Evolution of quantum discord and its stability in two-qubit NMR systems

We investigate evolution of quantum correlations in ensembles of two-qubit nuclear spin systems via nuclear magnetic resonance techniques. We use discord as a measure of quantum correlations and the Werner state as an explicit example. We, first, introduce different ways of measuring discord and geometric discord in two-qubit systems and then describe the following experimental studies: (a) We quantitatively measure discord for Werner-like states prepared using an entangling pulse sequence. An initial thermal state with zero discord is gradually and periodically transformed into a mixed state with maximum discord. The experimental and simulated behavior of rise and fall of discord agree fairly well. (b) We examine the efficiency of dynamical decoupling sequences in preserving quantum correlations. In our experimental setup, the dynamical decoupling sequences preserved the traceless parts of the density matrices at high fidelity. But they could not maintain the purity of the quantum states and so were unable to keep the discord from decaying. (c) We observe the evolution of discord for a singlet-triplet mixed state during a radio-frequency spin-lock. A simple relaxation model describes the evolution of discord, and the accompanying evolution of fidelity of the long-lived singlet state, reasonably well.

Figure 1

The Venn diagram representing total information $\mathcal{H}(A,B)$, individual informations $(\mathcal{H}\left(A\right)$, $\mathcal{H}\left(B\right))$, conditional informations $(\mathcal{H}\left(A\right|B)$, $\mathcal{H}\left(B\right|A))$, and mutual information $\mathcal{I}(A:B)=\mathcal{J}(A:B)$ in classical information theory.

Figure 2

Various correlations as functions of the purity $\epsilon$ for Werner states of the form given in Eq. (19). The inset shows the range of discord for the purity available in our NMR setup.

Figure 3

(a) The molecular structure of ${}^{13}$C-chloroform, and (b) the pulse sequence for discord preparation. In (b), PFG is the pulse field gradient operation which destroys the coherences and retains the diagonal elements of the density matrix.

Figure 4

(a) Fidelity relative to the Werner state of the experimental and the simulated states as a function of $\theta$, and (b) corresponding discord values in units of ${\epsilon }^2/\ln 2$. The maximum discord is obtained for the delay parameter $\theta =(2n+1)\pi /2$, corresponding to preparation of Bell-diagonal states.

Figure 5

(a) Fidelity of the experimental state relative to the Werner state for various DD schemes and (b) corresponding discord values in units of ${\epsilon }^2/\ln 2$.

Figure 6

The pulse sequence for preparing long-lived singlet states (a), molecular structure of 5-chlorothiophene-2-carbonitrile (b), and traceless real parts of the theoretical (c) and the experimental (d) density matrices. The experimental Werner state in (d) was obtained with a spin-lock of 16.4 s and has a fidelity of 0.99.

Figure 7

(a) Fidelity of the experimental state relative to the Werner state as a function of the spin-lock duration $\tau$, and (b) the corresponding values of discord in units of ${\epsilon }^2/\ln 2$ and geometric discord in units of ${\epsilon }^2/2$. Discord values were obtained using the methods described in Sec. 2.