Quantum entanglement and quantum discord in dimers in multiple quantum NMR experiments

TL;DR

This paper investigates quantum correlations, including entanglement and discord, in two-spin dimers during multiple quantum NMR experiments, revealing the emergence of specific coherences and quantum states.

Contribution

It provides a detailed analysis of MQ NMR dynamics in dimers, highlighting the appearance of entangled states and calculating quantum discord at high temperatures.

Findings

MQ NMR coherences of zeroth and second order emerge

Entangled states appear during system evolution

Quantum discord is quantified at high temperature

Abstract

Quantum correlations in multiple quantum (MQ) NMR experiments are investigated in two-spin systems (dimers). In the initial moment of time one spin is in a pure quantum polarized state and the other spin is in the thermodynamic equilibrium state defined by the temperature of the sample. MQ NMR dynamics of dimers is investigated. It is shown that MQ NMR coherences of only the zeroth and second orders emerge in such a system. The intensities of those coherences are calculated. Entangled states appear in the course of the system evolution in the MQ NMR experiment. The quantum discord is obtained in the high temperature approximation.

Equations

$${\rho }_0=\frac{(\alpha |0⟩+\beta |1⟩)({\alpha }^{\ast }⟨0|+{\beta }^{\ast }⟨1|)\otimes e^{\frac{\hslash {\omega }_0}{kT}{I}_{z,2}}}{Z}$$

$${\rho }_0=\left(\begin{array}{cccc}\frac{e^b|\alpha {|}^2}{e^b+1}& 0& \frac{e^b\alpha {\beta }^{\ast }}{e^b+1}& 0\\ 0& \frac{|\alpha {|}^2}{e^b+1}& 0& \frac{\alpha {\beta }^{\ast }}{e^b+1}\\ \frac{e^b\beta {\alpha }^{\ast }}{e^b+1}& 0& \frac{e^b|\beta {|}^2}{e^b+1}& 0\\ 0& \frac{\beta {\alpha }^{\ast }}{e^b+1}& 0& \frac{|\beta {|}^2}{e^b+1}\end{array}\right),$$

$$H_{MQ}=d\left(I_1^{+}{I}_2^{+}+I_1^{-}{I}_2^{-}\right),$$

$$\rho \left(t\right)=e^{-i{H}_{MQ}\tau }\rho (0)e^{i{H}_{MQ}\tau }.$$

$$\rho (\tau )=\left(\begin{array}{cccc}\frac{e^b|\alpha {|}^2{\cos }^2(d\tau )+|\beta {|}^2{\sin }^2(d\tau )}{e^b+1}& -\frac{i\beta {\alpha }^{\ast }\sin (d\tau )}{e^b+1}& \frac{e^b\alpha {\beta }^{\ast }\cos (d\tau )}{e^b+1}& \frac{i\sin (2d\tau )(e^b|\alpha {|}^2-|\beta {|}^2)}{2(e^b+1)}\\ \frac{i{\beta }^{\ast }\alpha \sin (d\tau )}{e^b+1}& \frac{|\alpha {|}^2}{e^b+1}& 0& \frac{\alpha {\beta }^{\ast }\cos (d\tau )}{e^b+1}\\ \frac{e^b{\alpha }^{\ast }\beta \cos (d\tau )}{e^b+1}& 0& \frac{e^b|\beta {|}^2}{e^b+1}& \frac{i{e}^b\beta {\alpha }^{\ast }\sin (d\tau )}{e^b+1}\\ -\frac{i\sin (2d\tau )(e^b|\alpha {|}^2-|\beta {|}^2)}{2(e^b+1)}& \frac{\beta {\alpha }^{\ast }\cos (d\tau )}{e^b+1}& -\frac{i{e}^b{\beta }^{\ast }\alpha \sin (d\tau )}{e^b+1}& \frac{e^b|\alpha {|}^2{\sin }^2(d\tau )+|\beta {|}^2{\cos }^2(d\tau )}{e^b+1}\end{array}\right).$$

$$\rho (\tau )={\rho }_0(\tau )+{\rho }_2(\tau )+{\rho }_{-2}(\tau ).$$

$$\begin{array}[]{l}G_{0}(\tau)=\mathrm{Tr}\{\rho_{0}(\tau)\rho_{0}^{ht}(\tau)\},\\ G_{\pm 2}(\tau)=\mathrm{Tr}\{\rho_{2}(\tau)\rho_{-2}^{ht}(\tau)\},\end{array}$$

$${\rho }^{ht}(\tau )={\rho }_0^{ht}(\tau )+{\rho }_2^{ht}(\tau )+{\rho }_{-2}^{ht}(\tau )=e^{-i{H}_{MQ}\tau }{I}_z{e}^{i{H}_{MQ}\tau },$$

$$G_0(\tau )=\frac{e^b|\alpha {|}^2-|\beta {|}^2}{e^b+1}{\cos }^2(2d\tau ),$$

$$G_{\pm 2}(\tau )=\frac{e^b|\alpha {|}^2-|\beta {|}^2}{2(e^b+1)}{\sin }^2(2d\tau ).$$

$$\tilde{\rho }(\tau )=({\sigma }_y\otimes {\sigma }_y){\rho }^{\ast }(\tau )({\sigma }_y\otimes {\sigma }_y),$$

$$C=\max \{0,2\lambda -{\lambda }_1-{\lambda }_2-{\lambda }_3-{\lambda }_4\},\lambda =\max \{{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\},$$

$$C=\frac{|(e^b|\alpha {|}^2-|\beta {|}^2)\sin (2d\tau )|}{e^b+1}$$

$$C=\sqrt{\frac{|(e^b|\alpha {|}^2-|\beta {|}^2)(G_2(\tau )+G_{-2}(\tau ))|}{e^b+1}}.$$

$$\begin{array}[]{l}\Pi_{+}=\frac{1}{2}+\frac{n_{x}I_{x}+n_{y}I_{y}+n_{z}I_{z}}{2},\\ \Pi_{-}=\frac{1}{2}-\frac{n_{x}I_{x}+n_{y}I_{y}+n_{z}I_{z}}{2},\end{array}$$

$$n_x^2+n_y^2+n_z^2=1.$$

$$n_x=0,|n_y|=1,n_z=0.$$

Full text

multiple quantum NMR experiment, quantum correlations, entanglement, quantum discord

I Introduction

Multiple quantum (MQ) NMR baum is an effective method for investigation of various problems of quantum informatics. MQ NMR creates multiple-qubit coherent states which can be used for the investigation of the quantum state transfer cappelaro , quantum correlations pyrkov ; kuznetsova , and decoherence processes on a basis of the study of relaxation of MQ NMR coherences doronin ; kaur ; bochkin .

Dimers are very suitable systems for a study of quantum correlations. Crystalline hydrates are examples of such systems. It is important that MQ NMR dynamics can be investigated analytically at different initial states of the system furman .

Dimers allow an investigate of not only entanglement, which is a measure of quantum correlations in pure states zenchuk , but also the quantum discord henderson ; zurek which is a measure of quantum correlations both in pure and mixed states. Those measures of quantum correlations are closely connected with the intensities of MQ NMR coherences which can be observed in MQ NMR experiments baum .

The paper is organized as follows. MQ NMR dynamics of dimers is investigated in section 2. It is shown that MQ NMR coherences of the zeroth and second orders only emerge in dimers. The intensities of those coherences are calculated. In section 3 we show that entangled states appear in the course of the system evolution in the MQ NMR experiment with dimers. We find that the concurrence hill ; wootters which is the important characteristics of entanglement is connected with the intensity of the MQ NMR coherence of the second order. The quantum discord in dimers is investigated in the high temperature approximation in section 4. The optimization of the quantum conditional entropy is performed with ’’Mathematica’’. We briefly summarize our main results in section 5.

II MQ NMR dynamics of dimers

We consider a system of two spins connected by the dipole-dipole interaction in an external magnetic field. Initially one spin is in a pure state and the second spin is in a thermodynamic equilibrium state. The initial density matrix $\rho_{0}$ of the system can be written as

[TABLE]

where $|0\rangle$, $|1\rangle$ form a basis in the Hilbert space of the first spin, $\alpha$, $\beta$ are complex numbers ($|\alpha|^{2}+|\beta|^{2}=1$), $\omega_{0}$ is the Larmor frequency. $T$ is the temperature, $I_{z,2}$ is the $z$-projection operator of the second spin, and $Z$ is the partition function ($Z=\mathrm{Tr}\{e^{\frac{\hbar\omega_{0}}{kT}I_{z,2}}\}$). In the standard basis $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$ nielson the density matrix (1) can be written as

[TABLE]

where $b=\frac{\hbar\omega_{0}}{kT}$. For a two-spin system the Hamiltonian $H_{MQ}$, describing MQ NMR dynamics on the preparation period of the MQ NMR experiment baum , can be presented in the following form

[TABLE]

where $d$ is the dipolar coupling constant and $I_{i}^{+}$ and $I_{i}^{-}$ are the raising and lowering operators. In the end of the preparation period of the MQ NMR experiment the density matrix $\rho(\tau)$ is

[TABLE]

Using the technique of Ref. pyrkov one can obtain the following expression for the density matrix

[TABLE]

The diagonal elements $\rho_{0}(\tau)$ of the density matrix (5) are responsible for the MQ NMR coherence of the zeroth order. The non-diagonal elements $\rho_{2}(\tau)$, $\rho_{-2}(\tau)$ of $\rho(\tau)$ are responsible for the MQ coherences of the plus/minus second orders. It means that the density matrix $\rho(\tau)$ can be represented as

[TABLE]

It is natural that in two-spin systems only MQ NMR coherences of the zeroth and second orders emerge. An analogous situation takes place feldlacele ; feldlacele2 ; maximov ; feldman in linear spin chains in the approximation of the nearest neighbor dipolar interactions goldman . The intensities of the MQ NMR coherences of the zeroth ($G_{0}(\tau)$) and plus/minus second ($G_{\pm 2}(\tau)$) orders, which can be observed experimentally, are determined by

[TABLE]

where $\rho_{0}^{ht}(\tau)$ and $\rho_{\pm 2}^{ht}(\tau)$ are the diagonal and non-diagonal parts of the high temperature density matrix feldman2002

[TABLE]

where $I_{z}=I_{1z}+I_{2z}$. Tedious calculations with (5)-(8) allow us to obtain

[TABLE]

III Entanglement of dimers in MQ NMR experiments

In order to calculate entanglement in dimers in MQ NMR experiments we use the criterion of wootters . To this end we consider the matrix

[TABLE]

where $\sigma_{y}$ is the Pauli matrix. The concurrence, which is closely connected with entanglement hill ; wootters , is determined wootters by the relation

[TABLE]

where $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ are the square roots of the eigenvalues of the matrix product $\rho(\tau)\tilde{\rho}(\tau)$.

After tedious calculations one obtains

[TABLE]

Comparing Eq. (10) and Eq. (13) one can find an important connection of the concurrence and the intensities of the MQ NMR coherence of the second order

[TABLE]

Such a comparison is shown in Fig.1.

IV The quantum discord in the high temperature approximation

According to the current understanding aldoshin , total (quantum and classical) correlations in a system are defined by the mutual information. The problem is how to separate the classical correlations from quantum ones. The problem was solved in henderson ; zurek . The classical correlations in a bipartite system are determined by a complete set of projective measurements carried out only over one of the subsystems zurek . We take the corresponding projectors $\Pi_{+}$, $\Pi_{-}$ in the form doronin2015

[TABLE]

where $I_{\alpha}$ ($\alpha=x,y,z$) is the spin projection operator and the parameters (the directional cosines) $n_{x},n_{y},n_{z}$ are connected by the relation

[TABLE]

After performing projective measurements over the density matrix (5) we can calculate the quantum conditional entropy doronin2015 . However, this entropy depends on the parameters $n_{x},n_{y},n_{z}$. At the same time, we need to find the minimal value of the quantum conditional entropy. We solved the problem only in the high temperature approximation goldman when $b<<1$. Using ’’Mathematica’’ we obtain that the entropy reaches it minimal value at

[TABLE]

Then the quantum discord can be found by the standard method doronin2015 . The dependence of the quantum discord on the dimensionless time of the preparation period of the MQ NMR experiment is given in Fig. 2.

V Conclusions

We consider MQ NMR dynamics of dimers and show that the MQ NMR coherences of the zeroth and second orders only emerge in dimer systems. We also explore quantum correlations for dimers. We have found a connection between entanglement and the intensity of the MQ NMR coherence of the second order. We have also investigate the quantum discord.

Acknowledgements.

The work is supported by the Russian Foundation of Basic Research (Grant 16-03-00056) and the Program of RAS ’’Element base of quantum computers (Grant No. 0089-2015-0220). We thank Professor E.B. Fel’dman for stimulating discussions.