Monogamy relations of entropic non-contextual inequalities and their experimental realization

An experiment corresponding to a contextuality inequality can be represented by a graph  Cabello et al. (2014b). Mathematically, a graph $G$ is defined by a set of vertices $V$ and and a set of edges $E$, such that $G=(V,E)$. In relation to the contextuality inequality, the set $V$ can represent either events, projectors or observables depending upon the scenario, while the set $E$ represents a relationship between different elements of $V$, such as orthogonality, exclusivity or commutativity Cabello et al. (2014b). In this paper we use commutativity graphs to study ENC inequalities, in which the set of vertices corresponds to observables and two vertices are connected by an edge if they commute with each other Kurzyński et al. (2012).

Consider an $n$-cycle commutation graph in which the $n$ vertices represent observables $X_{i}$ and the existence of an edge indicates that the corresponding observables commute. An example of one such graph for $n=4$ is given in Fig. 2 (top panel).

We assume that a non-contextual joint probability distribution exists over the entire set of observables considered, even though most of them do not commute. It is our aim to construct a condition based on the preceding assumption, for which a violation would indicate that such a non-contextual joint probability distribution does not exist.

The existence of a non-contextual joint probability distribution over the observables $X_{i}$ implies that it is possible to define a joint Shannon entropy $H(X_{0},...,X_{n-1})$ of them. We can then write

where the relationship $H(X)\leq H(X,Y)$ is physically motivated by the fact that two random variables cannot contain less information than a single one of them. Furthermore, with repeated application of the chain rule $H(X,Y)=H(X|Y)+H(Y)$, where $H(X|Y)$ denotes the conditional entropy of observable $X$ given information about observable $Y$, the right hand side of the inequality can be re-written as,

The latter inequality in Eq. (2) is a consequence of the relationship $H(X|Y)\leq H(X)$, which implies that conditioning cannot increase the information content of a random variable. Plugging Eq. (2) in Eq. (1) and using $H(X_{0}|X_{n-1})=H(X_{0},X_{n-1})-H(X_{n-1})$, we finally get the required entropic non-contextuality inequality,

A violation of Eq. (3) would then indicate that a non-contextual joint probability distribution over the corresponding set of observables does not exist.

It should be noted that no assumptions have been made regarding the nature of the observables $X_{i}$. For all intents and purposes they can correspond to projective measurements or POVMs with any number of outcomes. Furthermore, the observables could correspond to local scenarios or non-local, in which case the entropic inequality would be termed as non-contextual or Bell non-local. Since we consider generalized scenarios, we term all entropic inequalities as non-contextual as it subsumes the non-local scenarios as well.

One of the many interesting cases arises for $n=4$ observables, which corresponds to the well known Bell-CHSH scenario. Consider two parties, Alice and Bob, each having two observables labelled $\{A_{0},A_{1}\}$ and $\{B_{0},B_{1}\}$ respectively, such that observables of any one party do not commute, while observables of different parties commute. Differing from the traditional Bell-CHSH scenario, it is also assumed that a measurement of the observables can have an arbitrary number of outcomes. The corresponding entropic inequality is written as,

A violation of the above inequality implies a non-existence of a joint probability distribution over all the observables $A_{x}$ and $B_{y}$ $\forall x,y\in\{0,1\}$. In the next section, we show that ENC inequalities admit a monogamous relationship which can be derived using a graph theoretic formalism. We particularly focus on the entropic Bell-CHSH scenario described above and show that it admits a monogamous relationship.

We now elucidate the formalism to check and derive a monogamous relationship for any arbitrary scenario using the graph theoretic formalism. It should be remembered that a violation of the inequality (4) implies that a joint probability distribution cannot exist over the given observables.

The derived monogamy relationship (17) imposes severe restrictions on the violation of Alice-Charlie entropic CHSH inequality.

We experimentally implement the monogamy inequality (17) using a mixed tripartite state which is a classical mixture of two pure maximally entangled states given as:

where

and $p\in\left[0,1\right]$. As can be seen, the state $\rho$ physically implies that Alice and Bob share a maximally entangled state with probability $p$ while Charlie is separable, and with probability $1-p$, Alice and Charlie share a maximally entangled state while Bob is separable.

The observables of Alice, Bob and Charlie are assumed to lie in the $X-Z$ plane and correspond to Pauli spin measurements along the unit vectors $\bm{a},\bm{a^{\prime}},\bm{b},\bm{b^{\prime}}$ and $\bm{e},\bm{e^{\prime}}$ respectively. The vectors $\bm{a},\bm{b^{\prime}},\bm{a^{\prime}}$ and $\bm{b}$ are successively separated by an angle $\frac{\theta}{3}$, while the vectors $\bm{e}$ and $\bm{e^{\prime}}$ are taken to be the same as Bob’s. The corresponding Pauli observables are then given as:

while the observables of Charlie ($E_{0},E_{1}$) are the same as Bob’s but acting in a different Hilbert space. All of the aforementioned observables have eigenvalues $a_{i},b_{i},e_{i}\in\{-1,+1\}$ and follow the commutativity conditions as shown in Fig. 2.

For the set of observables given in Eq. (20), the maximum violation of the Alice-Bob entropic CHSH inequality $H_{K1}=0.237$ bits is found to be at $\theta=0.457$ radians when the parties share the state $\ket{\psi_{1}}$. The same also holds for Alice-Charlie entropic CHSH inequality when the parties share the state $\ket{\psi_{2}}$.

We experimentally implemented the monogamy relation given in Eq. (17) on an eight-dimensional quantum system. We used a molecule of ¹³C -labeled diethyl fluoromalonate dissolved in acetone-D6, with the ¹H, ¹⁹F and ¹³C spin-1/2 nuclei being encoded as ‘qubit one’, ‘qubit two’ and ‘qubit three’, respectively (see Fig. 3(a)). The system was initialized in the pseudopure state (PPS) $|000\rangle$ using the spatial averaging technique Mitra et al. (2007) and the corresponding NMR spectra and experimental tomographs are given in Fig. 3(a),(b).

Experiments were performed on a Bruker Avance III 600-MHz FT-NMR spectrometer equipped with a QXI probe. Local unitary operations were achieved by RF pulses of suitable amplitude, phase, and duration and nonlocal unitary operations were achieved by free evolution under the system Hamiltonian. The $T_{1}$ and $T_{2}$ relaxation times of ¹H, ¹⁹F, ¹³C spin-1/2 nuclei range from 4.16 sec to 7.16 sec and 0.99 sec to 3.56 sec, respectively. The duration of the $\frac{\pi}{2}$ pulses for ¹H, ¹⁹F, and ¹³C nuclei are 9.5 $\mu$s at 18.14 W power level, 22.2 $\mu$s at a power level of 42.27 W, and 15.2 $\mu$s at a power level of 179.47 W, respectively. At room temperature, NMR experiments are only sensitive to the deviation density matrix and the initial state is prepared from the thermal equilibrium into a PPS:

where $\epsilon\approx 10^{-6}$ and $I_{8}$ is $8\times 8$ identity operator. The system is initialized in the PPS state i.e.$|000\rangle$ using the spatial averaging technique Mitra et al. (2007), which is based on dividing the system in sub-ensembles which can be accessed independently in NMR by using a combination of RF pulses and pulsed magnetic gradients. The state tomography was performed using the least square optimization techniqueGaikwad et al. (2021) with an experimental state fidelity of 0.98.

We began by preparing the mixed tripartite state given in Eq. (18) for different values of $p$. In order to achieve this, we utilized the temporal averaging technique Oliveira et al. (2007); Knill et al. (1998). Using this technique, it is possible to prepare arbitrary mixed states on an NMR quantum information processor, by applying suitable unitary transformations on some common initial PPS. The different experiments are performed on common initial states, the results of which are independently stored. Finally, these results are combined to produce an average state which simulates the behavior of a mixed state.

In our case, we prepared the mixed state given in Eq. (18), which is a mixture of two pure states $|\psi_{1}\rangle$, $|\psi_{2}\rangle$. We first prepared these two states by applying suitable unitaries on the initial state $|000\rangle\langle 000|$ in two different and independent experiments Singh et al. (2019a). The states of these two experiments are then added with appropriate probabilities to achieve the desired mixed state given in Eq. (18). To demonstrate the monogamy relation, we chose different values of $p$ to experimentally prepare the desired mixed state with experimental state fidelities $\geq 0.956$. The tomograph of one such experimentally prepared state with $p=1$ is shown in Fig. 4.

After experimentally preparing the states, we measured the desired probabilities in order to calculate the entropies involved in the inequality given in Eq. (17). In order to calculate the probabilities, we transformed the required probabilities in terms of expectation values. This is necessary because experiments on an NMR quantum information processor yield only expectation values of the observables. These expectation values are evaluated by decomposing the observables in terms of linear combinations of Pauli operators which can be mapped to the single-qubit Pauli $Z$ operator. This mapping is particularly useful in the context of an NMR experimental setup where the expectation value of the $Z$ operator is easily accessible and corresponds to the observed $z$ magnetization of a nuclear spin in a particular quantum state. The normalized experimental intensities of the NMR signal then provide an estimate of the expectation value of the Pauli $Z$ operator in that quantum state Singh et al. (2019b).

The probabilities can be written as $P(A_{i}=a_{i},B_{j}=b_{j})=\text{tr}(\rho|a_{i}\rangle\langle a_{i}|\otimes|b_{j}\rangle\langle b_{j}|\otimes I)$ and $P(A_{i}=a_{i},E_{j}=e_{j})=\text{tr}(\rho|a_{i}\rangle\langle a_{i}|\otimes I\otimes|e_{j}\rangle\langle e_{j}|)$ where $|a_{i}\rangle$, $|b_{j}\rangle$, $|e_{j}\rangle$ are the eigenvector of the observables corresponding to Alice, Bob and Charlie, respectively. The observables ($|a_{i}\rangle\langle a_{i}|\otimes|b_{j}\rangle\langle b_{j}|\otimes I$) , ($|a_{i}\rangle\langle a_{i}|\otimes I\otimes|e_{i}\rangle\langle e_{i}|)$ are decomposed in terms of linear combinations of Pauli operators and details are given in Appendix-A. The idea is to unitarily map the state $\rho$ to another state $\rho^{\prime}$, such that $\langle X\rangle_{\rho}=\langle I_{iz}\rangle_{\rho^{\prime}}$ where $X$ is the observable to be measured in the state $\rho$ and $I_{iz}$ is the z-spin angular momentum of the qubit. This can be achieved by measuring the $I_{iz}$ on the state $\rho^{\prime}$. For example, one can find the expectation values of $\langle\sigma_{x}\otimes\sigma_{x}\otimes I\rangle_{\rho}$ and $\langle\sigma_{x}\otimes I\otimes\sigma_{x}\rangle_{\rho}$ which are involved in the evaluation of probabilities (see Appendix-A), by using the quantum circuit and corresponding NMR pulse sequence given in Fig.5(a) and Fig.5(b) respectively, where the implementation is followed by a measurement of the spin magnetization of the second and third qubits, respectively.

We experimentally calculated the ENC inequalities $H_{K1}$, $H_{K2}$ and the ENC monogamy relation $H_{K1}+H_{K2}$ for the tripartite mixed state given in Eq. (18) for different values of $p$. Experimental values of $H_{K1}$, $H_{K2}$, $H_{K1}+H_{K2}$ with respect to various values of $p$ are plotted in Fig. 6. It can be seen that $H_{K1}$ is violated for the tripartite state with $p=1$, while $H_{K2}$ is violated for the state with $p=0$. The monogamy relation $H_{K1}+H_{K2}$ is never violated for any value of $p$ and the results are in good agreement with the theoretical predictions.

It is seen that the experimental values are always lower than the corresponding theoretical values. This is due to the fact that the maximum value of the inequality is always achieved for a pure state and any addition of noise makes it a mixed state. Therefore, the value is always observed to be lower than the theoretical one. The difference between them also increases with increasing values of $p$ upto $p=0.5$. For this region, the state prepared for $p=0.0$ is the dominant one which is shown to violate $H_{K_{1}}$. Furthermore, the corresponding inequalities are logarithmic in nature, which also compounds the errors for states till $p=0.5$. However, after this value the state prepared for $p=1.0$ dominates and the errors again follow a similar trend. It should be noted that the experimental curve follows the same trend as the theoretical curve.

In this subsection, we experimentally test the monogamy relation Eq. (17) for a pure tripartite state, having two parameters which we can vary. We show that the monogamy relation holds and is in good agreement with theoretical predictions.

We take a pure tripartite state of the form,

where $N=\frac{1}{\sqrt{p_{1}^{2}+p_{2}^{2}+(p_{1}+p_{2})^{2}}}$ is the normalization factor. We experimentally prepared five different states corresponding to various values of $p_{1}$ and $p_{2}$. The quantum circuit and the corresponding NMR pulse sequence is given in Fig. 7(a), (b). Different pure states corresponding to various values of $p_{1}$ and $p_{2}$ were generated by suitably choosing the values of $\theta_{1},\theta_{2}$ and $\theta_{3}$. Tomograph of one such experimentally prepared state with $p_{1}=0.25$ and $p_{2}=0.50$ is depicted in Fig. 8, with an experimental state fidelity of $0.93$. After preparing the states, we measured the desired probabilities by mapping the state onto the Pauli basis operators in order to calculate the entropies involved in the inequality Eq. (17) as discussed earlier in Sec. III.1.

We experimentally evaluate $H_{K_{1}}$, $H_{K_{2}}$ and $H_{K_{1}}+H_{K_{2}}$ with respect to the different values of $p_{1}$ and $p_{2}$ and the results are given in Table-1. We see that the inequality $H_{K_{1}}$ is violated for $p_{1}=1$ and $p_{2}=0$ and $H_{K_{2}}$ is violated for $p_{1}=0$ and $p_{2}=1$, while the monogamy relationship $H_{K_{1}}+H_{K_{2}}$ is never violated for any value of $p_{1}$ and $p_{2}$. This shows that the monogamy relation between the ENC inequalities is obeyed for all such possible states.