Continuous measurement of entangled qubits

The problem of quantum measurement of a qubit ~twolevel system! received renewed attention recently in relation to its importance for quantum computing. The case of sufficiently fast ~instantaneous! measurement can be readily described by ‘‘orthodox’’ collapse postulate @1#, and this is the case assumed at present by all quantum algorithms. However, in practice, especially for solid-state qubits, the act of measurement is not instantaneous. Because of typically low coupling between a solid-state qubit and a detector, it takes a considerable time before the qubit state is completely destroyed by the act of measurement. Correspondingly, because of fundamentally unavoidable noise of the detector, the information about the state of the measured qubit is available not immediately, but only after some time sufficient to get an acceptably large signal-to-noise ratio. It is important that the time scale of measurement ~and collapse! process may be comparable to the time scale of ‘‘free’’ qubit evolution ~e.g., due to Rabi oscillations! or to the duration of the detector on-off operation sequence. ~For example, if the detector is switched off when signal-to noise ratio is still on the order of unity, the measurement is only partially completed.! So, for practical needs we should be able to describe the measurement of a solid-state qubit as a continuous process. The formalism suitable to describe a continuous measurement of an ensemble of qubits has been developed two decades ago @2# ~for its use in quantum computing problems see, e.g., Ref. @3#!. In contrast, the formalism describing the process of measurement of a single qubit have been presented only recently @4–6# and is still in the stage of active development. ~In fact, it can be considered as a direct continuation of the well-developed field of selective or conditional quantum measurements — see, e.g., Refs. @7–12# and references in Ref. @5#!. This formalism is called Bayesian ~because of the essential role of the Bayes formula @13#! and combines advantages of the ‘‘orthodox’’ approach @1# ~the ability to treat single quantum systems! and the Leggett’s approach @2# ~the ability to treat continuous measurement!. The Bayesian approach has been applied so far only to the continuous measurement of a single qubit. @4–6,14–16# In this paper we apply it to derive the equations describing continuous measurement of entangled qubits. Let us consider first the case of two entangled qubits, one of which is continuously measured by a detector ~Fig. 1!. As a main example, we consider qubits made of double quantum dots while the detector is a quantum point contact @realizations based on single-electron transistors and superconducting quantum interference devices ~SQUIDs! are also possible — see Ref. @5##. Let us denote four basis vectors characterizing the state of two qubits as u1&[ u↑↑&,