Probabilistic quantum control via indirect measurement

The most basic scenario of quantum control involves the organized manipulation of pure dynamical states of the system by means of unitary transformations. Recently, Vilela Mendes and Man’ko have shown that the conditions for controllability on the state space become less restrictive if unitary control operations may be supplemented by projective measurement. The present work builds on this idea, introducing the additional element of indirect measurement to achieve a kind of remote control. The target system that is to be remotely controlled is first entangled with another identical system, called the control system. The control system is then subjected to unitary transformations plus projective measurement. As anticipated by Schrödinger, such control via entanglement is necessarily probabilistic in nature. On the other hand, under appropriate conditions the remote-control scenario offers the special advantages of robustness against decoherence and a greater repertoire of unitary transformations. Simulations carried out for a two-level system demonstrate that, with optimization of control parameters, a substantial gain in the population of reachable states can be realized.

Figure 1

Coherent-vector representation of quantum dynamics for a two-level system. (a) The effect of entanglement on a pure state. (b) Unitary transformation of a mixed state. (c) Projective measurement on either system of an entangled pair. In each operation, the initial vector [final vector] is drawn as an arrow with a white (black) head.

Figure 2

(a) Control and target system become entangled. (b) A unitary transformation is applied to the control system. (c) A projective measurement is performed on the control system.