Noisy quantum measurement of solid state qubits

Noisyquantummeasurementofsolid-statequbitsAlexanderN. KorotkovDepartment of Electrical Engineering, Universityof California, Riverside,CA92521ABSTRACTThe quantum evolution of an individual solid-state qubit during the pro cess of its continuous measurement canbe described by the recently develop ed Bayesian formalism.In contrast to the conventional ensemble-aeragedformalism, it takes into account the measurement record (in a way similar to the standard Bayesian analysis) andtherefore is able to consider individual realizations of the measurement pro cess.The formalism provides testableexp erimental predictions and can b e used for the analysis ofa quantum feedbackcontrol of solid-state qubits.The Bayesian formalism can b e also applied to the continuous measurementof entangled qubits; in particular,it shows how to create a fully entangled pair of qubits without their direct interaction, just by measuring themwith an equally coupled detector.Keywords:quantum measurement, qubit, quantum noise, quantum feedback, entanglement1. INTRODUCTIONThe approachof Bayesian analysis to the problem of continuous quantum measurement is a relatively new sub jectfor the solid-state community,even though this approach has a long history1as a general quantum frameworkand is rather well develop ed, for example, in quantum optics2, 3(for more references, see Ref.4) The main problemconsidered in this pap er is a very simple question:whatis theevolutionofa quantum two-levelsystem (qubit)duringtheprocess ofits measurement bya solid-state detector(Fig. 1)?In spite of simplicity of the question,the answer is not trivial.Thetextb o ok\ortho dox"quantummechanics5saysthatthemeasurementshouldinstantlycollapsequbitstate,so thatafter themeasurement thequbit state iseitherj1ior2,dep endingon thetoutcome.[The measurement basis is obviously de ned by the detector; in particular, it is a charge basis for theexamples of Figs. 1(a) and 1(b).]Such answer is sucient for typical optical exp eriments when the measurementis instantaneous (a scintillator ash or a photo counter click).However, for typical solid-state setups (as well asfor some more advanced setups in quantum optics2, 3) the instantaneous collapse is not a sucient answer.Inparticular, in the examples of Fig. 1 typically the detector is weakly coupled to the qubit, so the measurementpro cesscantakeasigni canttimeandthereforethecollapseshouldbeconsideredascontinuouspro cess.The notion of a continuous evolution due to measurementiswell accepted in the solid-state community and isusuallyconsidered withintheframework ofLeggett's formalism.6Thisformalism gives thedecoherence-based answer to the question p osed ab ove.It says thatthe nondiagonal matrix elementsof thequbit densitymatrix (obtained by tracing over the detector degrees of freedom) gradually decay to zero, while thediagonalmatrix elements do not evolve (assuming that the qubit do es not oscillate by itself,H=0, wheredescrib esthe tunneling b etweenj1iand2).So, after the completed measurementwehave an incoherent mixture of thestatesj1iand2.Letusnoticethatthesetwoanswerstoourquestionobviouslycontradicteachotherandthe\ortho dox"answercannotbeobtainedassomelimitingcaseofthedecoherenceer.Theresolutionapparentcontradictionissimple:twoapproachesconsiderdi erenob jects.Thedecoherencehdescrib estheaverageevolutionoftheensemblequbits,while\ortho dox"quantummechanicsisdesignedtotreatasinglequantumsystem.Thisdi erencealsoexplainstheinabilityofdecoherenceformalismtotakemeasurement outcome into account.Obviously,itis desirable to have a formalism whichwould combine advantages ofthetwo approaches anddescrib e thecontinuous measurement of a single qubit.Then the \ortho dox" result would b e a limiting case forvery fast (and \strong") measurement, while the decoherence result could b e obtained by an ensemble aeraging.TheBayesianformalism4, 7whichisthesub jectofthispap erhasb eendevelop edexactlyforthatpurp ose