Optimal bounds on state transfer under quantum channels with application to spin system engineering

Modern applications of quantum control in quantum information science and technology require the precise characterization of quantum states and quantum channels. In particular, high-performance quantum state engineering often demands that quantum states are transferred with optimal efficiency via realizable controlled evolution, the latter often modeled by quantum channels. When an appropriate quantum control model for an interested system is constructed, the exploration of optimal bounds on state transfer for the underlying quantum channel is then an important task. In this work, we analyze the state transfer efficiency problem for different classes of quantum channels, including unitary, mixed unitary, and Markovian. We then apply the theory to nuclear magnetic resonance (NMR) experiments. We show that two most commonly used control techniques in NMR, namely gradient field control and phase cycling, can be described by mixed unitary channels. Then we show that employing mixed unitary channels does not extend the unitarily accessible region of states. Also, we present a strategy of optimal experiment design, which incorporates coherent radio-frequency field control, gradient field control, and phase cycling, aiming at maximizing state transfer efficiency and meanwhile minimizing the number of experiments required. Finally, we perform pseudopure state preparation experiments on two- and three-spin systems, in order to test the bound theory and to demonstrate the usefulness of nonunitary control means.

Figure 1

For an $N$-dimensional density operator $\rho$, its universal bound on spin dynamics is geometrically a permutohedron of order $N$. Here shows the first examples of permutohedrons: (a) $N=2$ (line segment); (b) $N=3$ (hexagon); (c) $N=4$ (truncated octahedron). The center point $O$ denotes the maximally mixed state $\mathit{1}/N$; here $\mathit{1}$ means the identity operator. If some of the eigenvalues of $\rho$ are equal to each other, then the corresponding permutohedron degenerates.

Figure 2

Statistical average of a series of unitary transform experiments gives a mixed quantum channel $\widehat{\mathcal{E}}$. Here ${\mu }_k$ is the weight or probability of applying $U_k$.

Figure 3

Geometric illustration of how to determine a minimal set of unitaries to prepare a target operator $\sigma$ with optimal efficiency.

Figure 4

Illustration of the nonunitary evolution under gradient field control. Because the dephasing is proportional to the gradient strength, the coherences vary from zero to $2\pi$ along the sample.

Figure 5

Permutohedron constructed by taking the convex hull of all four-vectors from the $4!=24$ diagonal permutations of ${\rho }_{\text{eq}}$.

Figure 6

Pulse sequence for two-qubit PPS preparation with two controlled-transfer gates.

Figure 7

Experimental spectra (in Hz) of pseudopure states prepared using previous approaches and our approaches. From left to right, (a)–(g) are spectra obtained by applying readout pulses to the thermal equilibrium state (a), PPS prepared by spatial averaging method (b), by line selective method (c), by controlled-transfer gate method (d), by periodic control (e), and by line selective saturation method (f), respectively. The small signal visible at the center of the ${}^1\mathrm{H}$ spectrum arises from an impurity (unlabeled chloroform). All spectra are plotted on the same vertical scale.

Figure 8

Experimental results of labeled-PPS preparation using conventional phase cycling method and our optimal strategy. (a) Full NMR equilibrium spectrum, where the chemical shifts (in Hz) of ${\mathrm{C}}_1$, ${\mathrm{C}}_2$, and ${\mathrm{C}}_3$ nuclei are marked. (b),(c) ${\mathrm{C}}_2$ NMR spectra for labeled PPS obtained by the phase cycling method (b) and by our optimal strategy (c). The middle three spectra in (b) and two spectra in (c) are obtained from implementing each unitary operation of the mixed unitary channel described in the main text. The bottom spectra in (b) and (c) are thermal equilibrium spectra of ${\mathrm{C}}_2$ nucleus and the top ones are the spectra of labeled PPS, which are obtained by averaging the middle spectra.