Qubit feedback and control with kicked quantum nondemolition measurements: A quantum Bayesian analysis

The informational approach to continuous quantum measurement is derived from positive operator-valued measure formalism for a mesoscopic scattering detector measuring a charge qubit. Quantum Bayesian equations for the qubit density matrix are derived, and cast into the form of a stochastic conformal map. Measurement statistics are derived for kicked quantum nondemolition measurements, combined with conditional unitary operations. These results are applied to derive a feedback protocol to produce an arbitrary pure state after a weak measurement, as well as to investigate how an initially mixed state becomes purified with and without feedback.

Figure 1

Weak entangling of a logical qubit with many ancilla qubits, followed by projective measurement on the ancilla qubits can be mapped onto the measurement of a quantum double dot by the transport electrons of a quantum point contact. Two given measurement realizations are shown, depending on whether the logical qubit state is either ∣1⟩ or ∣0⟩. The random ancilla result 1 is mapped onto measuring an electron in the current collector (denoted with a filled circle), while the random ancilla result 0 is mapped onto not measuring an electron in the current collector, or equivalently, measuring a hole (denoted with an empty circle). The fact that the states ∣1⟩ and ∣2⟩ of the logical qubit alter the probability of projecting the ancillas to 0 or 1, allows a weak POVM measurement on the logical qubit.

Figure 2

(Color online) (After Ref. 11.) Visualization of the kicked QND measurement scheme. A voltage pulse is applied to the quantum point contact on a time scale ${\tau }_V⪡{\tau }_q$, followed by a quiet period of zero voltage bias, lasting for a Rabi oscillation period ${\tau }_q$, followed by another pulse, and so on. The up/down variation is depicted, where the kicks come every half period, and the sign of the voltage pulse alternates with every kick. In this scheme, qubit read-out is by simply measuring the sign of the current, and corresponds to an elementary quantum pump.

Figure 3

(Color online) (After Ref. 11.) The conditional evolution of all initial pure states, represented on the Bloch sphere, under continuous QND measurement by an efficient detector. From (a)–(e), the rapidity of the measurement is $\gamma =N\mathcal{I}∕D=(-1,-.5,0,.5,1)$, respectively. As the detector obtains more information about the quantum state, we can with greater statistical certainty distinguish the post-measurement quantum state, so the Bloch sphere is more and more red ($\mid 1⟩)$ or blue ($\mid 2⟩)$, depending on the value of the rapidity measured. The conditional evolution of several representative states is also indicated with black arrows. The view-point is parallel to the equator of the Bloch sphere, so $\mid 1⟩\to \text{North}$ pole, and $\mid 2⟩\to \text{South}$ pole.