Thus the state of system A is set back to\<j>)every after r, while that ofB just evolves dynamically on the basis of the total HamiltonianH.Werepeat the same measurement, represented by (3),Ntimes and collect onlythose events in which system A has been found in state\<j>)consecutivelyNtimes;other events are discarded. The state of system B is then describedby the density matrixP{B\N)=( W ) % ( 0 ) ( l ^ ( r ) )V( r )W ,(4)whereV*{T)= (<f>\e-iH^)(5)is an operator acting on B andP(r>(N) =Tr\(Oe-iHT0)Np0(OeiHTO)N= TrB[(V4>(r))NpB(0)(Vl(r))N](6)is the success probability for these events to occur (yield). This normaliza-tion factor in (4) reflects the fact that only right outcomes are collected inthis process.In order to examine the asymptotic state of system B, consider thespectral decomposition of the operatorV^(T),which is not hermitian,V^(T)^Vl(r).We therefore need to set up both the right- and left-eigenvalue problemsV^(r)|u„) =\„\un),{vn\Vd>{r)-A„(t>„|.(7)The eigenvalue A„ is complex valued in general, but its absolute value isbounded120 < |A„| < 1,(8)which is a reflection of the unitarity of the time evolution operatore~lHr.These eigenvectors are assumed to form a complete orthonormal set in thefollowing sense^2\un)(Vn\=I s ,(Vn\um)-8nm.(9)

262Then the operator V^(rNitself is expanded in terms of these eigenvectors^ ( r ) = ^ A > „ ) ( i ;n| .(1 0)nIt is now easy to see that the TVth power of this operator is expressed asnand therefore it is dominated by a single term for largeN(V^f^^X^uoKvol(12)when the largest (in magnitude) eigenvalue Ao isdiscrete, nondegenerateand unique.If these conditions are satisfied, the density operator of systemB is driven to a pure statepW(N)1^1^|«o)(«0|/(uo|«o)(13)with the probabilitypW(iV)^^U\\o\2N(u0\u0)(v0\pB(0)\v0).(14)The pure state |u0), which is nothing but the right-eigenvector of the op-erator V^(r) belonging to the largest (in magnitude) eigenvalue Ao, is thusdistilled in system B. This is the purification scheme proposed in9.A few comments are in order. First, the final pure state |«o) towardwhich system B is to be driven is dependent on the choice of the state\<p)on which system A is projected every after measurement, the measurementintervalrand the HamiltonianH,but does not depend on the initial stateof system B at all. In this sense, the purification is accomplished irrespec-tively of the initial (mixed) state ps(0). Second, as is clear in the aboveexposition, what is crucial in this purification scheme is the repetition ofone and the same measurement (more appropriately, spectral decomposi-tion) and the measurement interval T need not be very small. It remains tobe an adjustable parameter. Third, if we can make other eigenvalues thanAo much smaller in magnitude|A„/A0|<1forn^ 0,(15)by adjusting parameters, we will need fewer steps (i.e., smallerN)to purifysystem B.

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Term

Fall

Professor

Natalia Ramos

Tags

L Accardi