On the Decompositions of a Quantum State

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 10, ARTICLE NO. AY On the Decompostons of a Quantum State G. Cassnell,* E. De Vto, and A. Levrero Dpartmento d Fsca, Unersta ` d Genoa, I.N.F.N., Sezone d Genoa, Va Dodecaneso 33, 16146, Genoa, Italy Submtted by James S. Howland Receved May 13, 1996 We classfy all the decompostons of a quantum state as a weghted sum of one dmensonal projectors. In partcular we descrbe explctly the set of rreducble decompostons. The physcal nterest n ths problem rests on the possblty of nterpretng the decomposton n terms of a classcal mxture Academc Press 1. PRELIMINARIES In the ordnary Hlbert space formulaton of quantum mechancs the states of a quantum system are the postve trace one operators on H, where H s the complex separable Hlbert space Žwth nner product Ž,.. assocated to the quantum system. The set S of states has the followng propertes: Ž. 1 t s a closed subset of the Banach space of trace class operators wth respect to the trace norm; Ž. t s convex, that s, f W, W S, then W ww Ž 1 ww. 1 1 s n S for all 0 w 1. We say that W s a conex combnaton of W 1 and W ; Ž. 3 f Ž W. s a famly of states and Ž w. I I s a famly of weghts,.e., 0 w 1 and Ý w 1, then Ž ww. I I s summable wth respect to the trace norm topology and ts sum s a state. Snce w 0 and Ž w. I s summable, then the ndex set I s necessarly fnte or countable. In ths paper we use the word famly to mean ether a fnte or countable famly; * E-mal address: cassnell@genova.nfn.t. E-mal address: devto@genova.nfn.t. E-mal address: levrero@genova.nfn.t X97 $5.00 Copyrght 1997 by Academc Press All rghts of reproducton n any form reserved. 47

2 DECOMPOSITIONS OF A QUANTUM STATE 473 Ž. 4 the extremal ponts of S, that s, the states that are not convex combnatons of two dfferent states, are exactly the one dmensonal projectors Ž the pure states.. The prevous propertes are well known, classcal facts n the theory of bounded and trace class operators n Hlbert spaces. In ths paper we shall use freely some basc, elementary results of ths theory. Let W be a state and P a set of pure states. For the sake of clarty we label P wth an ndex set I, so that P P 4 I and P Pj f j. We say that P decomposes W f there exsts a famly of weghts Ž w. I such that the Ž summable. famly Ž wp. has sum W Ž I wth respect to the trace norm.; n ths case the set Ž P, w.4 I s a decomposton of W. We stress that snce the famly Ž wp. I s summable, the fact the P decom- poses W does not depend from the ndex set I labellng P. The spectral theorem for selfadjont compact operators assures that a decomposton exsts for any state. However, f W s not a pure state, many decompostons correspond to the same state. Ths fact can be easly seen by drect computaton n the elementary case H. The study of the possble decompostons of a state has at least two physcal motvatons. Ž. 1 The central objects of nterest n quantum mechancs are the expectaton values TrŽ AW. where A ranges over the selfadjont bounded operators representng the physcal quanttes and W s the state of the system. Gven a decomposton W Ý I wp the prevous expectaton values can be expressed as TrŽ AW. w TrŽ AP.; Ý I dfferent decompostons of W can be suggested by the concrete physcal problem at hand, see, for example, Remark 5. Ž. The non-unqueness of the decomposton of a state W s at the root of Ž many of. the dffcultes n the nterpretaton of quantum mechancs, as wtnessed, for example, n two recent textbooks, 1, ; the problem of characterzng the possble decompostons of a gven state s thus an mportant ssue n the foundaton of quantum mechancs. In ths physcal framework, gven a state W we can sngle out three mathematcal problems: Ž. a classfy all the decompostons of W; Ž b. determne whether W can be decomposed on a gven set of pure states; Ž. c f a gven set of pure states decomposes W, determne whether the famly of weghts s unque.

3 474 CASSINELLI, DE VITO, AND LEVRERO The present paper solves the frst problem by gvng a complete charactersaton of all the decompostons of a gven state; ths s done n Secton. In Secton 3 we descrbe explctly a partcular class of decompostons that have relevance both from the physcal and the mathematcal pont of vew; for them we can answer also to the remanng two questons. To the best of our knowledge, the frst author who posed the problem Ž. a consdered n ths paper was Jaynes 5. He consdered only the case of fnte decompostons of a state W wth fnte dmensonal range. Ths case was completely worked out by Hughston et al. 4. Nevertheless ths result s not exhaustve, snce a state wth fnte dmensonal range can have countable, non-fnte, decompostons. A frst partal result on the general case e., countable decompostons for any state. was gven by Hadjsavvas 3.. THE DECOMPOSITIONS OF A STATE Let W be a state on H, K the closure of the range of W, and N ts kernel. We denote by P the projector onto K. AsW s selfadjont we have H N K. Possbly embeddng H n a bgger space, we can assume that N s nfnte dmensonal Ž see Remark 3 nfra.. We recall that Ran W Ran W K Ran W, where Ran denotes the closure n H of the range. Snce the restrcton of W to K s njectve, there exsts a unque Ž n general unbounded. selfadjont operator T actng n K, wth doman Ran W, such that TW K W T Ran W. Let e 4 be a set of vectors n H. We say that e 4 I I s nondegenerate wth respect to P when Ž. 1 e 4 I s orthonormal; Ž. spane : I4 Ran P where span denotes the closure n H of the subspace algebracally spanned; Ž. 3 Pe 0 for all I; Ž. 4 for all pars, j IPe s not collnear wth Pe. THEOREM 1. Let W be a state on H and let e 4 I be nondegenerate wth respect to P. For all I defne w Ž e, We. W Pe 0 w W e, j

4 DECOMPOSITIONS OF A QUANTUM STATE 475 then Ž w. I s a famly of weghts, are normalzed ectors, and ŽP,w.4 I s a decomposton of W. Conersely all decompostons of W can be obtaned n ths way Žproded that the kernel of W s nfnte dmensonal.. Proof. Snce W s a trace one operator and spane : I4 K, then Ž w. s a famly of weghts and the famly wp. I I s summable. To prove that ts sum s W t s suffcent to observe that for all K W K, so that ž Ý / Ý, wp,w e I I Ý W, e I W Ž,W. whch proves the frst clam. Now we must prove that all decompostons of W are of ths form. Let Ž P,w.4 be a decomposton of W, that s, I Ý I W wp. Ž 1. We prove that, for all I, Ran W Dom T. In fact, for all Dom T and I we have w Ž T,. Ž T,WT.. Ths shows that s n the doman of T* T. For all I let w T. Let J be a fnte subset of I, for all Dom T we have that Ý J Ý J Ž,. w Ž T,. Ý J w Ž,T. Ý I w Ž,T. Ž T,WT. Ž, P..

5 476 CASSINELLI, DE VITO, AND LEVRERO Snce Dom T s dense n K and K, relaton Ž. shows that the net ŽÝ Ž,.. J s a monotone ncreasng net of contnuous operators bounded by P, so that t converges weakly to a bounded operator and, usng Ž. 1 and the frst two lnes of Ž., ts lmt s P, that s, Ý I Ž,. P. From ths relaton t follows that the map j from K to l Ž I. j Ž,., K s a well defned sometry. Moreover the adjont of j s the map from l I onto K explctly gven by Ž I. Ý I Ž x. x, Ž x. l Ž I., I I where the sum s n the weak topology of H. We observe that j s the projector onto jž K.. If Ž f. s the canoncal bass of l Ž I. I, then f for all I. Snce l Ž I. jž K. jž K., H K N, and dm N, there exsts an Ž n general not unque. sometry U from l Ž I. to H such that PU. For all I let e Uf. Snce K Ul Ž Ž I.., then e 4 I s nondegener- ate wth respect to P. Moreover, for all I, W e W Pe W PUf W f W w, 4 Ž.4 so that e gves the decomposton P, w, as clamed. I I Remark 1. Gven two dfferent sets nondegenerate wth respect to P, e 4 and f 4, let Ž P, w.4 and Ž Q,.4 I I I I be the correspondng decompostons of W. Hence P Q for all I f and only f Pe Pf for some. In ths case the two decompostons are equal f and only f 1 for all I. The f parts of both clams are trval computatons. Conversely, f P Q for all I, then there exst a famly of nonzero numbers Ž. I such that W f W e, for all I. The clam follows observng that TW P. Moreover, f w for all I, then hence 1. w W f W e w,

6 DECOMPOSITIONS OF A QUANTUM STATE 477 Remark. Obvously a nondegenerate set can be completed to a Hlbert bass of H. Conversely, from any Hlbert bass we can extract a nondegenerate set. In fact, let f 4 be a bass of H and J k 1: Pe 04 k k1 k. Defne for all k J Ž f, Wf. k k k W f k k k Mmckng the frst part of the proof of Theorem 1, we have that Ý k kj W P, so that the set P P : k J4 k decomposes W, but t can happen that P P j for some j J. A nondegenerate set e 4 I gvng rse to the prevous decomposton set P can be constructed wth the followng procedure. Let J 4 I be the partton of J such that Ž. 1 f k, kj then Pek Pek for some ; Ž. If kj, kj,, then Pe s not collnear wth Pe. k k k For all I defne V spanf : k J 4 k and Q the orthogonal projector onto V. For all I choose an ndex k J, snce Pfk 0 we have that QPf k 0. Defne QPf k e, QPf then e 4 Is obvously nondegenerate wth respect to P and, by construc- ton, generates the decomposton ŽP, w.4 k I where w Ýk J k. Moreover P 4 P as clamed. I Remark 3. Theorem 1 and the prevous remark gve a classfcaton of the decompostons of W by means of the Hlbert bases of H. The same results could be obtaned by replacng H wth any other complex, separable, nfnte dmensonal Hlbert space contanng K as a closed subspace and havng codm K. Ž 3. Ths would only ntroduce notatonal complcatons. Moreover the frst statement of Theorem 1 holds wthout the assumpton Ž. 3 on the codmenson of K. Hence, a decomposton of W corre- k

7 478 CASSINELLI, DE VITO, AND LEVRERO sponds to any bass of any complex separable Hlbert space contanng K as a closed subspace. The condton Ž. 3 assures that the Hlbert space s bg enough to gve all decompostons of W. Remark 4. It follows from the prevous theorem that one pure state P s an element of some decomposton of W f and only f Ran W. Ths partal result was found by Hadjsavvas n 3. Remark 5. Let A be a smple selfadjont operator on H wth a pure pont spectrum and e 4 a bass of egenvectors of A Ž Ae e. 1. Then, due to Remark, e 4 gves rse to a decomposton Ž 1 possbly wth some repeated pure states. of the state W whose weghts are just the probabltes of obtanng the value whle measurng the physcal quantty A when the system s prepared n the state W. Ths result holds wthout assumptons on the codmenson of K Ž see Remark IRREDUCIBLE DECOMPOSITIONS In ths secton we descrbe a partcular class of decompostons, already studed n 3 and called rreducble. For fnte decompostons, they are exactly the ones wth the same number of elements of the spectral one. We begn wth the noton of rreducble famly of vectors. A famly Ž. n H s rreducble f I 4 span : j I, j I; j n the mathematcal lterature an rreducble famly s often called topologcally free. The rreducble famles are easly characterzed by the followng condton: LEMMA 1. Let Ž. Ibe a famly n H. The followng facts are equa- lent: Ž. 1 Is rreducble; Ž. there s a famly Ž. n H such that I Ž,j. j, ji Žwe call Ž. a dual famly of Ž... I I Moreoer we can always choose the ectors, so that span 4 I Ž 4. and, n ths case, the famly Ž. s unquely determned by Ž.. I k I

8 DECOMPOSITIONS OF A QUANTUM STATE 479 Proof. For all I let V span 1,..., 1, 1, Suppose that V for all I. Ths mples that V 0 4. Defne P as the projector onto V and P. By assumpton, the vector s nonzero. For all I, let, then we have that Ž,j. j I, whch proves the exstence. Moreover, by constructon, the vectors satsfy the condton Ž. 4. Conversely, suppose that Ž. Ihas a dual famly Ž., then V, snce Ž,. 0f j. Moreover Ž,. I j 1, so that V. Now, let Ž. be another dual famly of Ž. I I satsfyng Ž. 4. Then for all I, Ž, j. 0, j I, so that span 4. Due to Ž. 4, t follows that. k Remark 6. Let Ž. be an rreducble famly and K span 4 I. Due to Lemma 1, there exsts a unque dual famly Ž. of Ž. I I n K. Obvously Ž. s a dual famly of Ž. I I; nevertheless t can happen that span 4 s a proper subspace of K, so that Ž. I s not the dual famly of Ž. wth the property span 4 Ž I k see counterexample n Remark 9.. A decomposton Ž P, w.4 I of a state W s rreducble f there s an rreducble famly Ž. of unt vectors n H such that P P I for all I. In ths case W w I. Ths proves that an rreducble set P 4 I decomposes W wth a unque famly of weghts. The followng proposton characterzes the rreducble decompostons of a state n terms of a property of the nondegenerate sets that generate them va Theorem 1. Let W be a state, K the closure of the range of W, and P the projecton onto K. As n Secton, we assume that codm K. Let e 4 I be nondegenerate wth respect to P and ŽP, w.4 I the correspondng decomposton of W gven by Theorem 1. The followng statements are equalent: Ž. 1 ŽP,w.4 s an rreducble decomposton of W; PROPOSITION 1. I Ž. for all I, e Ran W.

9 480 CASSINELLI, DE VITO, AND LEVRERO Moreoer, n ths case, e 4 I s a Hlbert bass of K unquely defned by the famly Ž. I. Proof. If ŽP, w.4 I s an rreducble decomposton of W, then there s a dual famly Ž. of Ž. I I and W w ; hence, for all I, Ran W Dom T, where T s defned n Secton. By con- structon w W e so that Pe w T Ran W. Moreover Pe wž T,T. wž,t. Ž,. 1, so that Pe e for all I and ths proves the frst mplcaton. Con- Ž versely, f e Ran W for all I, then w Te. I s a dual famly Ž of w W e. Ž., provng that Ž. I I I s rreducble. Fnally, snce Ran W K, e K for all I. Snce e 4 I s a nondegenerate set, spane 4 K so that e 4 I s a bass of K and t s clearly unquely defned by the famly Ž.. I Remark 7. Usng ths result we have a one to one correspondence Žup to phase factors. between the rreducble decompostons of W and the Hlbert bases of K contaned n Ran W. Due to ths property, when dealng wth rreducble decompostons we can drop the assumpton on the codmenson of K. Remark 8. As a partcular case of the prevous proposton we obtan the followng result of Hadjsavvas, 3 : one pure state P s an element of some rreducble decomposton of W f and only f Ran W. We now turn to the problem Ž b. posed n the ntroducton: gven a state W and an rreducble famly Ž. I of unt vectors n H, determne whether W can be decomposed n terms of the set of pure states ŽP. I. If ths s the case then, as one readly verfes, span 4 Ran W and the correspondng famly of weghts Ž w. Is unque. Due to Remark 7 we can assume wthout loss of generalty that the state s njectve. Let W be an njectve state, T W, Ž. Ibe an rreducble famly of unt vectors of H such that span 4 H, and Ž. I be ts unquely defned dual famly n H. The followng theorem solves the problem of the decomposablty of W on the set P 4 I under the very weak assump- ton span 4 H. Ž 5.

10 DECOMPOSITIONS OF A QUANTUM STATE 481 THEOREM. Wth the preous assumpton Ž. 5, the followng facts are equalent: Ž. 1 the set P 4 I decomposes W; Ž. W for all I, wth ; Ž. 3 Ran W and Ž T, T. 0 f j. j In ths case the famly of weghts of the decomposton s exactly. I Proof. Ž. 1 Ž. 3. Snce W wp I for some famly of weghts w I, we have that W w, whch n turn mples W wt. The clam follows from these relatons. Ž. Ž. 3. Snce Ran W Dom T Dom T and T s njectve, then T 0 and 1 T T, j j. As the dual famly s unque, we have that T T I. The thess follows wth 1T. Ž. 1 Ž.. Snce W s postve and njectve, then Ž, W. 0. By hypothess Dom T, hence e T s well defned and n Dom T. It s easy to prove that Ž e. Is an orthonormal famly, so that, for any fnte subset J of I, Ý Ý J P W e, W e J ž Ý / J W Ž e,. e W. Ths shows that the net ŽÝ P. J converges weakly to a bounded operator A. By constructon A equals W on the algebrac span of the vectors, hence also on ts closure that, by assumpton, s H. To show that Ž P,.4 I s a decomposton of W we observe that, for any fnte subset J of I, Ý J P W. Takng the trace of both sdes one has Ý 1 whch shows that Ž. s summable, hence Ž P. J I I s summable n trace norm and ts sum s necessarly W. From the contnuty of the trace t follows that Ž. s a famly of weghts. I We observe that, wthout the hypothess Ž. 5, the relatons among the three statements of Theorem are Ž 1. Ž 3.. Ž. Ž. In partcular, the followng counterexample shows that 1.

11 48 CASSINELLI, DE VITO, AND LEVRERO Remark 9. Let W be an njectve state on an nfnte dmensonal separable Hlbert space H. Let f 4 1be a bass of egenvectors of W and Ž. 1be the correspondng famly of egenvalues. Snce W s a postve, trace class operator wth trace one, the vector e Ý f 1 1 s a well defned unt vector n H. A trval computaton shows that e Ran W Ž here we use the fact that dm H.. For all, let 1 f f, 1 1 and V be the algebrac span of the vectors. Due to the fact that W s njectve, the egenvalues are nonzero for all 1; from ths t follows that V Ran W and the set of vectors : 4 s lnearly ndepen- dent. Let e 4 be the orthonormal set n V obtaned usng the GramSchmdt procedure on the set 4. Then V spane, 4 and, for all, e Ran W. Moreover, by an easy calculaton, e1 s orthogonal to for all, so that e 4 1 s an orthonormal set. We prove that e 4 1 s a bass of H. In fact, let x H be such that x s orthogonal to e for all 1, then x s orthogonal to for all. Usng ths condton t follows that, f a Ž f, x., 1 1 a a. Moreover, snce e, x 0, we have that a 0, so that x 0. 1 Now, e 4 1s a bass of H, hence, due to Theorem 1 and Remark 3, t defnes a decomposton ŽP, w.4 of W where 1 w W e. Snce e Ran W, the famly Ž. 1 1s not rreducble, due to Proposton 1. For all, e Ran W, so that the vectors w Te are well defned and the followng relatons hold Ž, j. j, 1, j W w,. Snce W s njectve, span, 14 H. Due to Ž.Ž 6,. Ž 6. s rreducble,

12 DECOMPOSITIONS OF A QUANTUM STATE whereas s not rreducble, so that span,, hence 1 1 span, 4 H. Now, we have an rreducble famly Ž. span, 4 H such that W w, nevertheless W Ý wp. Moreover, due to Ž.Ž 6,,. 0 for all, so that 1 span, 4 H. Remark 10. It s worth notng that the statement Ž. 1 of Theorem does not mply the assumpton Ž. 5. In fact, let Ž. be the rreducble famly of the prevous remark and Ž. be a famly of weghts, then, obvously, ŽP,. 4 s an rreducble decomposton of the state Ý P, but the condton Ž. 5 does not hold because span, 4 H. As a partcular case, we observe that f the famly Ž. Is a Schauder bass of H, then both Ž. s rreducble and the condton Ž 5. holds. I ACKNOWLEDGMENT We thank Professor Huzhro Arak for useful comments and suggestons. In partcular he found a gap n the statement of Theorem 1 n a prevous verson of the paper. REFERENCES 1. P. Bush, P. J. Laht, and P. Mttelstaedt, The Quantum Theory of Measurement, nd ed., Sprnger-Verlag, Berln, B. C. van Frassen, Quantum Mechancs: An Emprcst Vew, Oxford Unv. Press, Oxford, N. Hadjsavvas, Propertes of mxtures of non-orthogonal states, Lett. Math. Phys. 5 Ž 1981., L. P. Hughston, R. Jozsa, and W. K. Wootters, A complete classfcaton of quantum ensembles havng a gven densty matrx, Phys. Lett. A 183 Ž 1993., E. T. Jaynes, Informaton theory and statstcal mechancs, II, Phys. Re. 108 Ž 1957.,