# Bell Diagonal and Werner State Generation: Entanglement, Non-Locality, Steering and Discord on the IBM Quantum Computer

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## Abstract

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## 1. Introduction

## 2. Bell Diagonal and Werner States and Their Properties

#### 2.1. The Octahedron of Separable States

#### 2.2. The Line of Werner States

## 3. Quantum Circuits for BDS and Werner States

#### 3.1. Four-Qubit Circuits and Relevant BDS Parameterizations

#### 3.2. Two-Qubit Circuits

## 4. Qiskit Implementation

## 5. Entanglement Measures and Discord

#### 5.1. Entanglement Measures for BDS

#### 5.1.1. Entanglement of Formation and Concurrence

#### 5.1.2. CHSH-Non-Locality

- We have $M\left(\rho \right)={\tau}_{1}+{\tau}_{2}$ the sum of the two largest eigenvalues (among three) of ${T}^{\u2020}T$ where $T=\left({t}_{ij}\right)$.
- CHSH-local states naturally satisfy $M\left(\rho \right)\le 1$.
- The maximum possible value of $M\left(\rho \right)=2$ is attained for pure Bell states.
- For BDS from Equation (5) ${T}^{\u2020}T=\mathrm{diag}({t}_{1}^{2},{t}_{2}^{2},{t}_{3}^{2})$ so ${\tau}_{1}+{\tau}_{2}=\parallel \overrightarrow{t}{\parallel}^{2}-{t}_{\mathrm{min}}^{2}$ and (26) becomes$$L\left(\rho \right)=max\left(\right)open="("\; close=")">0,\frac{\sqrt{{\parallel \overrightarrow{t}\parallel}^{2}-{t}_{\mathrm{min}}^{2}}-1}{\sqrt{2}-1}$$

#### 5.1.3. Steering

#### 5.1.4. Hierarchy between Quantum Correlation Measures: Entanglement, Steering and CHSH-Non-Locality

#### 5.2. Discord for BDS

#### 5.2.1. Quantum Discord

#### 5.2.2. Asymmetric Relative Entropy of Discord

#### 5.2.3. Application to BDS

#### 5.3. The Particular Case of Werner States

- Non-separability and entanglement of formation ${E}_{F}\left(\rho \right)$. The PPT criterion shows that Werner states are separable for $w\in [0,\frac{1}{3}]$ and display entanglement for $w\in (\frac{1}{3},1]$. The same threshold applies to concurrence (see (22)) and entanglement of formation (see (23)). The details are given in Appendix C.
- CHSH-non-locality. CHSH-non-locality $L\left(\rho \right)$ vanishes for $w\in [0,\frac{1}{\sqrt{2}}]$. Please note that $\frac{1}{\sqrt{2}}$ corresponds to the only points in the common intersection of the three unit cylinders oriented along the main axes.
- Steering. The threshold for 2-steering is identical to the one of CHSH-non-locality [53,57,59]. On the other hand, 3-steering ${S}_{3}\left(\rho \right)$ vanishes for $w\in [0,\frac{1}{\sqrt{3}}]$ and states with larger w are 3-steerable. We point out that [27] proved that Werner states cannot be replaced by a LHS model if an only if $w>\frac{1}{2}$ (this is the fundamental threshold below which Werner states are not steerable).
- Discord and classical correlation. From (41), the classical correlation is simply ${\mathcal{C}}_{W}=1-{h}_{2}\left(\frac{1-w}{2}\right)$ and using (40) we find the discord$$\begin{array}{cc}\hfill {\mathcal{D}}_{W}=& \frac{1}{4}(1-w){log}_{2}(1-w)-\frac{1}{2}(1+w){log}_{2}(1+w)\hfill \\ & +\frac{1}{4}(1+3w){log}_{2}(1+3w)\hfill \end{array}$$We note that discord is strictly bigger than classical correlation for all w except $w=0$ and 1.

## 6. IBM Q Results

#### 6.1. Simulations with Noise Models from IBM Q Quantum Devices

#### 6.1.1. Fidelity in Noisy Simulations on the Werner Line

#### 6.1.2. Noisy Simulation in the Whole Tetrahedron: Expected Quantum Mutual Information and Discord

`ibmq_london`(respectively generated on 2019–12–03 and 2019–12–10 with 1000 shots). Figure 17 and Figure 18 display the result for the quantum mutual information and the discord on the whole tetrahedron. In general, we observe that noise reduces these quantities to almost half their theoretical value close to the corners of the tetrahedron. Interestingly, in the corners and along the edges we observe that the 2-qubit circuit is slightly more faithful to the ideal results of Figure 11 and Figure 13 in Section 5.1. The same observations hold also for the classical correlation (not shown here). These results are consistent with the corresponding observations on the Werner line discussed in the previous paragraph.

#### 6.2. Experiments on IBM Q Quantum Devices

#### 6.2.1. Fidelity of Experimental Density Matrices

`ibmq_athens`and

`ibmq_santiago`with 5000 shots. This is compared to the ideal noiseless simulation with qasm-simulator, and the one using a Qiskit noise model based on the properties of each real hardware. The density matrix reconstructed using both

`ibmq_athens`and

`ibmq_santiago`with 5000 shots is close to the ideal one for small w, proving the performance of the real quantum computer in the corresponding domain, although it drops below $85\%$ and $75\%$, respectively, for $w=1$. At the time, this result suggests that it is necessary to improve the current noise model to describe the fidelity drop in a more faithful way. In this run, we see that slightly higher fidelity was obtained by

`ibmq_athens`compared to

`ibmq_santiago`.

`ibmqx2`backend, running the circuit of Figure 3b with 1000 shots per measurement. We see that the fidelity is fairly high, ∼0.9, over the whole domain which allows us to proceed with the calculations of quantum correlations in the following.

#### 6.2.2. Experimental Classical Correlations, Quantum Mutual Information and Discord

`ibmqx2`backend, running the circuit of Figure 3b with 1000 shots per measurement.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Solution for Parameters

## Appendix B. Quantum Implementation of Classical Operations

**Figure A1.**Fully quantum version of the circuit of Figure 6 for preparing Werner states.

## Appendix C. Separability and Entanglement of Formation for Werner States

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**Figure 1.**Geometrical representation of the BDS tetrahedron $\mathcal{T}$ bounded by the four planes ${t}_{1}-{t}_{2}+{t}_{3}\ge -1$, ${t}_{1}+{t}_{2}-{t}_{3}\ge -1$, ${t}_{1}-{t}_{2}-{t}_{3}\le 1$, ${t}_{1}+{t}_{2}+{t}_{3}\le 1$. The octahedron $\mathcal{O}$ defined by $|{t}_{1}|+|{t}_{2}|+|{t}_{3}|\le 1$ contains all the separable BDS. Accordingly, four entangled regions can be identified outside of the octahedron, in each of which a Bell state is located at the corresponding summits of $\mathcal{T}$. We have the correspondence $|{\beta}_{00}\rangle \leftrightarrow (1,-1,1)$, $|{\beta}_{01}\rangle \leftrightarrow (1,1,-1)$, $|{\beta}_{10}\rangle \leftrightarrow (-1,1,1)$, $|{\beta}_{11}\rangle \leftrightarrow (-1,-1,-1)$. The red line ${t}_{1}={t}_{2}={t}_{3}=-w$, $0\le w\le 1$, along the negative diagonal, represents Werner states (10).

**Figure 2.**The generalized four-qubit preparation circuit as in ref. [40]. Only the subcircuit G which encodes the probabilities ${\left\{{p}_{jk}\right\}}_{j,k=0}^{1}$ must be corrected. Qubits are then copied by CNOT gates. Subcircuit B finally entangles into the Bell basis.

**Figure 3.**Three different versions of the probability-encoding subcircuit G. (

**a**) Encoder G: incomplete circuit of [40], which involves only two parameters $\alpha $ and $\theta $ (definition in [40]), and the gate ${R}_{y}$ given by Equation (15). (

**b**) Encoder G: compact circuit which generates the whole class of BDS with the three parameters $\alpha $, $\beta $ and $\gamma $ appearing in Equation (16). (

**c**) Encoder G: complete three-parameter circuit based on canonical coordinates $\psi $, $\theta $ and $\phi $ on the unit 3-sphere appearing in Equation (17).

**Figure 4.**A two-qubit replacement for the circuit in Figure 2.

**Figure 7.**Entanglement of formation ${E}_{F}\left(\rho \right)$ of BDS, calculated from the noiseless simulation of compact circuit (Figure 3b). In the gray region entanglement of formation identically vanishes.

**Figure 9.**${S}_{3}\left(\rho \right)$, 3-steering of BDS, calculated from the noiseless simulation of compact circuit (Figure 3b). In the gray region 3-steering identically vanishes.

**Figure 10.**Hierarchy of regions of separability (red), vanishing 3-steering (orange), vanishing CHSH-non-locality (yellow) and the rest of the BDS tetrahedron. The Werner line is also shown.

**Figure 13.**Discord ${\mathcal{D}}_{\mathrm{BDS}}={\mathcal{I}}_{\mathrm{BDS}}-{\mathcal{C}}_{\mathrm{BDS}}$, calculated with (40) and (41) from the noiseless simulation of compact circuit (Figure 3b). Here the range is the natural range $[0,1]$. Discord does not identically vanish on any extended domain, note the three-pointed star pattern on the faces.

**Figure 14.**Correlations of Werner states as a function of w along the Werner line. The vertical bars mark the following critical values: entanglement of formation ${E}_{F}\left(\rho \right)$ vanishes for $w\le \frac{1}{3}$ and states are separable, 3-steering ${S}_{3}\left(\rho \right)$ vanishes for $w\le \frac{1}{\sqrt{3}}$, CHSH-non-locality $L\left(\rho \right)$ and 2-steering both vanish for $w\le \frac{1}{\sqrt{2}}$. Discord ${\mathcal{D}}_{W}$ and classical correlations ${\mathcal{C}}_{W}$ are always positive, and discord is always bigger than classical correlations. Both are monotonously increasing on $[0,1]$.

**Figure 15.**Simulated fidelity curves $F({\rho}^{\mathrm{W}\phantom{\rule{0.166667em}{0ex}}\mathrm{circ}},{\rho}^{\mathrm{W}\phantom{\rule{0.166667em}{0ex}}\mathrm{theo}})$ of Werner states as a function of the w parameter, for density matrices produced by two-qubit and four-qubit versions of the circuits described in Section 3. For comparison, the black dashed line corresponds to a maximally mixed state. These results are based on tomography with ${2}^{15}$ shots under a Qiskit noise model generated for

`ibmq_athens`on 16 May 2021.

**Figure 16.**(Above) Fidelity (43) of experimental Werner state density matrices reconstructed by Qiskit on

`ibmq_athens`and

`ibmq_santiago`for 5000 shots, and compared with the results from Qiskit simulator and noise models provided by Qiskit. The black dashed line corresponds to the extreme worst case where an identity matrix would be produced by the simulations (cf. Equation (44) and its discussion). (Below) Standard deviation of fidelity over 10 simulations for

`ibmq_athens`(red solid line) and its noise model (yellow dash-dotted line).

**Figure 17.**Quantum mutual information ${\mathcal{I}}_{BDS}$ on its natural scale $[0,2]$, as expected from noisy simulation of 4-qubit (

**left**) and 2-qubit (

**right**) circuits.

**Figure 18.**Discord ${\mathcal{D}}_{BDS}$ on its natural scale $[0,1]$, as expected from noisy simulation of 4-qubit (

**left**) and 2-qubit (

**right**) circuit. The three-pointed star pattern is visible, but deteriorating.

**Figure 22.**Experimental discord ${\mathcal{D}}_{BDS}$ on the scale $[0,0.7]$. The three-pointed star pattern is barely visible.

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**MDPI and ACS Style**

Riedel Gårding, E.; Schwaller, N.; Chan, C.L.; Chang, S.Y.; Bosch, S.; Gessler, F.; Laborde, W.R.; Hernandez, J.N.; Si, X.; Dupertuis, M.-A.;
et al. Bell Diagonal and Werner State Generation: Entanglement, Non-Locality, Steering and Discord on the IBM Quantum Computer. *Entropy* **2021**, *23*, 797.
https://doi.org/10.3390/e23070797

**AMA Style**

Riedel Gårding E, Schwaller N, Chan CL, Chang SY, Bosch S, Gessler F, Laborde WR, Hernandez JN, Si X, Dupertuis M-A,
et al. Bell Diagonal and Werner State Generation: Entanglement, Non-Locality, Steering and Discord on the IBM Quantum Computer. *Entropy*. 2021; 23(7):797.
https://doi.org/10.3390/e23070797

**Chicago/Turabian Style**

Riedel Gårding, Elias, Nicolas Schwaller, Chun Lam Chan, Su Yeon Chang, Samuel Bosch, Frederic Gessler, Willy Robert Laborde, Javier Naya Hernandez, Xinyu Si, Marc-André Dupertuis,
and et al. 2021. "Bell Diagonal and Werner State Generation: Entanglement, Non-Locality, Steering and Discord on the IBM Quantum Computer" *Entropy* 23, no. 7: 797.
https://doi.org/10.3390/e23070797