two-level DV systems, for example, where the generalized parity, P â , for a single, two-level, system is s P = + 3 2 a z  (ˆˆ) [23,24,28], for a full derivation of the kernel see [58]. Note that the subscript a here indicates that this is a state for the 'atom', or DV system. The analogue of the displacement operator, q f F U , , ( ), given in terms of Euler angles, is for the standard Pauli matrices s ŷ and s ẑ . Note as the parity operator commutes with s ẑ , the Φ term does not contribute, and the DV Wigner function depends only on θ and f, allowing it to be plotted on the surface of a sphere. Note that by DV Wigner function, we mean the Wigner function for DV systems; the Wigner function used here is however parameterized over the continuous variables θ and f. figure 1 shows examples of the DV Wigner function generated by(5) for some simple qubit states. Each of the DV Wigner functions presented infigure 1 is plotted following [59], using the Lambert azimuthal equalarea projection [60]. This projection is area preserving and maps the surface of a sphere to polar coordinates, with the north pole mapped to the centre of the disc and the south pole to the outer boundary. The equator of the sphere is projected onto a concentric circle, with a radius 1 2 times the radius of the entire circle, this is explicitly seen as the white circle infigure 1(f). This means that the Lambert azimuthal equal-area projection allows us to view the entire surface of the sphere as a circle. The reason for using this area-preserving mapping, rather than an angle-preserving mapping, is because we are dealing with a probability distribution function. By definition, the integral over a volume determines the probability; area-preserving therefore translates into probability-preserving. A consequence of this mapping is that in some regions of phase space, the quasiprobability distribution appears warped. For instance, the first three states in figures 1(a)-(c) are all rotations of one another on a sphere.
The DV Wigner functions presented in figures 1(a)-(c) are standard two-level quantum states, where figures 1(a) and (b) are the ±1 eigenstates of the s ẑ operator, ñ | a and ñ | a respectively. The state infigure 1(c) is the equal superposition of ñ | a and ñ | a , or the positive eigenstate of s x . In all the presented states, there are negative values in the DV Wigner function. Importantly, in the DV Wigner function for qubits, negative volume, as well as being an indicator of non-classicality, is also a measure of purity [37]. This is because discrete system coherent states are fundamentally quantum; regardless of whether the system is the polarization of a photon or the direction of spin in an electron.
More generally, in both CV and DV Wigner functions, negative values arise as a consequence of selfinterference. In the CV Wigner function this arises from non-Gaussianity [61], and can be seen in the Fock states (excluding the vacuum state) or in superpositions of Gaussian states, see figure 3 for an example, discussed later in the paper. This explains why negative values have been used as a measure of quantumness, however there is one notable exception, the non-negative, entangled, Gaussian CV two-mode squeezed state.
Since the Gaussian states of a DV Wigner function can be visualized on a sphere, the emergence of selfinterference is now inevitable, due to the inherent geometry of the sphere. For example, the Wigner function for the state ñ | a has a Gaussian distribution centred at the north pole; as this Gaussian distribution tends towards zero, near the south pole, there is an emergence of negative quasi-probabilities. This negativity in the Wigner function is manifested as a result of self-interference, as the quantum coherences interfere with each other at the south pole. As the number of levels is increased (from the two-level system) in the DV Wigner function and take the infinite limit 6 , the SU 2 ( ) DV Wigner function tends towards the Heisenberg-Weyl group, returning to the standard CV Wigner function. This is because the effective size of the sphere increases, decreasing the relative size of the Gaussian. In the infinite limit, the negativity in the Wigner function is completely eliminated, since the Gaussian can no longer interact with itself on the opposite side of the sphere.
Although the example states so far have been density operators for pure states, the general formalism in (1) allows for the Wigner function to be generated for any arbitrary operator. To emphasize this, in figures 1(d)-(f) are the DV Wigner representation of each of the three Pauli operators. In general, Wigner function exhibit the normalization condition For normal density operators, this yields unity, as would be expected for any probability distribution function. For the Pauli operators however, , where i={x, y, z}, therefore ò W . The tracelessness of these matrices can be seen in figures 1(d)-(f) by noting that the negative and positive volumes are equivalent and therefore cancel. This feature will be key to several of our observations later in this work.
For a CV-DV hybrid system, the total displaced parity operator is simply the tensor product of the displaced parity operator for each subsystem [23,24,28] yielding a hybrid Wigner function for a density matrix r Hybrid systems generated with (9) usually have more degrees of freedom than is convenient to plot. For this reason, many approaches that use phase-space methods to treat hybrid systems use reduced Wigner functions, rather than considering the full phase space of the composite system. To give a full picture of the quantum correlations found between the two systems, a method similar to that introduced in[38] can be used. As an example of the utility of this method, the fully separable state, ñ ñ 0 f a | | , is shown in figure 2. The reduced Wigner functions for CV and DV degrees of freedom are presented in figures 2(a) and (b) respectively. In figure 2(c) we apply the method first presented in [38] to plot the phase-space representation of this state.
Specifically, figure 2(c) was created by first dividing the CV phase space into discrete points on a rectangular map. Each of these discrete points is then associated with a discrete complex value α, equally spaced across the phase space grid. For each set point α, the values of the Wigner function for θ and f degrees of freedom are calculated, with the Wigner function at that point plotted using the Lambert projection. This produces a DV Wigner function at each α in CV phase space. The transparency of each disc is then set proportional to the absolute maximal value of the phase space at that point, For example, to generate the disc at the centre of figure 2(c), we calculate , resulting in a DV Wigner function for ñ | a , and then modify the amplitudes of the quasi-probabilities using the value of α. This is then repeated for every α. Note that a main difference between the plots presented here and in [38] is that the transparency of the DV Wigner functions in [38] is set proportional to a W f | ( )|. Using the method presented here allows for a clearer view of the quantum correlations that manifest.
Since the state being plotted here is a pure separable state, the Wigner function can be expressed as , , 10 here r f and r â are the reduced density matrices for the CV and DV systems respectively. As a result, figure 2(c) has the same form as a coherent state, dictated by the CV Wigner function, with every point in phase space having an ñ | a DV Wigner function. The difference in this method, in comparison to [38], is that here the transparency is not set by integrating out the qubit degrees of freedom; such an approach leads to a loss of quantum correlations in the systems of interest.

Visualizing correlations in hybrid quantum systems
Quantifying different types of correlations in quantum systems is a key area of research that has received a great deal of attention [62][63][64][65][66][67][68][69]. In parallel, phase-space methods have been utilized as a tool to identify and categorize quantum correlations [41,[70][71][72][73]. Further, these methods have been used to generate measures based on the emergence of negative quasi-probabilities in the Wigner function [37,[74][75][76]. However, due to the higher number of degrees of freedom, visually representing correlations in composite systems is more difficult. We now show how our technique produces definite signatures of both quantum and classical correlations, that can be discerned for hybrid quantum systems. When dealing with quantum information processing with two coupled qubits, the distinction between these two types of correlations is important. Beginning with how correlations that arise from superposition appear, we will describe our choices of DV and CV qubits and how the encoding of quantum information is represented on these qubits.
Certain similarities are seen between DV and CV systems, whether in structure, choice in qubit, or in appearance of the quantum correlations that manifest. These similarities will be demonstrated here, by showing how quantum information can be encoded onto different types of state. Encoding quantum information onto quantum states can be done in various ways, including a variety of approaches even within the same system [45]. We will therefore begin by using the simplest case of a DV qubit for quantum information processing. Since the DV systems used here are two-level systems, the encoding of quantum information is straightforward; a bit value 0 or 1 is simply assigned to each of the two levels, ñ | a and ñ | a respectively. The DV 0bit is now represented visually by figure 1(a), likewise the 1 bit value is represented by figure 1(b). Furthermore, a general pure superposition state | | | | , allowing any weighted superposition between 0 and 1.
, an equal superposition is yielded and is represented visually by figure 1(c).
This binary choice becomes more complicated when assigning bit values to a CV qubit. Although, there are various ways to encode quantum information onto a CV system creating similarities between CV and DV systems. Since the Hilbert space is infinite, there are different constraints on assigning qubit values. We will now demonstrate two examples of CV qubits, comparing the results with the DV qubits 3.1. Fock state qubits Fock states are orthogonal and therefore a natural choice for quantum information processing. For simplicity we consider the vacuum and one-photon Fock states, ñ 0 f | and ñ 1 f | respectively. We can now form the analogy with the DV qubit state by assigning bit values to these states  ñ 0 0 f | and  ñ Having established that the hybrid Wigner function allows local correlations to be discerned reliably, we now demonstrate how quantum correlations arising between subsystems in this type of hybrid system manifest. Entanglement in Fock hybrid states, a Bell-Fock state 7 , ñ ñ + ñ ñ , is shown in figure 3(i). The full Wigner functions for bipartite Bell-Fock states have a distinctive pattern, reminiscent of the spin-orbit coupled state from [38], where there is a twisting of the DV Wigner functions dependent on the point in CV phase space. This DV dependence on the CV Wigner function is indicative that (10) does not hold for this state. This means that the state in question is not separable, and since this state is a pure state this indicates coupling between the two subsystems. This is a signature one should look for when investigating quantum correlations in this type of hybrid state.
Comparing the hybrid Wigner function in figure 3(i) to the reduced Wigner function for the CV and DV qubits in figures 3(g) and (h) respectively, we see the importance in considering the full phase space for entangled states such as this. It can be seen in figures 3(g) and (h) how correlations between the two systems are lost when considering the reduced Wigner functions, leaving only statistical mixtures of the basis states in each case. ) for hybrid Wigner function. 7 Bell state for an entangled DV qubit with a CV Fock qubit.

Coherent state qubits
Another choice in creating a CV qubit is to encode quantum information onto coherent states [5,6]. Unlike with the Fock CV qubit, the coherent state basis is an overcomplete basis where there is some degree of overlap between any two coherent states. However with sufficient distance between two coherent states, this overlap is negligible. For simplicity, our example states will be real values of β, where the two levels are set to the values b b b = -= .
Since the full system is a simple tensor product of the two qubits, the subsystems are separable, resulting in a full Wigner function that obeys (10). The separability between these states is seen in the full Wigner function in figure 4(c). The image of the CV Schrödinger cat state is visible as a discrete grid, with the DV Wigner function for the state at every point. Given the difference in the local correlations between the two choices of CV qubit, it is now worthwhile to demonstrate how the signature of the non-local correlations differ for the coherent state CV qubits. The hybrid analogue of a Bell state for coherent states, the Bell-cat state, is Since many of the correlations in this state are due to entanglement, the standard approach of using reduced Wigner functions is insufficient, as seen in figures 5(a) and (b). Neither reduced Wigner function has visible quantum correlations, yielding two mixed states. This issue motivated other approaches to tomography and state verification for such states, for instance [40] used reduced CV Wigner functions in different Pauli bases to show Bell's inequality. Other tomography methods for entangled hybrid systems, such as [47], also take into consideration the problems of a reduced phase-space representation of a hybrid entangled state. Although approaches such as these give a better appreciation of the quantum correlations, they still only provide glimpses of the nature of the full quantum state.  (14) is shown in figure 5(c). Comparing our representation with the reduced Wigner function treatment, the quantum correlations are now visible, manifesting as interference terms between the two coherent states. The nature of these quantum correlations is completely lost when the full Wigner function is not generated. Further, within the quantum correlations, the qubit states approach traceless states, as in figures 1(d)-(f), where the state at the very centre, α=0, is in fact the s x Pauli matrix. It is important to note at this point that the manifestation of traceless here, found only in the hybrid phase-space picture, are a signature of quantum correlations. Some existing tomography methods can pick up these correlations, however their full nature is not captured. For example, measuring the reduced Wigner functions results in a loss of quantum and classical correlations, as demonstrated in figures 5(a) and (b). This makes classical and quantum correlations, for this kind of state, indistinguishable. The ability to obtain signatures to distinguish between classical and quantum correlations is important in determining the suitability of states in quantum information processing.

The hybrid Wigner function for
To highlight this, we now consider two further examples of states that have the same reduced CV and DV Wigner functions. Though the degree of quantum correlations differ for each state. The general state is ñá + ñá-ñá + -ñá ñá + -ñá-ñá e e e g g e g g 1 2 , where η determines the purity of the state. When η=1 (15) reduces to (14). Changing the value of the loss to η=0.5 and then to η=0, figures 5(d) and (e) are, respectively, generated. In both, it is clear that the quantum correlations are slowly lost. The loss of quantum correlations means these states are less useful for quantum information purposes, and analyzing the reduced Wigner functions, unlike our approach, does not provide any insight to this loss. By using our method to represent the full Wigner function, it is not only possible to distinguish the strength of the quantum correlations but, the signature of classical correlations is revealed. In figure 5(e) is the state ñá + -ñá-ñá e e g g 1 2 16 that describes the equal classical probability of finding an excited state at β and a ground state at b -. The classical correlations that correspond to this probability is shown in our full picture of the Wigner function, where the bñ f | coherent state is correlated with ñ | a states, likewise the b -ñ f | coherent state is correlated with ñ | a states. This process not only reveals that this is the signature of classical correlations, it verifies the case that the traceless states between the two states are a result of the quantum correlations within the hybrid system.

The Jaynes-Cummings model
Light-matter interaction in the form of quantum electrodynamics (QED) has been an experimental cornerstone in understanding quantum effects. It has also given a helping hand in the development of quantum information applications, such as single-photon quantum non-demolition measurements acting as two-qubit gates between microwaves and atoms [35]. The standard example of a QED interaction between a two-level DV system and a CV field is the Jaynes-Cummings model [36]. Jaynes-Cummings type interactions are the basis for the generation of non-Gaussian states and are well known for showing the collapse and revival of Rabi oscillations [66,77,78] throughout its evolution. During this evolution, quantum information is transferred back and forth between the CV and DV systems; through this process, quantum information can then manifest as a Schrödinger cat state or generate Bell pairs of the sort shown in figure 3(i). By using our methods, the transfer of quantum information can be visualized as is swaps between the microwave field and the atom.
The interaction picture of the Jaynes-Cummings model  The first snapshot is early on in the evolution, » t t 9 r , where there is a high degree of coupling between the two systems. The reduced Wigner functions in figures 7(b) and (c), indicate that something approaching a Schrödinger cat state forming in the CV system; where the DV qubit is in a highly mixed state. All that can be deduced from the reduced Wigner functions then is that there are correlations between the qubit and the field mode; the nature of the quantum correlations remains hidden.
Evaluating the full Wigner function in figure 7(d), a better appreciation of the quantum correlations at this point in the evolution can be obtained. The DV spin direction at the top of the CV Wigner functions are orthogonal to those in the bottom of the CV Wigner function. Where at the top, the spins point in the direction of the negative eigenstate of s ; x at the bottom they point in the positive eigenstate of s x . The correlations found in the middle in figure 7(d) match the quantum correlation signature for a coherent state qubit, as they are of a form similar to the traceless states in figure 1.
The second snapshot of the Jaynes-Cummings model, » t t 2 r , is where the field mode and the qubit disentangle, transferring the quantum correlations to form a CV Schrödinger cat state. Presence of this Schrödinger cat state is immediately visible in the reduced CV Wigner function in figure 7(e). The reduced DV Wigner function in figure 7(f) has now increased in both negative and positive amplitudes, rotating to the eigenstate of s ŷ with eigenvalue −1. The return of coherence of the DV qubit is a good indication that the correlations between the two systems have decreased.
Both of the reduced Wigner functions in figures 7(e) and (f) suggest that this state is similar to the example state in figure 4, which is approximately separable. Observation of figure 7(g) confirms this suggestion, but more detail can still be found. Although very few correlations appear between the two subsystems, some residual quantum correlation has remained between the two. These correlations are found in the slight twisting of the qubits around the two cats and within the quantum correlations in between.
The final snapshot occurs at the revival of the Rabi oscillations, » t t r , where the qubit state is closest to the initial state within the revival. In figure 7(i) the average spin is pointing in the direction of an excited state ñ | a , however, there is a loss of coherence associated with the decrease in amplitudes and no negative values. The full Wigner function reveals why the coherences in the reduced DV Wigner function have formed. At most points in the full Wigner function, the DV Wigner function is in the excited state, however at many points there are rotations in the qubit Wigner functions, indicating some residual quantum correlations. The strongest coherent states are found on the left-hand side, where it appears the state is returning to the initial state of a coherent state coupled to ñ | a . The quantum correlations that accompany the two choices of CV qubits have a somewhat different nature however their signatures are distinguishable when considering the full Wigner function. The correlations for the Fock state qubits show a dependence on each other, arising due to the non-separability of the state. This closely resembles the pattern found in spin-orbit coupled states [38], and is comparable to spin texture images. The fundamental signatures come from the behaviour of the coherences and correlations within and between the systems. The form of the Wigner function of a two-mode squeezed state, although lacking negative values due to it being Gaussian, resembles the signature identified for the Bell-Fock states; the spatial dependence of one system affecting the state in the other system.

Conclusions
By plotting the information generated by calculating the Wigner function for a CV-DV hybrid system, we have shown that the usual techniques for visualizing these systems misses the full nature of the quantum correlations that arise. For example, the most common technique of generating the reduced Wigner function causes the correlations that arise between the systems to be traced out. Tracing out a system results in a loss of correlations that can be found between the two systems. A method to overcome this loss of information was presented in [38], but an envelope was applied setting the transparency of the points in phase space according to the reduced Wigner function for the CV degrees of freedom a W f ( ). Here this method has been developed, changing the envelope to instead be proportional to a q f q f r W max , , , | ( )| at each point in CV phase space. This adjustment further allows us to visualize the quantum correlations present in CV-DV hybrid states, such as those that manifest between the two coherent states in a hybrid Schrödinger cat state. Doing this means it is possible to gain a more full picture of the correlations that arise in the interaction between CV and DV systems. As well as allowing us to characterize signatures of quantum correlations found in certain systems; a result that promises potential usefulness in analyzing the correlations in maximally entangled states and entanglement as a result of squeezing. Being able to visually determine the level of quantum correlations, not always clear in coupled systems, gives significant advantage over reduced Wigner function methods that do not always detect the purity of Bell-cat like entanglement.
By demonstrating these methods within the Jaynes-Cummings model, we show how excitations are shared and swapped, demonstrating a visual representation of the transfer of quantum information between systems. Extending these methods to different systems, will allow for a more intuitive picture of how quantum information moved around coupled systems, providing further insight into the inner process of quantum processes and algorithms.
There have been previous experimental examples which have used phase space to investigate the types of state considered in this paper. One notable example is [47], from a sequence of measurements of the expectation values of the qubit in different bases, they have been able to recreate the CV Wigner function. Using a similar procedure with our generalized displaced parity operator, it should be possible to extend this to produce experimental results equivalent to those in this paper. This technique could be considered to be a form of quantum state spectroscopy.