Representations of two-qubit and ququart states via discrete Wigner functions

TL;DR

This paper develops a theoretical framework using discrete Wigner functions for two-qubit and ququart states, linking phase space representations with experimental quantum state tomography and quantum correlations.

Contribution

It introduces a detailed study of discrete SU(2)⊗SU(2) and SU(4) Wigner functions, connecting them with experimental density matrix reconstructions and quantum correlation visualization.

Findings

Established a formal correspondence between different phase space descriptions.

Analyzed quantum correlation effects in two-qubit X-states.

Proposed extensions to other distribution functions like Husimi and Glauber-Sudarshan.

Abstract

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 tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.

Equations

$$\mathscr{L}_{+,1}(\mathcal{H}_{N})=\left\{\mbox{$\hat{\rho}$}\in\mathscr{L}(\mathcal{H}_{N})\;|\;\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}]=1,\;\mbox{$\hat{\rho}$}=\mbox{$\hat{\rho}$}^{\dagger},\;\rho_{\ell}\geq 0\;\;(\ell=1,\ldots,N)\right\}$$

$$\mathtt{(i)}\;\hat{g}_{i}=\hat{g}_{i}^{\dagger},\quad\mathtt{(ii)}\;\mbox{${\rm Tr}$}[\hat{g}_{i}]=0,\quad\mathtt{(iii)}\;\mbox{${\rm Tr}$}[\hat{g}_{i}\hat{g}_{j}]=2\delta_{ij};$$

$$\mathtt{(iv)}\;[\hat{g}_{i},\hat{g}_{j}]=2{\rm i}\sum_{k=1}^{N^{2}-1}\mathscr{F}_{ijk}\hat{g}_{k}\;\Rightarrow\;\mathscr{F}_{ijk}=-\frac{{\rm i}}{4}\mbox{${\rm Tr}$}[[\hat{g}_{i},\hat{g}_{j}]\hat{g}_{k}]\;\;(\mbox{antisymmetric tensor});$$

$$\mathtt{(v)}\;\{\hat{g}_{i},\hat{g}_{j}\}=\frac{4}{N}\delta_{ij}\hat{\mathds{I}}_{N}+2\sum_{k=1}^{N^{2}-1}\mathscr{D}_{ijk}\hat{g}_{k}\;\Rightarrow\;\mathscr{D}_{ijk}=\frac{1}{4}\mbox{${\rm Tr}$}[\{\hat{g}_{i},\hat{g}_{j}\}\hat{g}_{k}]\;\;(\mbox{symmetric tensor}),$$

$$(\mathtt{vi})[{\hat{g}}_i,[{\hat{g}}_j,{\hat{g}}_k]]+[{\hat{g}}_j,[{\hat{g}}_k,{\hat{g}}_i]]+[{\hat{g}}_k,[{\hat{g}}_i,{\hat{g}}_j]]=0,$$

$$(\mathtt{vii})[{\hat{g}}_i,\{{\hat{g}}_j,{\hat{g}}_k\}]+[{\hat{g}}_j,\{{\hat{g}}_k,{\hat{g}}_i\}]+[{\hat{g}}_k,\{{\hat{g}}_i,{\hat{g}}_j\}]=0;$$

$$\displaystyle\mathtt{(viii)}\;\mbox{${\rm Tr}$}[\hat{g}_{i}\hat{g}_{j}\hat{g}_{k}]=2\mathscr{J}_{ijk}\quad\mbox{with}\quad\mathscr{J}_{ijk}=\mathscr{D}_{ijk}+{\rm i}\mathscr{F}_{ijk},$$

$$\displaystyle\mathtt{(ix)}\;\mbox{${\rm Tr}$}[\hat{g}_{i}\hat{g}_{j}\hat{g}_{k}\hat{g}_{l}]=\frac{4}{N}\delta_{ij}\delta_{kl}+2\sum_{p=1}^{N^{2}-1}\mathscr{J}_{ijp}\mathscr{J}_{pkl}.$$

$$\hat{O}=\frac{1}{N}\mbox{${\rm Tr}$}[\hat{O}]\mathds{I}_{N}+\frac{1}{2}\sum_{i=1}^{N^{2}-1}\mathcal{O}_{i}\hat{g}_{i},$$

$$\mbox{$\hat{\rho}$}=\frac{1}{N}\mathds{I}_{N}+\frac{1}{2}\sum_{i=1}^{N^{2}-1}\langle\hat{g}_{i}\rangle\hat{g}_{i},$$

$$\mathbf{g}=\left(⟨{\hat{g}}_1⟩,\dots ,⟨{\hat{g}}_{N^2-1}⟩\right)\in {\mathbb{R}}^{N^2-1}.$$

$$\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]=\frac{1}{N}+\frac{1}{2}|\mathbf{g}|^{2}\leq 1$$

$$\hat{G}(\mu ,\nu ):=\frac{1}{\sqrt{N}}\sum _{\eta ,\xi =0}^{N-1}{\omega }^{-(\mu \eta +\nu \xi )}{\omega }^{\frac{1}{2}N\Phi (\eta ,\xi ;N)}{\hat{S}}_{\mathrm{S}}(\eta ,\xi )(\mu ,\nu =0,\dots ,N-1)$$

$${\hat{S}}_{\mathrm{S}}(\eta ,\xi )=\frac{1}{\sqrt{N}}{\omega }^{\frac{1}{2}\eta \xi }{\hat{U}}^{\eta }{\hat{V}}^{\xi },$$

$$\hat{O}=\frac{1}{N}\sum _{\mu ,\nu =0}^{N-1}\mathrm{O}(\mu ,\nu )\hat{G}(\mu ,\nu ).$$

$$\mbox{$\hat{\rho}$}=\frac{1}{N}\sum_{\mu,\nu=0}^{N-1}W(\mu,\nu)\hat{G}(\mu,\nu)$$

$$W(\mu ,\nu )=\frac{1}{N}+\frac{1}{2}\sum _{i=1}^{N^2-1}⟨{\hat{g}}_i⟩\left({\hat{g}}_i\right)(\mu ,\nu ),$$

$$W(\mu ,\nu )=\frac{1}{2}\left[1+(-1{)}^{\nu }{P}_x+(-1{)}^{\mu +\nu +1}{P}_y+(-1{)}^{\mu }{P}_z\right],$$

$$\mbox{$\hat{\rho}$}=\frac{1}{4}\left[\hat{\mathds{I}}_{4}+\sum_{i=x,y,z}a_{i}\hat{\sigma}_{i}^{(1)}\otimes\hat{\mathds{I}}_{2}^{(2)}+\sum_{j=x,y,z}b_{j}\hat{\mathds{I}}_{2}^{(1)}\otimes\hat{\sigma}_{j}^{(2)}+\sum_{i,j=x,y,z}c_{ij}\hat{\sigma}_{i}^{(1)}\otimes\hat{\sigma}_{j}^{(2)}\right]$$

$$\displaystyle\left\{\begin{array}[]{lll}\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]\leq 1,\\ \mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{3}]\geq\frac{3}{2}\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]-\frac{1}{2},\\ \mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{4}]\leq\frac{1}{6}-\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]+\frac{1}{2}\left(\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]\right)^{2}+\frac{4}{3}\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{3}].\end{array}\right.$$

$$\mbox{${\rm Tr}$}[\mbox{$\hat{\rho}$}^{2}]=\frac{1}{4}\left[1+\sum_{i=x,y,z}\left(a_{i}^{2}+b_{i}^{2}\right)+\sum_{i,j=x,y,z}c_{ij}\right]\leq 1$$

$$\mbox{$\hat{\rho}$}_{\mathtt{R}}^{(1)}\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}_{2}[\mbox{$\hat{\rho}$}]=\frac{1}{2}\left[\hat{\mathds{I}}_{2}^{(1)}+\sum_{i=x,y,z}a_{i}\hat{\sigma}_{i}^{(1)}\right]$$

$$\mbox{$\hat{\rho}$}_{\mathtt{R}}^{(2)}\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}_{1}[\mbox{$\hat{\rho}$}]=\frac{1}{2}\left[\hat{\mathds{I}}_{2}^{(2)}+\sum_{j=x,y,z}b_{j}\hat{\sigma}_{j}^{(2)}\right].$$

$$W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{1},\nu_{1})\otimes\hat{G}^{\dagger}(\mu_{2},\nu_{2})\mbox{$\hat{\rho}$}]\qquad(0\leq\mu_{1},\nu_{1},\mu_{2},\nu_{2}\leq 1).$$

$$\displaystyle W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})=\frac{1}{4}\left[1+\sum_{i=x,y,z}a_{i}\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{1},\nu_{1})\hat{\sigma}_{i}^{(1)}]+\sum_{j=x,y,z}b_{j}\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{2},\nu_{2})\hat{\sigma}_{j}^{(2)}]\right.$$

$$\displaystyle\hskip 108.12054pt+\left.\sum_{i,j=x,y,z}c_{ij}\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{1},\nu_{1})\hat{\sigma}_{i}^{(1)}]\,\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{2},\nu_{2})\hat{\sigma}_{j}^{(2)}]\right],$$

$$\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{a},\nu_{a})\hat{\sigma}_{\ell}^{(a)}]=(-1)^{\nu_{a}}\delta_{\ell x}+(-1)^{\mu_{a}+\nu_{a}+1}\delta_{\ell y}+(-1)^{\mu_{a}}\delta_{\ell z}.$$

$$\displaystyle W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})=\frac{1}{4}\bigl{[}1+(-1)^{\nu_{1}}a_{x}+(-1)^{\mu_{1}+\nu_{1}+1}a_{y}+(-1)^{\mu_{1}}a_{z}+(-1)^{\nu_{2}}b_{x}$$

$$+(-1{)}^{{\mu }_2+{\nu }_2+1}{b}_y+(-1{)}^{{\mu }_2}{b}_z+(-1{)}^{{\nu }_1+{\nu }_2}{c}_{xx}+(-1{)}^{{\nu }_1+{\mu }_2+{\nu }_2+1}{c}_{xy}+(-1{)}^{{\nu }_1+{\mu }_2}{c}_{xz}$$

$$+(-1{)}^{{\mu }_1+{\nu }_1+{\nu }_2+1}{c}_{yx}+(-1{)}^{{\mu }_1+{\nu }_1+{\mu }_2+{\nu }_2}{c}_{yy}+(-1{)}^{{\mu }_1+{\nu }_1+{\mu }_2+1}{c}_{yz}$$

$$\displaystyle\hskip 28.45274pt+(-1)^{\mu_{1}+\nu_{2}}c_{zx}+(-1)^{\mu_{1}+\mu_{2}+\nu_{2}+1}c_{zy}+(-1)^{\mu_{1}+\mu_{2}}c_{zz}\bigr{]},$$

$$\frac{1}{4}\sum _{{\mu }_1,{\nu }_1,{\mu }_2,{\nu }_2}W({\mu }_1,{\nu }_1,{\mu }_2,{\nu }_2)=1$$

$$\displaystyle\mathcal{W}_{\mathtt{R}}(\mu_{1},\nu_{1})\mathrel{\mathop{:}}=\mbox{${\rm Tr}$}[\hat{G}^{\dagger}(\mu_{1},\nu_{1})\mbox{$\hat{\rho}$}_{\mathtt{R}}^{(1)}]=\frac{1}{2}\left[1+(-1)^{\nu_{1}}a_{x}+(-1)^{\mu_{1}+\nu_{1}+1}a_{y}+(-1)^{\mu_{1}}a_{z}\right],$$

Full text

Quantum Information and Computation, Vol. 0, No. 0 (2019) 000–000

©Rinton Press

REPRESENTATIONS OF TWO-QUBIT AND QUQUART STATES

VIA DISCRETE WIGNER FUNCTIONS

MARCELO A. MARCHIOLLI

Avenida General Osório 414, centro, 14.870-100 Jaboticabal, São Paulo, Brazil

E-mail: marcelo$\_$march@bol.com.br

DIÓGENES GALETTI

Instituto de Física Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271

Bloco II, Barra Funda, 01140-070 São Paulo, São Paulo, Brazil

E-mail: diogaletti@hotmail.com

Received (received date)

Revised (revised date)

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 tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.

*Keywords: Finite-dimensional Discrete Phase Spaces, Discrete Wigner Function,

                Two-qubit States, Ququart States, Two-qubit X-states, Entanglement*

Communicated by: to be filled by the Editorial

1 Introduction

Despite the heated debates occurred in 1935 on the “spooky” feature of quantum theory [2, 3], nowadays it is well-accepted that entanglement essentially corresponds to a holistic property of multipartite quantum systems whose genuinely quantum correlations represent not only an important physical resource for many quantum processes (such as, for instance, quantum cryptography, quantum teleportation, and dense coding), but also a fundamental feature for our understanding on how Nature behaves at the microscopic and/or mesoscopic levels. Due its nontrivial structure, quantum entanglement suffers of certain inexorable effects (with deleterious consequences) and intrinsic limitations: it is highly sensitive to environment and does not increase on average when systems are distributed in spatially separated regions. Despite these inherent difficulties, both the theoretical advances [4, 5, 6, 7, 8, 9] and the outstanding experimental achievements[10, 11, 12, 13, 14, 15, 16] in the last decades reveal a very promising future for new quantum technologies based on the entanglement effects.

One of the fundamental concerns of quantum information theory, in particular the chapter associated with quantum entanglement, refers to characterize, control and quantify entanglement [4, 17, 18, 19, 20, 21, 22, 23, 24]. In particular, let us focus on bipartite systems constituted by subsystems of low dimensionality in Hilbert space, where some well-known entanglement measures (namely, concurrence, negativity, and relative entropy of entanglement) are widely used to characterize the underlying quantum states [25, 26]. Moreover, let us also adopt the theoretical framework established in [27] for finite-dimensional discrete phase spaces, which permits us to construct discrete Wigner functions related to the special unitary group $\mathrm{SU(N)}$. So, a pertinent question then naturally emerges from these considerations: “Can the discrete Wigner function be used as an effective mathematical tool in the analysis of quantum correlation effects in bipartite systems or even in single systems described by a finite number of levels?”

In this paper we introduce both the discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions associated with spin representations of two two-level systems and a single four-level system, respectively, which lead us to describe general two-qubit and ququart states. The theoretical background used to obtain these discrete distribution functions is based on a well-grounded mapping scheme between Schwinger unitary operators and generators of $\mathrm{SU(N)}$, which provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described essentially by finite-dimensional state vector spaces. This particular mapping scheme is performed by means of the $\mathit{mod(N)}$-invariant operator basis introduced in [28] and whose mathematical properties were extensively explored in [29], which allowed us to formulate a finite-dimensional phase space labelled by genuinely discrete variables [30].

Initially, we adopt the well-established Fano’s prescription for two-qubit states as pairs of two-level systems [31, 32] to obtain the corresponding discrete Wigner function that obeys the criterion ‘easy-to-handle’. As a first immediate by-product the discrete Wigner functions associated with the reduced density operators are promptly determined. For the sake of simplicity, we briefly introduce the two-qubit density matrix in the computational-basis representation and, as a consequence, we derive its discrete Wigner function expressed in terms of the matrix elements. Furthermore, we apply these results to the Bell states [17, 18, 19, 20, 21, 22, 23] and Werner states [33] in order to establish the first compact expressions for the respective discrete Wigner functions. However, these functions suffer a minor problem: they cannot be visualized in a unique phase space due to the initial decomposition of the mapping kernel in two kernels, each one responsible for a particular subsystem. To conclude, we discuss possible experimental scenarios where the discrete Wigner functions can be used as an effective theoretical tool to monitorate the entanglement between both the qubits.

Following, we focus on the generators of $\mathrm{SU(4)}$ and their connections with the Schwinger unitary operators, where it is possible to show that all the fifteen generators can be expressed as specific combinations of these unitary operators in a one-to-one correspondence. Hence, the discrete $\mathrm{SU(4)}$ Wigner function obtained from this mathematical procedure is quite general since it is written as a function of the density-matrix elements associated with an arbitrary four-level system. As an interesting application, we investigate a recent experiment involving Nuclear Magnetic Ressonance (NMR) techniques used to implement an oracle based quantum algorithm that solves a black-box problem faster than any classical counterpart by means of a single ququart [12]. Here, we show that the discrete Wigner functions related to each stage of the experiment are not mere figurative mathematical tools, but rather valuable and important theoretical instruments which allow to expand our knowledge on the physical processes involved. Next, we adopt the theoretical prescription described in Refs. [11, 34] to establish an isomorphic correspondence between ququart and two-qubit states, which allows us to rewrite the Fano’s decomposition for two-qubit density matrix in terms of the $\mathrm{SU(4)}$ generators by means of an adequate change of basis. This procedure leads us to solve the previous problem associated with the visualization of two-qubit discrete Wigner functions in finite-dimensional discrete phase spaces. To corroborate these results, we revisit once again the two-qubit Bell and Werner states with the aim of searching for a noticeable signature of the entanglement effect through their respective discrete $\mathrm{SU(4)}$ Wigner functions. Besides, the two-qubit X-states [25] are also investigated in this manuscript, where the corresponding discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions reveal new quite interesting results: it is possible to define a functional onto this particular finite phase space, which is responsible for recognizing the quantum correlations present in these states. Some relevant points associated with possible extensions to the discrete Husimi and Glauber-Sudarshan quasiprobability distribution functions, as well as potential applications on spin chains and definitions of fidelity, were also included in our discussions.

This paper is structured as follows. In Section 2, we fix the quantum-algebraic framework that paves the way to establish an important set of solid mathematical results associated with the discrete $\mathrm{SU(N)}$ Wigner functions. In Section 3, we present two different but complementary group-theoretical approaches which lead us to describe both the two-qubit and ququart states through the discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions. In addition, we also apply these results to study the entanglement effect in the two-qubit Bell and Werner states, as well as to analyse all the experimental stages of a particular NMR experiment involving ququarts via discrete Wigner functions. Next, Section 4 is dedicated to the study of two-qubit X-states, whose results allow to define a functional on the corresponding discrete phase space responsible for recognizing the quantum correlations exhibited by the X-states under scrutiny. Finally, in Section 5 we summarize our main results and discuss some possible avenues for future research. To conclude, Appendix A contains technical details on the $\mathrm{SU(4)}$ generators, their relations with the Schwinger unitary operators, and the respective mapped expressions in the corresponding finite-dimensional discrete phase space.

2 Preliminaries on the discrete Wigner functions for SU(N)

Let us initially introduce the density-matrix space for $N$-level quantum systems, here related to the Hilbert space $\mathcal{H}_{N}$, through the mathematical definition [35, 36]

[TABLE]

which embodies, in principle, three important requirements on the matrix density $\hat{\rho}$:

• $\bullet$

\mbox{{\rm Tr}}[\mbox{\hat{\rho}}]=1: its normalization is preserved;

• $\bullet$

\mbox{\hat{\rho}}=\mbox{\hat{\rho}}^{\dagger}: by definition, $\hat{\rho}$ must be a Hermitian matrix; and finally,

• $\bullet$

$\rho_{\ell}\in\mathbb{R}_{+}(\forall\ell)$: it must be positive semidefinite [37] (i.e., all its eigenvalues are positive).

This last requirement has become of vital importance in quantum information theory [38], since it permits to identify entangled states [39, 40] and classify quantum channels [41, 42, 43] by means of positive and completely positive maps. The further requirement \mbox{{\rm Tr}}[\mbox{\hat{\rho}}^{2}]\leq 1 can be interpreted as a direct consequence of \mbox{\hat{\rho}}=\mbox{\hat{\rho}}^{\dagger} for any \mbox{\hat{\rho}}\in\mathscr{L}_{+,1}(\mathcal{H}_{N}), being the saturation reached in this case only for pure states. In fact, this condition characterizes the Bloch-vector space to be in a ball in $\mathbb{R}^{N^{2}-1}$ [35]. Furthermore, the notation $\mathscr{L}(\mathcal{H}_{N})$ represents the set of linear operators on $\mathcal{H}_{N}$.

The next step consists in considering the complete orthonormal operator basis constituted by $N^{2}-1$ generators $\{\hat{g}_{i}\}_{i=1,\ldots,N^{2}-1}$ associated with the special unitary group $\mathrm{SU(N)}$, which are characterized by $N\times N$ skew-Hermitian matrices satisfying a special set of mathematical relations [44, 45, 46]:

• $\bullet$

Basic rules

[TABLE]

• $\bullet$

Commutation relation

[TABLE]

• $\bullet$

Anticommutation relation

[TABLE]

where $\hat{\mathds{I}}_{N}$ denotes the $N$-dimensional unit matrix;

• $\bullet$

Jacobi identity

[TABLE]

• $\bullet$

Trace of products

[TABLE]

Note that $\mathscr{F}_{ijk}$ and $\mathscr{D}_{ijk}$ are well-known constants in literature [46, 47] and whose respective values can be found in tabulated form for different values of $N$. Therefore, any linear operator $\hat{O}$ can be decomposed in terms of the elements arising from such operator basis,

[TABLE]

with \mathcal{O}_{i}\equiv\mbox{{\rm Tr}}[\hat{g}_{i}\hat{O}] representing the $N^{2}-1$ coefficients of the expansion. In particular, if one considers the $N$-level quantum systems, the associated density matrix \mbox{\hat{\rho}}\in\mathscr{L}_{+,1}(\mathcal{H}_{N}) admits the following expression [44]:

[TABLE]

where the mean values \langle\hat{g}_{i}\rangle\equiv\mbox{{\rm Tr}}[\hat{g}_{i}\mbox{\hat{\rho}}] are the components of the generalized Bloch vector

[TABLE]

In this algebraic approach, the condition

[TABLE]

allows us to determine if a given density matrix (2) describes pure or mixed states: indeed, from the experimental point of view, it is sufficient to measure the length of the generalized Bloch vector in such a case [48]. The constructive aspects of the generators of $\mathrm{SU(N)}$ can be properly found in Refs. [44, 46].

Now, let us introduce the $\mathit{mod(N)}$-invariant unitary operator basis [28]

[TABLE]

written in terms of the discrete Fourier transform of the symmetrized basis

[TABLE]

where $\hat{U}$ and $\hat{V}$ correspond to the Schwinger unitary operators [49] defined in an $N$-dimensional state vector space, whose mathematical properties were extensively explored in Refs. [29, 50]. It is worth stressing that the extra phase $\Phi(\eta,\xi;N)=NI_{\eta}^{N}I_{\xi}^{N}-\eta I_{\xi}^{N}-\xi I_{\eta}^{N}$ appearing in (4) is responsible for the $\mathit{mod(N)}$-invariance property of this operator basis, $I_{\varepsilon}^{N}=\left[\frac{\varepsilon}{N}\right]$ being the integer part of $\varepsilon$ with respect to $N$. Thus, as expected from a well-grounded unitary operator basis, the decomposition of any linear operator can also be promptly established in this case, that is

[TABLE]

The coefficients \mathrm{O}(\mu,\nu)=\mbox{{\rm Tr}}[\hat{G}^{\dagger}(\mu,\nu)\hat{O}] show a one-to-one correspondence between operators and functions belonging to an $N^{2}$-dimensional discrete phase space. For \hat{O}\equiv\mbox{\hat{\rho}}\in\mathscr{L}_{+,1}(\mathcal{H}_{N}), this particular decomposition assumes the compact form

[TABLE]

with W(\mu,\nu)\mathrel{\mathop{:}}=\mbox{{\rm Tr}}[\hat{G}^{\dagger}(\mu,\nu)\mbox{\hat{\rho}}] formally defining the discrete Wigner function [28]. Henceforth, the finite-dimensional discrete phase space will represent a finite mesh with $N^{2}$ points labelled by genuinely discrete variables [30].

How the discrete Wigner function for $\mathrm{SU(N)}$ can be constructed out from these theoretical frameworks? This question was properly answered in [27] through the connections established between generators of the group $\mathrm{SU(N)}$ and Schwinger unitary operators via $\mathit{mod(N)}$-invariant operator basis, that is, for each generator $\hat{g}_{i}$ exists a one-to-one correspondence with a given decomposition of unitary operators reached, by its turns, through the discrete mapping kernel (4). Pursuing this guideline, the discrete Wigner function for $\mathrm{SU(N)}$ is defined as follows:

[TABLE]

where \left(\hat{g}_{i}\right)\!(\mu,\nu)=\mbox{{\rm Tr}}[\hat{G}^{\dagger}(\mu,\nu)\hat{g}_{i}] corresponds to the representatives in the finite-dimensional discrete phase space of the generators $\{\hat{g}_{i}\}_{i=1,\ldots,N^{2}-1}$. This elegant and compact mathematical result represents an alternative approach to those recently proposed by Tilma, Everitt et al. [51, 52] for Wigner functions with continuous representations (through coherent states or Euler angles): in fact, Eq. (7) describes the Wigner function defined upon a finite-dimensional phase space labelled by genuine discrete variables associated with spin representations.

To illustrate such argument, let us now consider, from the non-relativistic quantum theory point of view, the group $\mathrm{SU(2)}$ and its corresponding generators $\{\hat{\sigma}_{i}\}_{i=x,y,z}$, where $\hat{\sigma}_{i}$ denotes the Pauli matrices. Thus, it is possible to demonstrate that $\hat{\sigma}_{i}=\hat{V}\delta_{ix}-{\rm i}\hat{U}\hat{V}\delta_{iy}+\hat{U}\delta_{iz}$ indeed establishes the correspondence, once both the operators share the same set of orthonormal eigenvectorsaaaSimilar relations were already obtained for the Gell-Mann matrices $\lambda$’s associated with the group $\mathrm{SU(3)}$ – see Ref. [27] for technical details.. In this particular example, the discrete Wigner function is given by

[TABLE]

with P_{i}=\mbox{{\rm Tr}}[\mbox{\hat{\rho}}\hat{\sigma}_{i}]\in[-1,1] for $i=x,y,z$ corresponding to the polarization-vector components which obey the relation $P_{x}^{2}+P_{y}^{2}+P_{z}^{2}\leq 1$ (the saturation occurs only for pure states). Table 2 shows all the possible values of $W(\mu,\nu)$ in the finite-dimensional phase space here labelled by the discrete variables $0\leq\mu,\nu\leq 1$ with respect to (w.r.t.) $(P_{x},P_{y},P_{z})$ and also as a function of the matrix elements $(\rho_{11},\rho_{12},\rho_{22})$, once that (P_{x},P_{y},P_{z})=\left(2\mbox{{\mathtt{Re}}}(\rho_{12}),-2\mbox{{\mathtt{Im}}}(\rho_{12}),\rho_{11}-\rho_{22}\right). As expected, the absolute minimum values reached by (8) happen for pure states.

3 Description of two-qubit and ququart states via discrete Wigner functions

In this section, we establish two different but complementary group-theoretical approaches to describe two-qubit and ququart states through discrete Wigner functions: the first approach considers the Klein’s group $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ and encompasses the Fano’s description for two-qubit states as pairs of two-level systems [31, 32], whereas the second one embodies the group $\mathrm{SU(4)}$ in order to describe ququart states as a single four-level system. Besides, we obtain the respective exact discrete Wigner functions associated with a finite-dimensional discrete phase space.

3.1 The Klein’s group

Initially, let us mention that $\{\hat{\mathds{I}}_{2}^{(1)},\hat{\sigma}_{x}^{(1)},\hat{\sigma}_{y}^{(1)},\hat{\sigma}_{z}^{(1)}\}\otimes\{\hat{\mathds{I}}_{2}^{(2)},\hat{\sigma}_{x}^{(2)},\hat{\sigma}_{y}^{(2)},\hat{\sigma}_{z}^{(2)}\}$ represents the operator basis used to describe a completely general two-qubit state, where the superscripts $(1)$ and $(2)$ correspond to the qubits $1$ and $2$. Following, let us adopt the Fano’s prescription for the density matrix [32]

[TABLE]

with real coefficientsbbbAlthough this operator basis is expressed as a tensor product, this fact does not imply that $\hat{\rho}$ can be decomposed as \mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)}: indeed, only for $a_{i}=P_{i}^{(1)}$, $b_{j}=P_{j}^{(2)}$ and $c_{ij}=P_{i}^{(1)}P_{j}^{(2)}$, the condition \mbox{\hat{\rho}}=\mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)} is verified; otherwise, we obtain \mbox{\hat{\rho}}\neq\mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)}. This genuinely quantum property is essential for characterizing, in such a case, the bipartite states in separable states (\mbox{\hat{\rho}}=\mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)}) and entangled states (\mbox{\hat{\rho}}\neq\mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)}), with immediate implications in quantum mechanics and quantum information theory [4, 8]., where \mbox{\hat{\rho}}\in\mathscr{L}_{+,1}(\mathcal{H}_{2}\otimes\mathcal{H}_{2}). Note that the positivity of all four eigenvalues (necessary condition to ensure that $\hat{\rho}$ is positive semidefinite) is reached through the solution of the following nontrivial set of inequalities [36]:

[TABLE]

For the sake of completeness, the first inequality

[TABLE]

consists of a mathematical condition that distinguishes between mixed and pure states, with the saturation occurring only for pure states. In addition, the reduced density matrix related to the qubit $1(2)$ is obtained from $\hat{\rho}$ by taking the partial trace over the subspace of the qubit $2(1)$, namely,

[TABLE]

and

[TABLE]

These results determine only the partial information of the bipartite system $\hat{\rho}$ under scrutiny since they see only the quantum state of a given subsystem [18].

Now, let us calculate the discrete Wigner function associated with the density matrix (9). For such a particular task, we initially consider the mapping kernel $\hat{G}^{\dagger}(\mu_{a},\nu_{a})$, here related to the subspaces $a=1,2$ of each qubit, in order to obtain a first expression for

[TABLE]

Next, substituting Eq. (9) in this definition, we get the intermediate result

[TABLE]

where the terms \mbox{{\rm Tr}}[\hat{G}^{\dagger}(\mu_{a},\nu_{a})\hat{\sigma}_{\ell}^{(a)}] for $a=1,2$ and $\ell=x,y,z$ are determined as follows:

[TABLE]

So, after a few algebraic manipulations, the discrete Wigner function $W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})$ achieves the general expression

[TABLE]

whose normalization condition

[TABLE]

can be promptly verified through the results showed in Table 3.1. However, Eq. (3.1) presents an apparent disadvantage: the visualization of this function in the finite-dimensional phase space labelled by the discrete variables $(\mu_{1},\nu_{1},\mu_{2},\nu_{2})$ is not fully functional since only bidimensional projections are easily manageable.

As a further remark, let us now determine the discrete Wigner functions associated with the reduced density matrices (14) and (15), i.e.,

[TABLE]

These results present, as expected, a particular connection with the partial sums of Eq. (3.1) through the relations

[TABLE]

The restriction $c_{ij}=a_{i}b_{j}$ for $i,j=x,y,z$ implies that $W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})$ splits into the product $\mathcal{W}_{\mathtt{R}}(\mu_{1},\nu_{1})\mathcal{W}_{\mathtt{R}}(\mu_{2},\nu_{2})$, this condition being directly related to the separable states.

3.1.1 The computational basis

In general, an arbitrary two-qubit state is defined as a linear superposition of the (orthonormalized) computational-basis states [23]

[TABLE]

namely,

[TABLE]

with $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ and $|\alpha|^{2}+|\beta|^{2}+|\gamma|^{2}+|\delta|^{2}=1$. From an operational point of view, it is common to deal with the matrix representation for \mbox{\hat{\rho}}=|\Psi\rangle\langle\Psi| in the computacional basis, which leads us to obtain the positive semidefinite $4\times 4$ Hermitian matrix

[TABLE]

such that \mbox{{\rm Tr}}[\mbox{\hat{\rho}}]=1. Note that Eqs. (9) and (18) are connected by means of a change of basis: indeed, for $\hat{\mathds{I}}_{2}^{(a)}=|0_{a}\rangle\langle 0_{a}|+|1_{a}\rangle\langle 1_{a}|$, $\hat{\sigma}_{x}^{(a)}=|0_{a}\rangle\langle 1_{a}|+|1_{a}\rangle\langle 0_{a}|$, $\hat{\sigma}_{y}^{(a)}=-{\rm i}(|0_{a}\rangle\langle 1_{a}|-|1_{a}\rangle\langle 0_{a}|)$, and $\hat{\sigma}_{z}^{(a)}=|0_{a}\rangle\langle 0_{a}|-|1_{a}\rangle\langle 1_{a}|$ (with $a=1,2$), the first one can be written in a similar fashion as the second one or vice-versa, where now the elements $\rho_{ij}$ for $i\leq j$ are evaluated as follows:

[TABLE]

This system of linear equations is invertible and this fact allows us to rewrite $W(\mu_{1},\nu_{1},\mu_{2},\nu_{2})$ as a function of the matrix elements appeared in (18),

[TABLE]

where

[TABLE]

Table 3.1.1 shows all the possible values of Eq. (3.1.1) in the finite-dimensional discrete phase space labelled by $0\leq\mu_{1},\nu_{1},\mu_{2},\nu_{2}\leq 1$. Such a compilation of results allows us to demonstrate that this discrete quasiprobability distribution function obeys the criterion easy-to-handle.

It is worth stressing that Eq. (17) does not necessarily imply the decomposition

[TABLE]

which describes a separable pure two-qubit state. Indeed, this case occurs only for the coefficients $\alpha=a_{1}a_{2}$, $\beta=a_{1}b_{2}$, $\gamma=b_{1}a_{2}$, and $\delta=b_{1}b_{2}$; otherwise, $|\Psi\rangle$ characterizes an entangled two-qubit state (that is, $|\Psi\rangle\neq|\psi^{(1)}\rangle\otimes|\psi^{(2)}\rangle$). In addition, the associated density matrix (18) leads us to obtain

[TABLE]

the saturation arising only for pure states – note that both the discrete Wigner functions (3.1) and (3.1.1) can also be used to evaluate the expression for \mbox{{\rm Tr}}[\mbox{\hat{\rho}}^{2}] since that

[TABLE]

Summarizing, there are four different possibilities of describing an arbitrary bipartite quantum state [39, 40], being the two-qubit quantum state a legitimate representative of this general case: entangled pure state, entangled mixed state, separable pure state, and finally, separable mixed state. Next, we will illustrate these cases through well-established examples in the literature with the help of the associated discrete Wigner functions.

3.1.2 Applications

The first example of entangled pure two-qubit states are the Bell states $|\Psi_{\pm}\rangle$ and $|\Phi_{\pm}\rangle$, defined by [18, 23]

[TABLE]

whose respective density operators assume the following forms:

[TABLE]

Once the coefficients of (20) and (21) are completely determined, it turns immediate to obtain the corresponding discrete Wigner functions for each set of Bell states,

[TABLE]

By using the expressions obtained in the right-hand-side of Table 3.1.1, it is possible to show that Eqs. (22) and (23) assume only two unique values for any $0\leq\mu_{1},\nu_{1},\mu_{2},\nu_{2}\leq 1$: $+\frac{1}{2}$ and $-\frac{1}{2}$. In addition, note that $|\Psi_{\pm}\rangle$ and $|\Phi_{\pm}\rangle$ are indeed maximally entangled pure states since their associated reduced density matrices present the same value $\frac{1}{2}\hat{\mathds{I}}_{2}$, that is, the states $|0_{1(2)}\rangle$ and $|1_{1(2)}\rangle$ are equally likely to be found with the same probability $\frac{1}{2}$. In this regard, the discrete Wigner functions established in our analysis possess an important rule: they permit to show that the relationscccOriginally, these relations for the discrete Wigner functions are associated with the two-qubit density-matrix decomposition \mbox{\hat{\rho}}=\mbox{\hat{\rho}}^{(1)}\otimes\mbox{\hat{\rho}}^{(2)}+\hat{\Delta}, such that \mbox{{\rm Tr}}_{1}[\hat{\Delta}]=\mbox{{\rm Tr}}_{2}[\Delta]=0\,\hat{\mathds{I}}_{2}, where $\hat{\Delta}$ contains, in principle, all the possible classical and quantum correlations admitted by \mbox{\hat{\rho}}^{(1)} and \mbox{\hat{\rho}}^{(2)} [53].

[TABLE]

and

[TABLE]

prevail for all the finite-dimensional discrete phase space, which represent a genuine signature of the states under scrutiny [54].

As an interesting second example, let us consider the Werner states [33]

[TABLE]

which are defined as mixtures of the Bell states, where the parameter $\mathcal{F}$ determines the degree of mixing for these states. Substituting Eqs. (20) and (21) in this definition, it is quite easy to show that

[TABLE]

whose reduced density matrices coincide with $\frac{1}{2}\hat{\mathds{I}}_{2}$ and do not depend on $\mathcal{F}$. So, the expression for \mbox{{\rm Tr}}[\mbox{\hat{\rho}}_{\mathtt{W}}^{2}]=\frac{1}{3}(1-2\mathcal{F}+4\mathcal{F}^{2})\leq 1 leads us, in principle, to characterize this state as follows:

• •

$\mathcal{F}=1$: \mbox{\hat{\rho}}_{\mathtt{W}} describes a maximally entangled pure state;

• •

$\frac{1}{2}<\mathcal{F}<1$: according to the Peres-Horodecki criterion [39, 40], \mbox{\hat{\rho}}_{\mathtt{W}} basically depicts entangled mixed states;

• •

$\mathcal{F}=\frac{1}{2}$: \mbox{\hat{\rho}}_{\mathtt{W}} corresponds to a separable mixed state [18]; and finally,

• •

$\mathcal{F}=\frac{1}{4}$: the states $|0_{1}0_{2}\rangle$, $|0_{1}1_{2}\rangle$, $|1_{1}0_{2}\rangle$, and $|1_{1}1_{2}\rangle$ are equally likely to be found (in this case, with the same probability $\frac{1}{4}$), since \mbox{\hat{\rho}}_{\mathtt{W}} is given by $\frac{1}{4}\hat{\mathds{I}}_{4}$.

With respect to its discrete Wigner function

[TABLE]

it assumes two distinct values, $\frac{1}{6}+\frac{\mathcal{F}}{3}$ and $\frac{1}{2}-\mathcal{F}$, which change according to $\mathcal{F}\in[0,1]$ and $0\leq\mu_{1},\nu_{1},\mu_{2},\nu_{2}\leq 1$: for instance, $\mathcal{F}=0$ gives $W_{\mathtt{W}}=\frac{1}{6}$ or $\frac{1}{2}$, while $\mathcal{F}=1$ leads to $W_{\mathtt{W}}=-\frac{1}{2}$ or $\frac{1}{2}$; however, if one considers $\mathcal{F}=\frac{1}{2}$, we obtain $W_{\mathtt{W}}=0$ or $\frac{1}{3}$. Moreover, it is also important to observe that $\Delta_{\mathtt{W}}(\mu_{1},\nu_{1},\mu_{2},\nu_{2})$ admits the following situations:

• •

$\mathcal{F}=1$: $\Delta_{\mathtt{W}}=-\frac{3}{4}$ or $\frac{1}{4}$ (see previous example);

• •

$\mathcal{F}=\frac{1}{2}$: $\Delta_{\mathtt{W}}=-\frac{1}{4}$ or $\frac{1}{12}$ (separable mixed state);

• •

$\mathcal{F}=\frac{1}{4}$: $\Delta_{\mathtt{W}}=0$ ($W_{\mathtt{W}}$ constant and equal to $\frac{1}{4}$);

• •

$\mathcal{F}=0$: $\Delta_{\mathtt{W}}=-\frac{1}{12}$ or $\frac{1}{4}$; otherwise, $\Delta_{\mathtt{W}}=\frac{\mathcal{F}}{3}-\frac{1}{12}$ or $\frac{1}{4}-\mathcal{F}$.

Further results related to the Werner states in connection with its local and nonlocal properties can be promptly found in Ref. [55].

Similar analysis via discrete Wigner functions can also be applied to the isotropic states [56] and maximally entangled mixed states [57, 58, 59, 60], or even in the investigation on the connection between entangled states and the closest disentangled states [61]. On the other hand, we left behind the strong visual appeal associated with the discrete Wigner functions in this Klein’s group approach, at the cost of obtaining a theoretical framework completely compatible with the Fano’s prescription for two-qubit density matrix (9). This disadvantage does not represent a complicated problem for our considerations, since it can be apparently solved through the use of Eq. (7) and the generators of $\mathrm{SU(4)}$.

From a practical point of view, Nuclear Magnetic Ressonance (NMR) actually corresponds to one of the possible experimental tecniques [62] which can be used for reconstructing both the density matrices (9) and (18) associated with two-qubit states and consequently, to directly determine the respective discrete Wigner functions (3.1) and (3.1.1). Recently, Micadei et al. [15] conducted an important experimental investigation on the reversal of heat flow between two initially quantum-correlated qubits prepared in local thermal states at different temperatures, basically employing the aforementioned experimental technique. In fact, the authors employed the quantum-state tomography [62] in order to reconstruct the global two-qubit density matrix and then calculate the changes of internal energies of each qubit during the thermal contact. In this experimental approach, we argue that discrete Wigner functions should be used as an effective theoretical tool to monitorate the pre-existing correlations between both the qubits.

3.2 The group $\mathrm{SU(4)}$

Initially, let us consider the discrete Wigner function (7) with $N=4$ fixed, which is equivalent to construct the function

[TABLE]

for a given four-level quantum system, which is described by the density matrix \mbox{\hat{\varrho}}\in\mathscr{L}_{+,1}(\mathcal{H}_{4}) explicitly presented in appendix A. Note that $\{\hat{g}_{i}\}_{i=1,\ldots,15}$ denote the generators of the special unitary group $\mathrm{SU(4)}$ and represent the building blocks of this fundamental process. In addition, $\langle\hat{g}_{i}\rangle$ and $\left(\hat{g}_{i}\right)\!(\mu,\nu)$ were previously defined in section 2 and promptly calculated in appendix A. These results provide a completely general discrete Wigner function for $\mathrm{SU(4)}$, that is

[TABLE]

where the superscript $[4]$ on the Kronecker deltas denotes that this function is different from zero when its labels are $\mathit{mod}(4)$-congruent. Table 3.2 presents all the possible values that Eq. (28) assumes in the finite-dimensional discrete phase space.

Recently, Gedik et al. [12] showed that a single ququart is enough to implement an oracle based quantum algorithm that solves a black-box problem faster than any classical algorithm. In this experimental approach, the main idea is to determine the parity of a cyclic permutation of the elements $\{|0\rangle,|1\rangle,|2\rangle,|3\rangle\}$ through a single evaluation of the permutation function using as initial state the ququart $|\psi_{1}\rangle=\hat{\mathfrak{F}}|1\rangle=\frac{1}{2}\left(|0\rangle+{\rm i}|1\rangle-|2\rangle-{\rm i}|3\rangle\right)$, where

[TABLE]

denotes the discrete Fourier operator for $\mathrm{SU(4)}$ such that $\hat{\mathfrak{F}}^{4}=\hat{\mathds{I}}_{4}$ and $\hat{\mathfrak{F}}\hat{\mathfrak{F}}^{\dagger}=\hat{\mathfrak{F}}^{\dagger}\hat{\mathfrak{F}}=\hat{\mathds{I}}_{4}$. Now, let us discuss such experimental approach by means of the schematic diagram exhibited below:

[TABLE] \textstyle{\hat{\mathcal{U}}_{6}|\psi_{1}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\mathfrak{F}}^{\dagger}}$$\textstyle{\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{6}|\psi_{1}\rangle=\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{6}\hat{\mathfrak{F}}|1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|3\rangle\langle 3|\;(\texttt{negative})}$$\textstyle{|1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\mathfrak{F}}}$$\textstyle{\hat{\mathfrak{F}}|1\rangle\equiv|\psi_{1}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\mathcal{U}}_{6}}$$\scriptstyle{\hat{\mathcal{U}}_{2}}$$\textstyle{\hat{\mathcal{U}}_{2}|\psi_{1}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\mathfrak{F}}^{\dagger}}$$\textstyle{\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{2}|\psi_{1}\rangle=\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{2}\hat{\mathfrak{F}}|1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{|1\rangle\langle 1|\;(\texttt{positive})}

step 1.

Basically, this first step is responsible for creating the initial state $|1\rangle$;

step 2.

the next one applies the discrete Fourier operator $\hat{\mathfrak{F}}$ upon the initial state $|1\rangle$ in order to obtain $|\psi_{1}\rangle=\hat{\mathfrak{F}}|1\rangle$; following,

step 3.

two different pulses are then applied on $|\psi_{1}\rangle$ with the aim of producing the unitary matrices

[TABLE]

i.e., for $\hat{\mathcal{U}}_{2}|\psi_{1}\rangle$ we obtain $-{\rm i}|\psi_{1}\rangle$, while $\hat{\mathcal{U}}_{6}|\psi_{1}\rangle$ gives $-\frac{1}{2}\left(|0\rangle-{\rm i}|1\rangle-|2\rangle+{\rm i}|3\rangle\right)$;

step 4.

as a subsequent step we apply, once again, the operator $\hat{\mathfrak{F}}^{\dagger}$ on each one of the resulting states, which implies in $\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{2}|\psi_{1}\rangle=-{\rm i}|1\rangle$ and $\hat{\mathfrak{F}}^{\dagger}\hat{\mathcal{U}}_{6}|\psi_{1}\rangle=-|3\rangle$; finally,

step 5.

we measure both the possibilities through its respective density matrices. In particular, the measurements of $|1\rangle\langle 1|$ and $|3\rangle\langle 3|$ correspond to the positive and negative cyclic permutations.

In fact, all the measurements performed in the experiment are associated with the tomographic reconstruction of the density matrix for each aforementioned step with errors always smaller than $6\%$. In order to illustrate part of these experimental steps, we have performed numerical calculations that lead us to show three-dimensional plots of the discrete Wigner function (28) for all the range $0\leq\mu,\nu\leq 3$ – see Figure 3.2. These pictures clearly demonstrate the relevant role of discrete $\mathrm{SU(4)}$ Wigner functions in the comprehension of the physical processes involved in each step: for instance, the effects of unitary transformations $\hat{\mathfrak{F}}$ and $\hat{\mathcal{U}}_{6}$ on the respective states $|1\rangle$ and $|\psi_{1}\rangle$ are quite significant and visually different, i.e., the first unitary operation shuffles the state $|1\rangle$ (thereby producing state $|\psi_{1}\rangle$) – see picture (b) – while the second one promotes the exchange of states by means of displacements in the discrete phase space – see picture (c). In this context, these functions are not mere figurative mathematical tools, but rather valuable theoretical instruments that allow to increase our knowledge on the physical processes involved.

Nowadays, there are different experimental arrangements which produce ququarts of photons [10] as well as two entangled qudits [13] (each qudit encodes a ten-dimensional state), with the objective of performing information-processing tasks related to quantum information theory. Moreover, Hu and coworkers [14] used a path-polarization hybrid system to generate high-dimensional entangled states (in this case, two entangled ququarts with high quality) in order to beat the channel capacity limit for superdense coding. These different experimental frameworks represent an interesting scenario for future research in theoretical physics where the discrete $\mathrm{SU(N)}$ Wigner function takes place.

3.2.1 Change of basis

Let us consider once again the Fano’s prescription for the density matrix $\hat{\rho}$ and its corresponding discrete Wigner function (3.1), to solve an apparent difficulty associated with the visualization of this function in the finite-dimensional discrete phase space. For such a particular task, let us adopt the isomorphic correspondence between ququart and two-qubit states, in according to the theoretical prescription established in Refs. [11, 34], namely, in this case an isomorphic correspondence between the stationary energy states of a four-level system and the two-qubit computational basis: $|0\rangle\leftrightarrow|0_{1}0_{2}\rangle$, $|1\rangle\leftrightarrow|0_{1}1_{2}\rangle$, $|2\rangle\leftrightarrow|1_{1}0_{2}\rangle$, and $|3\rangle\leftrightarrow|1_{1}1_{2}\rangle$. Therefore, the task consists in establishing a connection between Eqs. (2) for $N=4$ and (9) under specific conditions; otherwise, this correspondence \mbox{\hat{\rho}}\leftrightarrow\hat{\varrho} must be clearly stated.

In this sense, let us perform a change of basis in (9) through the auxiliary results

[TABLE]

which were obtained with the help of Eq. (1). In particular, these results permit us to rewrite (9) in the compact form

[TABLE]

whose coefficients are given by

[TABLE]

As expected, the matrix elements in the computational basis reproduce exactly those obtained in Eq. (18), which leads us to establish the correspondence with (A.2).

3.2.2 Revisiting two-qubit Bell and Werner states

As a first application, let us consider once again the density matrices (20) and (21) associated with the Bell states $\Psi_{\pm}$ and $\Phi_{\pm}$, respectively. So, the corresponding discrete Wigner functions in these cases have the forms

[TABLE]

and

[TABLE]

whose three-dimensional representations in finite-dimensional discrete phase space were depicted by Fig. 3.2.2 as follows: (a) $W_{\Psi_{+}}(\mu,\nu)$, (b) $W_{\Psi_{-}}(\mu,\nu)$, (c) $W_{\Phi_{+}}(\mu,\nu)$, and (d) $W_{\Phi_{-}}(\mu,\nu)$. To start with the analysis of the numerical results sketched in these pictures, let us consider the aforementioned correspondence between two-qubit states and four-level systems with emphasis on two-qubit Bell states written in terms of the computational basis of a ququart,

[TABLE]

In this description, the states $\Psi_{\pm}$ correspond to a four-level system where only two levels are accessed: in such a case, the states $|1\rangle$ and $|2\rangle$ are equally populated in the ratio $\frac{1}{2}$ with the same transition rates. So, the maximum values observed in pictures (a) and (b) are coincident and equal to $\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2+\sqrt{2}}{2}}\approx 1.153$, while the minimum values assume $-\frac{1}{2}\sqrt{\frac{2-\sqrt{2}}{2}}\approx-0.271$. Table 3.2 allows us to show that contributions associated with $\mu=0,3$ fixed and $\nu=0,1,2,3$ have zero sum, namely, the non-accessed states $|0\rangle$ and $|3\rangle$ present only quantum fluctuations; in addition, the contributions for $\mu=1,2$ fixed and $\nu=0,1,2,3$ exhibit non-zero sum because they are directly connected in this case with the population and transition rates of the states $|1\rangle$ and $|2\rangle$. The negative signal present in $\Psi_{-}$ stands for an interchange between $\nu=0$ and $\nu=2$ for any $\mu=0,1,2,3$. Similar analysis can also be applied in pictures (c) and (d), where now only the states $|0\rangle$ and $|3\rangle$ are accessed, with the maximum and minimum values being given for $W_{\Phi_{\pm}}(\mu,\nu)$ by $\frac{1}{2}+\frac{1}{2}\sqrt{\frac{2-\sqrt{2}}{2}}\approx 0.771$ and $-\frac{1}{2}\sqrt{\frac{2+\sqrt{2}}{2}}\approx-0.653$. It is worth to observe the pronounced quantum fluctuations when we deal with the non-accessed states $|1\rangle$ and $|2\rangle$ for $\mu=1,2$ fixed and $\nu=0,1,2,3$.

Let us now consider the Werner states (24) written in terms of the $\mathrm{SU(4)}$ computational basis as follows:

[TABLE]

whose matrix representation assumes the form

[TABLE]

In this particular four-level system, we have the states $|0\rangle$ and $|3\rangle$ having the same population rate of $\frac{1-\mathcal{F}}{3}$, the states $|1\rangle$ and $|2\rangle$ with $\frac{1+2\mathcal{F}}{6}$, and both the transitions $1\leftrightarrow 2$ sharing the rate $\frac{1-4\mathcal{F}}{6}$. The corresponding discrete $\mathrm{SU(4)}$ Wigner function for $\mathcal{F}\in[0,1]$ is given by

[TABLE]

such that $\mathcal{F}=1$ implies in $W_{\mathtt{W}}(\mu,\nu)=W_{\Psi_{-}}(\mu,\nu)$. Thus, the analysis on this four-level system is very similar to that already performed in the previous example: the basic difference is the presence of contributions related to the states $|0\rangle$ and $|3\rangle$ with equal weights. Figure 3.2.2 exhibits the 3D plots of Eq. (42) for different values of $\mathcal{F}\in[0,1]$, where (a) $\mathcal{F}=0.35$, (b) $\mathcal{F}=0.50$, (c) $\mathcal{F}=0.75$, and (d) $\mathcal{F}=1$ were considered. The quantum fluctuation contributions observed in (d) for $\mu=0,3$ fixed and $\nu=0,1,2,3$ have null sum since only quantum effects inherent to the states $|1\rangle$ and $|2\rangle$ prevail.

The solution established for the visualization difficulty related to the discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ Wigner functions has, in the correspondence with four-level systems, an effective mathematical tool in the study of maximally entangled mixed states [57, 58, 59, 60] where, in particular, the two-qubit X-states take a special place [25, 63, 64, 65, 66, 67, 68]. Next, we will explore this fact to characterize the maximally entangled mixed two-qubit X-states through the discrete $\mathrm{SU(4)}$ Wigner functions.

4 Two-qubit X-states

Two-qubit X-states are an important family of quantum states belonging to a four-dimensional Hilbert space $\mathcal{H}_{4}$ characterized by a unique property: basically, they do not mix the subspaces $\mathsf{S}_{1}=\mathtt{Span}\left(|0_{1}0_{2}\rangle,|1_{1}1_{2}\rangle\right)$ and $\mathsf{S}_{2}=\mathtt{Span}\left(|0_{1}1_{2}\rangle,|1_{1}0_{2}\rangle\right)$. Then, if one considers the associated computational basis, these states have potentially nonzero density-matrix elements located on the main diagonal and antidiagonal as follows [69]:dddNote the resemblance of Eq. (43) with the alphabet letter X justifying, in this way, the nomenclature X-state. This particular characteristic can be extended in order to encompass more general situations, namely, every density matrix possessing nonzero terms along the main diagonal and antidiagonal refers to the X-form.

[TABLE]

It is worth stressing that two-qubit Bell and Werner states are particular examples of X-states. Morever, an important form of universality property with respect to two-qubit entanglement was properly established in [25] through a set of transcendental parameters inherent to the matrix elements present in (43): “for every two-qubit state, there is a corresponding two-qubit X-state of same spectrum and entanglement in accordance with three different entanglement measurements, that are concurrence, negativity, and relative entropy of entanglement.” Hence, there exists an entanglement-preserving unitary (EPU) transformation $\hat{\mathsf{U}}_{\mathrm{EPU}}$ that preserves the entanglement of the input state, that is \mbox{\hat{\rho}}_{\mathtt{X}}=\hat{\mathsf{U}}_{\mathrm{EPU}}\mbox{\hat{\rho}}\,\hat{\mathsf{U}}^{\dagger}_{\mathrm{EPU}}, this property being termed by ‘EPU equivalence’. After this, Hedemann [67] established a definitive prove on the existence of such transformations obtaining, in this way, a compact implicit solution for them; furthermore, the author also provided an explicit form for the corresponding two-qubit X-state family. Now, let us also mention that such X-states were recently used in the study of certain spin chains with emphasis on the thermal entanglement properties and quantum discord related to these models [70, 71, 72].

Following, it is worth stressing that \mbox{\hat{\rho}}_{\mathtt{X}} represents a particular case of those general two-qubit states studied until the present moment, once we have now $\rho_{12}=\rho_{13}=\rho_{24}=\rho_{34}=0$. So, in order to establish the discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ Wigner function for two-qubit X-states, let us rewrite Eq. (3.1.1) as follows:

[TABLE]

where the $\Gamma$’s functions have already been previously defined. Therefore, the discrete Wigner functions $\mathcal{W}_{\mathtt{R}}(\mu_{1},\nu_{1})$ and $\mathcal{W}_{\mathtt{R}}(\mu_{2},\nu_{2})$ assume in this context the simplified forms

[TABLE]

and

[TABLE]

which do not depend on the discrete variables $\nu_{1}$ and $\nu_{2}$, respectively.

However, if one considers the discrete $\mathrm{SU(4)}$ Wigner function, it can be promptly obtained from Eq. (28) through a similar mathematical procedure, that is $\varrho_{12}=\varrho_{13}=\varrho_{24}=\varrho_{34}=0$,

[TABLE]

The discrete marginal distribution functions

[TABLE]

complete our description of discrete Wigner functions related to the two-qubit X-states, when it is possible to show that

[TABLE]

and

[TABLE]

satisfy the relations

[TABLE]

It is important to stress that $Q_{\mathtt{X}}(\mu)$ only depends on the matrix elements of the main diagonal, while $R_{\mathtt{X}}(\nu)$ brings information on the antidiagonal matrix elements. This important property associated with X-states can help us to comprehend the preexisting quantum correlations in these states by means of the difference $\Delta_{\mathtt{X}}(\mu,\nu)\mathrel{\mathop{:}}=W_{\mathtt{X}}(\mu,\nu)-Q_{\mathtt{X}}(\mu)R_{\mathtt{X}}(\nu)$, this function being responsible for distinguishing the effects of the main-diagonal and antidiagonal matrix elements on the aforementioned quantum correlations – see Table 4 for calculational details.

To illustrate these results, let us now consider the two-qubit X-states introduced by Munro et al. [73] with maximal concurrence for a given fixed purity $\gamma$,

[TABLE]

With respect to the correspondence \mbox{\hat{\rho}}\leftrightarrow\hat{\varrho} (see appendix A), Eq. (50) can be written as

[TABLE]

where it is clear the physical rules employed by the coefficients $g(\gamma)$ and $\frac{\gamma}{2}$: for instance, the case $\gamma=1$ deals with maximally entangled pure states, since \mbox{\hat{\rho}}_{\mathtt{X}} coincides with the Bell state \mbox{\hat{\rho}}_{\Phi_{+}}; otherwise, (50) describes maximally entangled mixed states. For this particular example under scrutiny, the discrete Wigner function (47) assumes the compact form

[TABLE]

while the discrete marginal distribution functions are given by

[TABLE]

Figure 4 shows the three-dimensional plots of discrete Wigner function (51) and the difference function $\Delta_{\mathtt{X}}(\mu,\nu)=W_{\mathtt{X}}(\mu,\nu)-Q_{\mathtt{X}}(\mu)R_{\mathtt{X}}(\nu)$ as a function of $0\leq\mu,\nu\leq 3$ for $\gamma=\frac{3}{4}$ and $\frac{1}{2}$. Pictures (a) $W_{\mathtt{X}}(\mu,\nu;\frac{3}{4})$ and (c) $W_{\mathtt{X}}(\mu,\nu;\frac{1}{2})$ describe maximally entangled mixed states with different values of purity $\gamma$, where two physical effects can be promptly verified: as $\gamma$ increases, the quantum fluctuations due to the states $|1\rangle$ and $|2\rangle$ – see $\mu=1,2$ and $\nu=0,1,2,3$ – are more visible, while for $\gamma=1$ these fluctuations exhibit equally contributions since we are now describing maximally entangled pure states (in such a case, the Bell state $|\Phi_{+}\rangle$); on the other hand, the quantum correlations present in (b) $\Delta_{\mathtt{X}}(\mu,\nu;\frac{3}{4})$ and (d) $\Delta_{\mathtt{X}}(\mu,\nu;\frac{1}{2})$ become more prominent and symmetric as $\gamma$ goes to 1. Indeed, for $\gamma=\frac{1}{2},\frac{3}{4},1$ fixed, we obtain

• •

$\Delta_{\mathtt{X}}(1,0;\frac{1}{2})\approx 0.26$, $\Delta_{\mathtt{X}}(1,2;\frac{1}{2})\approx-0.26$, $\Delta_{\mathtt{X}}(2,0;\frac{1}{2})\approx 0.33$, $\Delta_{\mathtt{X}}(2,2;\frac{1}{2})\approx-0.33$,

• •

$\Delta_{\mathtt{X}}(1,0;\frac{3}{4})\approx 0.42$, $\Delta_{\mathtt{X}}(1,2;\frac{3}{4})\approx-0.42$, $\Delta_{\mathtt{X}}(2,0;\frac{3}{4})\approx 0.49$, $\Delta_{\mathtt{X}}(2,2;\frac{3}{4})\approx-0.49$,

• •

$\Delta_{\mathtt{X}}(1,0;1)\approx 0.65$, $\Delta_{\mathtt{X}}(1,2;1)\approx-0.65$, $\Delta_{\mathtt{X}}(2,0;1)\approx 0.65$, $\Delta_{\mathtt{X}}(2,2;1)\approx-0.65$,

which corroborate our considerations.

Finally, let us say some few words about the potential use of discrete Wigner functions in experiments involving the two-qubit and ququart states: the complete algebraic framework here developed for discrete $\mathrm{SU(4)}$ Wigner functions really works well, as expected, in detecting genuinelly quantum effects (for example, entanglement, among others); besides, experiments associated with NMR techniques, where the matrix elements of the density matrix are tomographycally reconstructed, can be considered as the best scenario to implement this important mathematical tool.

5 Concluding remarks

In this work, we have established an algebraic approach that allows us to describe in a general way both the two two-level and four-level quantum-mechanical systems through their respective discrete Wigner functions. For this specific task, we have employed the connection between $\mathrm{SU(N)}$ generators and Schwinger unitary operators that, in particular, paves the way to introduce a genuinely discrete finite-dimensional phase space [27]. So, the discrete Wigner function framework emerged from this approach is completely general since it allows, among other things, to describe arbitrary two-qubit and ququart states. Furthermore, experimental researches dealing with NMR techniques (or even dealing with different experimental arrangements) have, in our results on discrete Wigner functions, a new solid mathematical tool for searching on entanglement in analogous or even more complex systems [15, 12, 10, 13, 14, 11]. Next, we will discuss on effective gains and future perspectives derived from this manuscript.

The correspondence between discrete $\mathrm{SU(2)}\otimes\mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions not only solved the difficulty of visualizing general two-qubit states in finite-dimensional discrete phase spaces, but also introduced a new mathematical tool that provides qualitative information on the entanglement effects associated with two-qubit X-states [25] through the functioneeeIn particular, for the Bell states \mbox{\hat{\rho}}_{\Psi_{\pm}} and \mbox{\hat{\rho}}_{\Phi_{\pm}}, the corresponding functions $\Delta_{\Psi_{\pm}}(\mu,\nu)$ and $\Delta_{\Phi_{\pm}}(\mu,\nu)$ are restricted to the symmetric intervals

$-\frac{1}{4}\sqrt{2+\sqrt{2}}\,\leq\Delta_{\Psi_{\pm}}(\mu,\nu)\leq\frac{1}{4}\sqrt{2+\sqrt{2}}\quad\mbox{and}\;\,-\frac{1}{2}\sqrt{\frac{2+\sqrt{2}}{2}}\,\leq\Delta_{\Phi_{\pm}}(\mu,\nu)\leq\frac{1}{2}\sqrt{\frac{2+\sqrt{2}}{2}}\;,$

whose expressions are numerically equivalent to $|\Delta_{\Psi_{\pm}}(\mu,\nu)|\leq 0.46$ and $|\Delta_{\Phi_{\pm}}(\mu,\nu)|\leq 0.65$. Such results will be our guidelines for subsequent comparisons with different entangled X-states.

[TABLE]

where the discrete marginal distribution functions $Q_{\mathtt{X}}(\mu)$ and $R_{\mathtt{X}}(\nu)$ are then responsible for the main diagonal and antidiagonal matrix elements of \mbox{\hat{\rho}}_{\mathtt{X}}, respectively. It will be quite interesting to apply this result in different two-qubit X-states under kinematical and dynamical perspectives: for example, let us initially consider certain static states as those introduced by Peres-Horodecki (PH) [39, 40]

[TABLE]

for $x\in(0,1]$ (it is separable in $x=0$), as well as the Gisin (G) state [74]

[TABLE]

which basically depends on three intrinsic parameters $(a,b,\mathrm{x})$ such that $a>b$ and $\mathrm{x}\in[0,1]$. By varying the unique internal parameter present in \mbox{\hat{\rho}}_{\mathtt{PH}}, it is easy to verify that $\Delta_{\mathtt{PH}}(\mu,\nu)$ is null for $x=0$ and it attains its maximum value for $x=1$ when we get $|\Delta_{\mathtt{PH}}(\mu,\nu)|\leq 0.46$ (in such a case, $\Delta_{\mathtt{PH}}=\Delta_{\Psi_{-}}$). Now, if one considers the Gisin state $\hat{\varrho}_{\mathtt{G}}$ with $a^{2}-b^{2}=\frac{\sqrt{2}}{4}$, $ab=\frac{1}{2}$, and $\mathrm{x}=1$ fixed, we obtain $|\Delta_{\mathtt{G}}(\mu,\nu)|\leq 0.60$, which represents a value close to that reached by the Bell states $\Phi_{\pm}$. Therefore, these results lead us to establish a hierarchy relation among the static two-qubit X-states through the function $\Delta_{\mathtt{X}}(\mu,\nu)$; in addition, it can also be extended to include dynamic states where both the continuous [75, 76] and discrete [77] time descriptions take place.

Nowadays, it is well-known that fidelity corresponds to an important concept to quantum information since it provides an effective measurement of the degree of similarity between two quantum states [78]. From the experimental point of view, this specific measurement allows to quantify how close the state produced in any experimental apparatus – once this state is limited by imperfections and noise – stays from that intended one. A theoretical application comes from the entanglement quantification context, since it measures how close an entangled state is to the set of separable states. Hence, let us consider the definition of fidelity initially introduced in Ref. [79] as

[TABLE]

and later studied independently in [80] by the name of super-fidelity, where $\hat{\rho}$ and $\hat{\sigma}$ represent two density matrices that belong to $\mathscr{L}_{+,1}(\mathcal{H}_{N})$. With respect to \mathcal{F}_{\mathrm{N}}(\mbox{\hat{\rho}},\hat{\sigma}) the connection with the discrete Wigner functions $W_{\rho}(\mu,\nu)$ and $W_{\sigma}(\mu,\nu)$ can be promptly established as follows:

[TABLE]

For $N=4$, the fidelity (54) perfectly matches with the results established in this paper.

Finally, let us discuss on a possible extension of the mathematical framework exposed here, in order to include the discrete Husimi and Glauber-Sudarshan distribution functions [81, 82]. As mentioned in Ref. [27], the change $\hat{G}(\mu,\nu)\rightarrow\hat{T}^{(s)}(\mu,\nu)$ of operator bases in Eq. (4) permits to include a wide range of possibilities in what concerns the quasiprobability distribution functions defined over a finite-dimensional discrete phase space: indeed, for $s=-1,0,+1$ the parametrized function F^{(s)}(\mu,\nu)=\mbox{{\rm Tr}}[\hat{T}^{(s)}(\mu,\nu)\mbox{\hat{\rho}}] recovers the discrete Husimi, Wigner, and Glauber-Sudarshan distribution functions, respectively. To conclude, an interesting scenario of possible applications for discrete Wigner functions refers to the study of certain spin chains where two-qubit X-states have a key role [70, 71, 72], as well as the study on the separability of multi-qubit states [83].

References