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  ELSEVIER 16January1995 Physics Letters A 197 (1995) 83-87 PHYSICS LETTERS Optimal distinction between two non orthogonal quantum states Gregg Jaeger , Abner Shimony a Department of Physics and College of General Studies, Boston University, Boston, MA 02215, USA b Departments of Physics andPhilosophy, Boston University, Boston, MA 02215, USA Received 14 September 1994; revised manuscript received 9 November 1994; accepted for publication 9 November 1994 Communicated by P.R. Holland bstract Two procedures are developed for classifying an individual system as IP) or I q), non-orthogonal, given an ensemble with respective proportions r and 1 - r. One (generalizing Ivanovic, Dieks, and Peres) infallibly classifies some systems, leaving others unclassified. The second is statistically optimum, allowing individual errors. Ivanovic [ 1 ], Dieks[ 2 ], and Peres [ 3 ] consider the problem of determining what quantum state an individual system was prepared in, when it is given that it was prepared either in IP) or I q), in general nonorthogonal. The problem, thus stated, is vague and can be clarified in (at least) two distinct ways: (1) What procedure yields on the average a maxi- mum number of correct classifications, in an ensem- ble of such cases, assuming that for each member of the ensemble a definite classification is made? (2) What procedure enables one in a maximum number of cases to infer with certainty whether the system was prepared in IP) or I q), leaving a minimum number of cases undecided? The three authors mentioned are primarily inter- ested in the problem (2), though Dieks briefly con- siders problem ( 1 ). Under the assumption that half of the ensemble is prepared in IP) and half in I q), all three obtain the same evaluation of the maximum probability of correct classification and minimum probability of no decisions: namely, P= probability of correct classification =l-I(plq) l, (la) 1 - P= probability of no classification = I (Plq) I ã (lb) The procedures suggested by Dieks and Peres are es- sentially the same: prepare an auxiliary system in a state Is o) and choose a unitary evolution (equiva- lent to a choice of interaction) which yields IP So -~a [P~ s~ +fliP2 sa , (2a) [qso ~lq, sl +61q2 s2 , (2b) where Isl , Is2 , [p, , IP2), Lql , Iqz are allnor- malized, and (Pl ]ql )=0, (2c) (st Is2 ) =0, (2d) and I q2 is identical with IP2) except for a phase fac- tor. With this unitary evolution there is a measure- ment on the auxiliary system that decides unequivo- cally between Is1 ) and Is2); if the former occurs, then a measurement between IPl) and Iq~) decides be- tween IP) and I q , and if the latter occurs the state of the primary system is inevitably IP2) (or equiva- lently I q2 ), leaving the choice between I P) and I q) 0375-9601/95/ 09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601 ( 94 )00919-8  84 G. Jaeger, A. Shimony /Physics Letters A 197 1995) 83-87 undecided. The procedure of Ivanovic is more cum- bersome, requiring possibly an infinite number of steps. The present note extends the work of Ivanovic, Dicks, and Peres in several ways. The first is to allow a proportion r of systems of the ensemble to be in ]p } and s= 1 - r to be in [ q}, instead of requiring r=s= ½. With no loss of generality, r>s. (3) The procedure for solving problem ( 1 ) is the same as that of Dieks and Peres, except for the evaluation of the coefficients on the r.h.s, of (2a) and (2b). From Eqs. (2a) and (2b), and the assumed proportions r and s, we have P=r]ol]iq-s]y]2= 1 - rl/?12-sI6l 2 . (4) By unitarity and (2c), (2d) I(plq) l=r/?l 161 I(pelq2)l, (5a) and hence I/?1 161 ~>l(Plq) l ã (5b) The maximum of P in Eq. (4), subject to the con- straint of Eq. (5b) and to 0 ~ I/?12 ~< 1, 0 ~< 1612 ~< 1, is achieved when I/?[2=max{l(Plq}l(s/r) /2, I(P[q)[2} (6a) and IP2 = Iq2 ã (6b) (Note: I/?12 s not permitted to be less than I (Pl q) 12, for if it were, then 161 would exceed unity. ) If 1/712= I(plq) l(s/r)l/Z>_-l(plq)12 , (7a) then 1612= I (plq} l (r/s) 1/2 (8a) and P= 1-2(rs)t/21 (Plq) I , ( ) but if 1/712= I (P[q)12> l (Plq) l (s/r) 1/2 , (7b) then 1612=1 (8b) and P=r(1-[(Plq) [2) . (9b) Note that if r=s= ½, then Eq. (7a) holds and hence (9a), which in this case agrees with Eq. (la). A second extension of the results of Ivanovic, Dieks, and Peres is to assume that the dimensionality of the Hilbert space of the system of interest is greater than two. It is then possible to parallel the strategy of Dicks and Peres without introducing an auxiliary system. We show that because of the dimensionality of the Hilbert space, we can express [p) and I q) in the form IP} =a IP~ } +fliP2 , (lOa) Iq} =YlqL } -t- 61P2 }, (10b) where Ip~), Iqt) and IP2) are orthonormal and/? and 6 satisfy Eqs. (5b) and (6a), and oe and 7 are real. To achieve this expression we first write I q} = el°Nip} + ( 1 -N 2 1/21m } , ( 1 la) N=l(plq)[, (lib) where Im) is normalized and orthogonal to IP). Then Eq. (10b) is equivalent to ei°N[p)+(1-N 2)t/2 Im)=ylql)+6lp2). (10e) There exists in the Hilbert space a normalized vector I/} orthogonal to ]p) and I m), and we shall express [Pl ), I q,) and [P2) explicitly in terms of IP), I m) and I/). At this juncture we must treat the cases of (7a) and ( 7b ) separately. If (7a) holds, make the following identifications of /?, & o~, y, IPl), Iql), and IP2}, /?=N W2(s/r) /4, (12a) 6= eiON l/2(r/s ) i/4, (12b) oe = [ 1 - (s/r) /2N] 1/2 12c) 7= [ 1 - (r/s)l/2N] l/i, (12d) IPl ) = [ 1 - (s/r) 1/2N] 1/2 IP} -iO e ( 1 --N 2 1/2 N[ 1 - (s/r)I/2N] l/2lm } ( 1 -N 2) l/5 N l/2 s/r),/4[ I -- r/s)I/2N] 1/21l), (12e)  G. Jaeger, A. Shimony / Physics Letters A 197 (1995) 83-87 85 Iql )-- l__N2)I/2 [1-(r/s)l/ZN]l/2lm) e i0 + (1 --N 2) '/sN'/Z(r/s)l/4[1 -- (s/r)l/2N] ~/Zll) 12f) 11)2 ) =N'/Z(s/r)1/4 IP) e -io + (1-N 2) l/:iNl/2(r/s)l/4[1 -- (s/r)l/2N] m) ( 1 -N 2) ,/2 [ 1 - (r/s)l/2N] ,/2 X [ 1 - (s/r)'/2N l '/21l ) . 12g) Then eqs. (10a), (10b) are satisfied with the requi- site orthonormality of IP~ ), I q~ ), and IP2), while fl and ~ satisfy Eqs. ( 5b ) and (6a) and a and ~ are real. Hence we can optimally decide with certainty (but sometimes abstaining from a decision) whether the system was initially in IP) or in I q) by a procedure analogous to that given in Eqs. (4)-(9a). Specifi- cally, measure A=lpl)(pll+2lql)(qll+3lp2)(P2l . (13) If the measured value of A is 1, we know with cer- tainty that the system was prepared in IP) ; if the value is 2, we know with certainty that it was prepared in I q); and if the value is 3, we do not know and abstain from deciding. The probability of a correct classifi- cation is given by Eq. (9a). This agreement shows that the essence of the procedure is to increase the dimensionality of the relevant Hilbert space beyond the two dimensions determines by IP) and I q)- If the Hilbert space of the system of interest is intrinsi- cally greater than two, then one need not introduce an auxiliary system in the manner of Dieks and Peres. In the case of Eq. (7b), I~1 is unity and therefore Eqs. (10a), ( 10b ) reduce to IP) -- c~ Ip, ) +fliP2 ) , (14a) Iq) =~lp2) , (14b) where Now I P) can be rewritten [p) = e~°NI q) + ( 1 -N 2) 1/21/7 ) , 16 ) where In) is normalized and orthogonal to I q)- An optimum procedure for classifying a system as being in IP) or Iq) with certainty is to measure the observable B= In) (nl +21q) (ql ã (17) If the measured value is 1, then the system can be inferred with certainty to have been prepared in IP); if 2, then there is no decision. The probability of a correct classification is P=rlal2=r( 1- I (plq)12 ) , (18) in agreement with Eq. (9b). Note that the procedure can be carried out in a Hilbert space of dimension two and hence is more economical than the proce- dure in case (7a). A third extension is previous work is to solve prob- lem ( 1 ), by finding a procedure for maximizing on the average a correct classification. Again, we con- sider the general preparation in which a proportion r is prepared in IP) and s- 1 -r in I q). We seek a bi- valent procedure, which prescribes choosing IP) upon one outcome and I q) upon the other. There is no loss of generality in restricting the bivalent proce- dure to measuring a projection operator E on the Hil- bert space of the system, with eigenvalues 1 and 0. If the measured value of E is 1 choose IP), if0 choose I q). The probability of a correct choice is P=r(plEIp) +s( 1 - (qlEI q) ) ã (19) Decompose I q) into a superposition oforthonormal IP) and I m), as in Eq. ( 1 la), and consider the ac- tion of the projection operator E on these two vectors, EIp) =clp) +c' lm) +c ll) , (20a) El m) =dip) +d' Im) +d [l'), (20b) where II) and II ) are orthonormalto both IP) and [m). (Of course, if the Hilbert space is two-dimen- sional, there are no terms in I ) and I I' ). ) Then (pIEIp) =c (21a) IPl =N~ I<Plq>l, (15a) lal= (1-N2) 1/2 (15b) The procedure used here was presented in Section 2 ofRef. [ 4 ] by Jaeger et al. In that paper the procedure was used for predic- tion, whereas here it is used for retrodiction.  86 G. Jaeger, A. Shimony / Physics Letters A 197 (1995) 83-8 7 and ( qlEIq) =N2c+2 cos(0+ ¢~)N(1-N 2) t/21c' I +(1-NZ)d ' , (21b) where d*=c' = [c' I ei° . (21c) Since E2=E, (x[E[x) = (Ex[Ex) for any Ix), and therefore c and d are real and c=c2+ [c' [2+ [c ]2, (22a) d'= [d12+d'2+ [d [2. (22b) Hence P=rc+s[1-Nac - (1-N2)d ' 2 cos(0+ (~)N( 1 -N 2) 1/21c' [ l ã (23) When c, d', [c'] are fixed, P is maximized by cos(0+ ~) = - 1, P= (r-sN2)c+s-s( 1 -NZ)d +2sN(1 -N 2) 1/21c' I . (24) By (22a), (22b) c=½ +_ ½ 1-4(Ic' 12+ Ic [2 ]1/2, (25a) d'=½+½[1-4(lc'12+ld l 2)]1/2 (25b) and since r>~s we shall take the + for c, - for d'. Moreover, when I c' I is fixed, c is maximized and d' is minimized when c = d = 0, in other words, when E is a projection operator on the two-dimensional Hilbert space spanned by IP) and I q). Then c=½(l+x) (26a) d'=~(l-x), (26b) where x= (1-41c' 12) 1/2 (26c) I c'l =~(1-x2)~/2. (26d) Hence P= ½ +x(½ -sN 2) +sN( 1 -N 2) 1/2( 1 -to 2) 1/2. Maximum P yields (27) I _sN 2 x= [(~_sNZ)2+sZNZ(l_N2)]I/2 (28) and P=½+½(1-4rsl (p[q)12 ) 1/2 (29) The procedure for achieving this optimum probabil- ity of a correct classification is to measure the projec- tion operator E determined by the x of Eq. (28), and choose IP) if the outcome is 1, I q) if the outcome is 0. We emphasize that this procedure works whether the dimension of the Hilbert space of the system of interest is two or greater than two. For the case of r= s = ½, Eq. (29) implies P=½+½(1- I(plq)[2)1/2 (30) For this case, Dicks considers (without claiming op- timality) a procedure which yields a probability PD of correct classification, PD=I--½1(pIq) I 2. (31) But P-PD = ½ (1 -I (Plq)[ 2) 1/2_ (1 - [ (Plq) 12)] >/0, (32) and therefore our procedure is preferable to the one considered by Dicks. As a final remark, we consider a class of practical situations in some of which the procedure of problem (1) is appropriate, and in others the procedure of problem (2). Suppose that a wager can be made about each system of the ensemble, with a gain g> 0 if the classification is correct, a loss l> 0 if the classifica- tion is incorrect, and neither gain nor loss if the sub- ject refrains from betting. The average gain using the optimum procedure of problem (2) is G2 =g[1-2(rs)l/21 (p[ q) [] (33a) ifEq. (7a) holds, and G2=gr(l-I(Plq) [2) (33b) if Eq. (7b) holds. The average gain using the opti- mum procedure of problem ( 1 ) is GI =g[½ + ½ 1-4rs[ (Plq)12) 1 2] -l[½-½ (1-4rs] (p[q)[2)1/2] (34) If 2 ( rs ) 1 21 (P I q ) I is less than unity, then clearly for
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