Quantum phase space measurement and entanglement validation made easy

It has recently been shown that it is possible to represent the complete quantum state of any system as a phase-space quasi-probability distribution (Wigner function) [Phys Rev Lett 117, 180401]. Such functions take the form of expectation values of an observable that has a direct analogy to displaced parity operators. In this work we give a procedure for the measurement of the Wigner function that should be applicable to any quantum system. We have applied our procedure to IBM's Quantum Experience five-qubit quantum processor to demonstrate that we can measure and generate the Wigner functions of two different Bell states as well as the five-qubit Greenberger-Horne-Zeilinger (GHZ) state. As Wigner functions for spin systems are not unique, we define, compare, and contrast two distinct examples. We show how using these Wigner functions leads to an optimal method for quantum state analysis especially in the situation where specific characteristic features are of particular interest (such as for spin Schr\"odinger cat states). Furthermore we show that this analysis leads to straightforward, and potentially very efficient, entanglement test and state characterisation methods.


I. INTRODUCTION
In 1932, Eugene Wigner, in an attempt to link the physics of many particle systems (statistical physics) with quantum mechanics, defined a new way of describing the quantum state [1].It took the form of a probability density function in position and momentum but, interestingly, it could take on negative values.Now named after its creator, the Wigner function is usually presented in advanced quantum optics texts as an integral combining the notions of Fourier transformations and autocorrelations.The function rapidly established its usefulness when its ability to take on negative values enabled physicists to be able to visualise quantum correlations in ways that were not previously possible.This capability is most commonly seen in the superposition of two macroscopically distinct coherent states [2][3][4].In Fig. 1 we show an example of the Wigner function for such a superposition, the famous Schrödinger cat state.Such a state is very similar to those presented in [5] where it was demonstrated that non-classical states of light can be made.
Similar schemes to those used in [5] for the direct reconstruction of the Wigner function for light have been in existence for some time (see, for example, [6][7][8][9][10]).These schemes all have the same feature that they, either implicitly or explicitly, rely on the fact that the Wigner function can be written as the expectation value of an appropriately normalized displaced parity operator or, equivalently, the expectation of parity for a displaced 1.The iconic textbook example of a Wigner function for a Schrödinger cat state.The bell shapes represent the 'alive' and 'dead' possible states for the 'cat' and the oscillations between them indicate the quantum coherence between these states (i.e. the classic "both alive and dead" statement).A similar Wigner function without these interference terms would represent a state with a classical coin toss probability of being either "alive" or "dead" but not both.The presence of the interference terms indicates that this Wigner function represents a state that is in both states ("alive and dead") at the same time (a superposition).
established in the quantum optics community [12].
While it has been known for a long time that parity displacement could be done for continuous systems [13,14] it has only recently been proposed that any quantum system's Wigner function can be written as a function of a displaced and/or rotated generalized parity operator [15].Mathematically this can be expressed as where W is the Wigner function and Ω is the set of parameters over which displacement or rotations are defined (typically this would be position and momentum); ρ is the density matrix; U (Ω) is a general displacement/rotation operator, or collection of operators; and Π is motivated by the usual parity operator.The conventional Wigner function in position and momentum space is obtained if U is set to the displacement operator that defines coherent states, |α , from the vacuum state, |0 , according to D(α) |0 = |α and the parity operator Π is defined to be twice the usual phase space parity operator Π C |α = |−α [16].For a given system the choice of U (Ω) and Π is not unique but in [15] it was stipulated that a distribution W ρ (Ω) over a phase space defined by the parameters Ω is a Wigner function of ρ if there exists a kernel ∆(Ω) satisfying the following restricted version of the Stratonovich-Weyl correspondence (reproduced verbatim from [15]): S-W. 1 The mappings W ρ (Ω) = Tr [ρ ∆ (Ω)] and ρ = Ω W ρ (Ω) ∆ (Ω) dΩ exist and are informationally complete.Simply put, we can fully reconstruct ρ from W ρ (Ω) and vice versa.S-W.2 W ρ (Ω) is always real valued which means that ∆ (Ω) must be Hermitian.
S-W.3 W ρ (Ω) is "standardized" so that the definite integral over all space Ω W ρ (Ω) dΩ = Tr ρ exists and . This is a restriction of the usual Stratonovich-Weyl correspondence.
If we define U (Ω) as an element of a Special Unitary (SU) group that acts as a displacement or rotation and Π as an appropriately normalised identity plus a traceless diagonal matrix (i.e. an element of the Cartan subalgebra of the appropriate group) then, from [15], Eq. ( 1) is sufficient to generate Wigner functions for any finitedimensional, continuous-variable, quantum system.We note that beyond satisfying the Stratonovich-Weyl correspondence, we have yet to fully determine the level to which this definition is constrained.Because Π performs the same role as parity does in the standard Wigner function, we refer to it as a generalised parity, or parity, for brevity.

II. THE SCHEME
In this work we present a procedure for the direct measurement and reconstruction of the quantum state for a series of qubits from two different Wigner functions that both satisfy the above restricted Stratonovich-Weyl correspondence.We start by considering a Wigner function where the parity operator is defined with respect to the underlying group structure of the total system.We then proceed to investigate another Wigner function whose kernel comprises a tensor product of one qubit Wigner functions which is arguably a more natural way at looking at composite quantum systems.In both cases we apply our procedure to IBM's Quantum Experience five qubit quantum processor to demonstrate that we can directly measure and reconstruct the Wigner functions of two different Bell states and the five-qubit Greenberger-Horne-Zeilinger (GHZ) state.
While Wigner functions can be considered to be expectation values of displaced parity operators, this view does not necessarily lead to the best way to practically determine the Wigner function.As previously discussed, displacing the parity operator and taking its expectation value should be the same as displacing/rotating the state, i.e. creating a new "state" and calculating the expectation value of the unshifted parity operator.
Mathematically this is equivalent to our original expression for the Wigner function (Eq.( 1)) as trace is invariant under cyclic permutations of its arguments.Furthermore, it is possible, and in some cases (such as with the IBM Quantum Experience) easier, to make ρ(Ω) by performing local rotations on each qubit rather than displacing Π.
In the ideal case, the parity Π shown in Eq. (3) will be directly measurable, allowing for reconstruction of the quantum state via its Wigner function without any intermediate steps being needed.Even if it is not possible to measure the parity directly, such as with the IBM Quantum Experience, there is a simple alternative.Note that Π, as introduced in [15], is always a diagonal operator in the computational basis.The Wigner function is then easy to calculate according to To determine the Wigner function we are only required to measure the probability of the rotated system occupying each state of the computational basis.
For a set of qubits the rotation of the system can be intuitively defined in terms of rotation operators acting on each of the system's constituent parts.Explicitly, we can define a total rotation operator for N qubits as where U i (θ i , φ i , Φ i ) = e iσz i φi e iσy i θi e iσz i Φi is the rotation operator for each qubit in terms of the Euler angles In the following sections we discuss the Wigner functions defined through two different possible choices of Π.

III. A SPIN WIGNER FUNCTION WITH SU(•) PARITY
In this section we define and explore a Wigner function for N qubits where the parity operator reflects the underlying group structure of the total system.Here, parity is motivated by the idea of doing what amounts to a global π rotation on the hypersphere of the underlying SU 2 [N ] coherent state representation.This is achieved by defining our parity operator Π SU(2 for one qubit and for two qubits in the computational basis.
Combining this definition of parity with the composite rotation operator, U N we obtain the kernel that satisfies the restricted Stratonovich-Weyl correspondence given in the introduction.We note that the Φ i 's make no contribution as Π SU(2 [N ] ) commutes with σ zi .
This kernel defines our SU 2 [N ] , parity-based, Wigner function according to Let us now consider the specific case of the Wigner function W SU(2 [N ] ) for two qubits.Each qubit brings with it two degrees of freedom, expressed in terms of Euler angles Ω = (θ 1 , ϕ 1 , θ 2 , ϕ 2 ), thus the associated Wigner function takes the form of a four-dimensional pseudoprobability distribution W SU(2 [2] ) (θ 1 , ϕ 1 , θ 2 , ϕ 2 ).Fourdimensional functions are not easy to visualise, but we can take slices of the function in order to gain an appreciation of it as a whole.In Fig. 2 (a-d) we show some example Wigner function slices for two Bell states.Specifically Fig. 2 (c,d) shows the equal angle ("= ") slice ) (θ, ϕ, θ, ϕ) while Fig. 2 (c,d) Note that Fig. 2 (e,f) will be discussed in section IV.
In order to demonstrate that this function is indeed easy to construct we have taken advantage of IBM's Quantum Experience project.The project makes available through the Internet a five qubit processor based on a simple "star" topology: a central qubit is coupled to four other qubits.The machine has already been used to produce interesting results [18,19].Here we use it to directly measure and reconstruct the Wigner functions for the two Bell states |Φ + and |Ψ − as presented in Fig. 2. In this work, we are limited by the operations that IBM has made available to the user, operations that naturally focus on quantum computing applications.Nevertheless, following Eq.( 2), we are able to produce ρ(θ 1 , ϕ 1 , θ 2 , ϕ 2 ) using rotations generated by combinations of gate operations and readout state populations of ρnn (θ 1 , ϕ 1 , θ 2 , ϕ 2 ) via the standard output of the IBM processor.We then use Eq. ( 4) and Eq. ( 7) to reconstruct the Wigner function, Eq. ( 9), directly.
In Fig. 3 we plot the Wigner function W φi=0 SU(2 [2] ) slices comparing the ideal theoretical values of Fig. 2 (c,d), values generated by IBM's built in simulator (that models environmental effects), and real experimental data.The calibration data pertaining to the experiments is provided in Table I.In principle, to fully reconstruct the state requires us to measure 2 2N − 1 distinct points, the same number of points as needed to reconstruct the density matrix.In Fig. 3 we have actually measured more, ) (θ1, θ2), the slice where ϕ1 = ϕ2 = 0. We recommend that the reader see the supplementary material which expands on these figures and shows animations of the Deutsch-Jozsa algorithm [17] and the creation of all four Bell states (in the animations, for example, it becomes clear that the Wigner functions for the Bell states [or for that matter, any maximally entangled two qubit state] are simply rotations of the same function in four-dimensional space).Later in this work we will present experimental reconstructions of the θ1 versus θ2 plots.In understanding the form of these plots we note that the |Ψ+ state is one with total spin-angular momentum but zero total z spin-angular momentum.We thus expect to see the observed ring-like symmetry in for |Ψ+ (the symmetry of |Φ− follows from |Ψ+ as they are rotations of each other in four-dimensional space).This state is also an angular-momentum analogue of a photon number (Fock) state which shares a similar symmetry in its Wigner function [8][9][10].In (e,f ) we show W ϕ i =0 2 SU(2) (θ1, θ2) created using the alternative parity operator Π 2 SU(2) as discussed in section IV.The availability of more than one parity operator, which produces Wigner functions with qualitatively very similar features, opens up possible alternative paths for direct phase space reconstruction (note we have also included an animation of W 2 SU(2) for the creation of the Bell states in the supplementary material).and different, points than would be needed to fully reconstruct the state.This was done to demonstrate the ability to generate the Wigner function using a raster scan approach as this makes clear the direct measurement nature of our method.Due to finite computational resources, and the need to do rotations as outlined above, we are limited in our resolution.Nevertheless, we find good agreement between theory, simulation, and experimental data, demonstrating that our tomographic process is clearly able to distinguish between the two Bell states.
Bell states are interesting both as an example of ) slice of the Wigner function for two qubits; making use of the periodicity of the function at the edges of each plot for computational efficiency.We have included for comparison ideal theoretical values, numerical results using IBM's built in simulator, and real experimental data from IBM's quantum processor.The quantum circuit presented above is a screenshot taken directly from IBM's Quantum Experience web interface.It provides an example of the measurement protocol we used to obtain the diagonal elements of the rotated density matrix ρnn(θ1, ϕ1, θ2, ϕ2).The theoretical, simulated, and experimental data are all in very good agreement with each other.Slight differences exist due to imperfect implementation of needed rotations due to different gate operations having different levels of noise (decoherence).It should be straightforward to replace the "Gates for performing θ rotations" with generalized rotation operators on each qubit.Furthermore, if measurement of the parity operator (Π) were available, direct observation of the quantum state would be reduced to a two-stage process of rotate and measure.We believe such a protocol, because it would need fewer gate operations, would result in better agreement between theory and experiment than that seen in this figure.Note that in order to have good colour graduation in the transition from positive to negative values there is some color clipping for the very strong blue points.
maximally entangled states and for their usefulness in quantum information processing.Fortunately, for systems comprising more spins, we can extend this class of states to those that have a direct analogy with optical Schrödinger cat states as considered in [5] and others.Such states are termed "spin-cat states" of which the GHZ state [20] is an excellent example.In previous theoretical work, spin Wigner-like functions have been proposed as a mechanism for visualizing such cat states [21][22][23].Using our method we now directly construct the W SU(2 [5] ) Wigner function for a T1 and T2 are the usual relaxation times, g is the gate error, r is the readout error and i2 g is the C-NOT gate error between the qubit listed and qubit 2 (which is the target qubit for the C-NOT operation).
spin-cat of the form In Fig. 4 we show the θ 1 = θ 2 = θ 3 = θ 4 = θ 5 and ϕ 1 = ϕ 2 = ϕ 3 = ϕ 4 = ϕ 5 slice of the W SU(2 [5] ) Wigner function for |GHZ 5 which is the higher dimensional analogue of Fig. 2 (a,b).We show both theoretical predictions and, due to limited computational resource, as insets, simulation and experimental data obtained from the IBM machine.Once more the calibration data pertaining to the experiments is provided in Table I.We note that the θ 1 = θ 2 = θ 3 = θ 4 = θ 5 and ϕ 1 = ϕ 2 = ϕ 3 = ϕ 4 = ϕ 5 slice does not contain all the information needed to reconstruct the state; for full reconstruction we would need to measure and visualise all {θ i , θ j } i = j sets of angles for various values of ϕ i .We also note that, in order to optimise the color map in Fig. 4 some clipping of the very dark blue points was needed.For the top and bottom point the theoretical value is 2.7 while the simulated values are 1.64 and 1.70, and experimental values 1.16 and 1.22, respectively.Here simulation and experiment are in good agreement.The difference from the theoretical values for all four points indicates that there is some decoherence in the system, mostly accounted for in IBM's simulation, meaning that the observed state is not in an ideal GHZ state.

IV. A WIGNER FUNCTION FOR TENSOR PRODUCTS OF SPINS
The Stratonovich-Weyl conditions do not uniquely specify the parity operator Π and hence the Wigner function is also not uniquely defined.Because of this, it is natural to ask what difference choosing alternative Wigner functions will make.As our current focus is on experimental reconstruction of the quantum state in phase space, we believe that it is instructive to explore at least one alternative whose direct measurement may be more readily available to those working in quantum information.In the previous case, the definition of parity was motivated by the idea of a global π rotation on the hypersphere of the underlying SU 2 [N ] coherent state rep- This can be considered a qubit-system analogue of Fig. 1 and which was presented in [5] to reconstruct non-classical cavity field states.We note that in [5] the interference terms that were observed correspond to quantum coherence in macroscopically distinct superpositions of states.In this figure, the interference terms should be interpreted as a direct visualisation of the entanglement in the system.Here we show the ideal function, and as insets, show both simulated and experimental results from IBM's Quantum Experience project.In this figure we also show an example circuit used to generate simulated and experimental data.As with the circuits used to create the Bell states presented in Fig. 3, these gate operations ideally would be replaced by optimized, single-rotation, operations.We note that the two, non-polar, points can be obtained in a variety of ways.Specifically they could be found by using just θ rotations, or through a combination of θ and ϕ rotations.We have verified that the results that we obtained from the IBM Quantum Experience project are independent of the combination of rotations used.
resentation.In this case the notion of parity is motivated on an individual qubit level; a global π rotation on each qubit's Bloch sphere.This leads to a parity operator that is nothing more than the tensor product of the parities of individual qubits: (10) which for one qubit is equal to Eq. ( 6) but for two qubits takes the explicit form in the computational basis.When compared with Eq. ( 7) we see that this version of parity no longer treats onequbit and two-qubit contributions on an equal footing.
The definition of the Wigner function continues in the same way as before and, in terms of the rotated density matrix ρ = U † N ρU N takes the form Returning to Fig. 2 (e,f) we show example slices of , ϕ 2 = 0) that demonstrates this alternative Wigner function is qualitatively very similar to the equivalent slices of W SU(2 [N ] ) (Ω) shown in Fig. 2 (c,d).
In Fig. 5 (top) we show results for comparison with Fig. 4 and (bottom) with Fig. 3 which demonstrates that W N SU(2) is a Wigner function with qualitatively very similar features to W SU(2 [N ] ) that will be compared in the next section.Again we note that, in order to optimise the color map in Fig. 4 some clipping of the very dark blue points was needed.For the top and bottom point the theoretical value is 2.375.The simulated values are 1.13 and 1.11, and the experimental values are 0.8876 and 0.9006, respectively.

V. ON THE DIFFERENCES BETWEEN WIGNER FUNCTIONS
Each of the two cases we have considered here have their own strengths which will be expanded on in a later publication.However, we are including a brief discussion to highlight that there is some freedom in choosing parity in tensor product spaces.This should be of utility as it increases the number of available options in designing experiments for the direct measurement of a Wigner function.
The full-group Wigner function W SU (2 [N ] ) and the tensor-product Wigner function W N SU(2) are related to the density matrix by different, but still invertible, linear maps, and therefore both contain full information about the quantum state.The tensor-product form has the additional property of respecting the marginals in each subspace.We can see this is indeed the case by noting that the two qubit kernel separates leading to the result where ρ A is the reduced density matrix of subsystem A.
Note that extension to arbitrary number of qubits is a trivial extension of this argument.
As an example, consider the Bell state |Ψ + shown in Fig 2(b).Here our two Wigner function cases have the same structure, with the tensor-product form having a larger amplitude of modulation: where (x i , y i , z i ) is the unit vector in the direction Ω i .However, for the product state (|0 1 |0 2 ) we see a distinction in angular dependence: Note that the one-qubit and two-qubit angular terms carry coefficients of different magnitude in the tensorproduct Wigner function.
The above distinctions have led us to speculate that the two different forms of the Wigner function that we consider in this paper may be useful as a mechanism to differentiate (in representation) logical and physical qubit systems.That is, when there is a natural separation into physical qubits, into subsystems, or into a system and an environment, we choose the tensor product formulation.If, on the other hand, the system under consideration comprises a many-level quantum system constrained to act as logical qubits, it is less natural to impose a tensor product structure to the phase space representation than use the full-group form, which may be more natural.Furthermore, in systems that comprise a mixture of logical and physical qubits a tensor product of the different kernels could be used to maintain this distinction.While much further work needs to be done, it may well be that drawing such distinctions may help us understand separability from a phase space perspective, thus enabling derivation of new useful entanglement measures.

VI. CONCLUDING REMARKS
We have demonstrated a method for direct quantum state reconstruction that extends those previously known for quantum optical systems [5][6][7][8][9][10] to other classes of systems.Using IBM's Quantum Experience five qubit quantum processor, we have shown direct reconstruction of two Bell states and the five qubit GHZ spin Schrödinger cat state via spin Wigner function measurements.We note that our procedure could be made much more efficient by direct implementation of rotation operations and measurement of any suitable parity operator (or, if appropriate, direct measurement of the rotated parity).By doing so, the potential advantage of our procedure over other tomographic methods would be made much clearer in that fewer measurements would be needed to check certain properties of the quantum state.In such an instance, in verifying the fidelity of a high-quality GHZ state, only a small set of measurements that quantifies the strength of the interference terms is needed, providing an improvement over traditional quantum state tomography.Furthermore, this work demonstrates how phase space methods can be of utility in understanding processes such as decoherence and be useful in the "debugging" of quantum information processors.The utility of this work extends beyond metrology as the inclusion of tomography in device engineering will no doubt be of use in the development of quantum analogues for "Design for Test", debug, fault identification and system certification.
FIG. 1.The iconic textbook example of a Wigner function for a Schrödinger cat state.The bell shapes represent the 'alive' and 'dead' possible states for the 'cat' and the oscillations between them indicate the quantum coherence between these states (i.e. the classic "both alive and dead" statement).A similar Wigner function without these interference terms would represent a state with a classical coin toss probability of being either "alive" or "dead" but not both.The presence of the interference terms indicates that this Wigner function represents a state that is in both states ("alive and dead") at the same time (a superposition).

FIG. 3 .
FIG.3.Plots of the spin Wigner function for the two Bell states |Φ+ and |Ψ− .We plot θ1 versus θ2 for the W φ i =0 SU(2[2] ) slice of the Wigner function for two qubits; making use of the periodicity of the function at the edges of each plot for computational efficiency.We have included for comparison ideal theoretical values, numerical results using IBM's built in simulator, and real experimental data from IBM's quantum processor.The quantum circuit presented above is a screenshot taken directly from IBM's Quantum Experience web interface.It provides an example of the measurement protocol we used to obtain the diagonal elements of the rotated density matrix ρnn(θ1, ϕ1, θ2, ϕ2).The theoretical, simulated, and experimental data are all in very good agreement with each other.Slight differences exist due to imperfect implementation of needed rotations due to different gate operations having different levels of noise (decoherence).It should be straightforward to replace the "Gates for performing θ rotations" with generalized rotation operators on each qubit.Furthermore, if measurement of the parity operator (Π) were available, direct observation of the quantum state would be reduced to a two-stage process of rotate and measure.We believe such a protocol, because it would need fewer gate operations, would result in better agreement between theory and experiment than that seen in this figure.Note that in order to have good colour graduation in the transition from positive to negative values there is some color clipping for the very strong blue points.

FIG. 4 .
FIG. 4.Here we show the five-qubit GHZ spin Schrödinger cat state Wigner function W SU(2 5 ) for the θ1 = θ2 = θ3 = θ4 = θ5 and ϕ1 = ϕ2 = ϕ3 = ϕ4 = ϕ5 slice.This can be considered a qubit-system analogue of Fig.1and which was presented in[5] to reconstruct non-classical cavity field states.We note that in[5] the interference terms that were observed correspond to quantum coherence in macroscopically distinct superpositions of states.In this figure, the interference terms should be interpreted as a direct visualisation of the entanglement in the system.Here we show the ideal function, and as insets, show both simulated and experimental results from IBM's Quantum Experience project.In this figure we also show an example circuit used to generate simulated and experimental data.As with the circuits used to create the Bell states presented in Fig.3, these gate operations ideally would be replaced by optimized, single-rotation, operations.We note that the two, non-polar, points can be obtained in a variety of ways.Specifically they could be found by using just θ rotations, or through a combination of θ and ϕ rotations.We have verified that the results that we obtained from the IBM Quantum Experience project are independent of the combination of rotations used.

FIG. 5 .
FIG. 5.Here we reproduce Figs. 3 and 4 using the same data but now employing the Wigner function defined using the alternative parity operators as given in Eq. (12).In the top figure, for comparison with Fig. 4, we show the five-qubit GHZ spin Schrödinger cat state Wigner function W 5 SU(2) for the θ1 = θ2 = θ3 = θ4 = θ5 and ϕ1 = ϕ2 = ϕ3 = ϕ4 = ϕ5 slice.Again we show the ideal function, and as insets, show both simulated and experimental results from IBM's Quantum Experience project.On the bottom figure, for comparison with Fig. 3, we provide plots of W 2 SU(2) for the two Bell states |Φ+ and |Ψ− .We plot θ1 versus θ2 for the ϕ1 = ϕ2 = 0 slice of the Wigner function for two qubits.Once more, we have included for comparison ideal theoretical values, numerical results using IBM's built in simulator, and real experimental data from IBM's quantum processor.Again we see good agreement between theory, simulation and experiment and note that using a different parity operator provides an alternative path to direct measurement of phase space.

TABLE I .
Calibration data for the experimental results contained within this paper.Data for the Bell state and GHZ Wigner functions were taken on 16 th and 17 th June 2016 when the fridge temperature was 18.25 mK and 17.916 mK respectively.