State tomography of a chain of qubits embedded in a spin field-effect transistor via repeated spin-blockade measurements on the edge qubit

As a possible physical realization of a quantum information processor, a system with stacked self-assembled InAs quantum dots buried in GaAs adjacent to the channel of a spin field-effect transistor has been proposed. In this system, only one of the stacked qubits, i.e., the edge qubit (the qubit closest to the channel), is measurable via “spin-blockade measurement.” It is shown that the state tomography of the whole chain of the qubits is still possible even under such a restricted accessibility. The idea is to make use of the entangling dynamics of the qubits. A recipe for the two-qubit system is explicitly constructed and the effect of an imperfect fidelity of the measurement is clarified. A general scheme for multiple qubits based on repeated measurements is also presented.

Figure 1

(Color online) Spin FET embedded with quantum dots.

Figure 2

(Color online) Mechanism of the tomography for the sequences of operations for (a) $p_{\uparrow }^{\left(1\right)}$, (b) $p_{\downarrow }^{\left(1\right)}$, and (c) $p_{\downarrow }^{\left(2\right)}$. The curves describe how the components of a state evolve and entangle in time according to Hamiltonian (3.1), while the “walls” represent the projective measurements on qubit $X$. The survived curves which are not shut off by the walls correspond to the events where every measurement gives the desired result. The probabilities of such events are nothing but the probabilities $p_{\uparrow (\downarrow )}^{\left(n\right)}$ for the sequences of operations and are equivalent (in the simple cases shown here) to the occupations of the survived states in the given initial state $\varrho$, yielding its relevant matrix elements, (a) $⟨\uparrow \uparrow |\varrho |\uparrow \uparrow ⟩$, (b) $⟨\uparrow \downarrow |\varrho |\uparrow \downarrow ⟩$, and (c) $⟨\downarrow \uparrow |\varrho |\downarrow \uparrow ⟩$.

Figure 3

(Color online) State tomography of $|{\Psi }^{-}⟩=\left({|\uparrow \downarrow ⟩}_{XA}-{|\downarrow \uparrow ⟩}_{XA}\right)/\sqrt{2}$ with partially polarized spin channel. The polarization of the channel is defined by $r=\left(N_{\uparrow }-N_{\downarrow }\right)/\left(N_{\uparrow }+N_{\downarrow }\right)$ with $N_{\uparrow (\downarrow )}$ as the number of spins in the $|\uparrow (\downarrow )⟩$ state in the channel.

Figure 4

(Color online) State tomography of $|{\Phi }^{-}⟩=\left({|\uparrow \uparrow ⟩}_{XA}-{|\downarrow \downarrow ⟩}_{XA}\right)/\sqrt{2}$ with partially polarized spin channel.