Physics Letters A 309 (2003) 24–28

www.elsevier.com/locate/pla

A general scheme for ensemble purification

Angelo Bassia, GianCarlo Ghirardib,∗

a The Abdus Salam International Centre for Theoretical Physics, Trieste, and Istituto Nazionale di Fisica Nucleare, sezione di Trieste, Italyb Department of Theoretical Physics of the University of Trieste, the Abdus Salam International Centre for Theoretical Physics, Trieste,

and Istituto Nazionale di Fisica Nucleare, sezione di Trieste, Italy

Received 6 November 2002; accepted 20 November 2002

Communicated by P.R. Holland

Abstract

We exhibit a general procedure to purify any given ensemble by identifying an appropriate interaction between the physicalsystemS of the ensemble and the reference systemK . We show that the interaction can be chosen in such a way to lead to aspatial separation of the pairS–K . As a consequence, one can use it to prepare at a distance different equivalent ensembles.The argument associates a physically precise procedure to the purely formal and fictitious process usually considered in theliterature. We conclude with an illuminating example taken from quantum computational theory. 2003 Elsevier Science B.V. All rights reserved.

1. Introduction

A statistical ensembleE of physical systemsS ischaracterized by a (finite, countable or continuous) setof positive numberspi summing up to 1 and by acorresponding set of normalized vectors|ψi〉 of theHilbert spaceHS associated to the systemS, so thatwe will write E(pi, |ψi〉) to represent it. The statisticaloperatorρE (a trace-class, trace one, semipositive def-inite operator) associated toE(pi, |ψi〉) is defined as:

(1.1)ρE =∑i

pi |ψi〉〈ψi |.

A point of great conceptual relevance which marksa radical difference between the classical and quantum

* Corresponding author.E-mail addresses: bassi@ictp.trieste.it (A. Bassi),

ghirardi@ts.infn.it (G.C. Ghirardi).

cases is that, while in classical mechanics the assign-ment of the statistical operatorρ(r,p) uniquely iden-tifies the ensemble, within quantum mechanics, as itis well known, the correspondence between statisticalensembles and statistical operators is infinitely manyto one.

With reference to this point, let us consider theset of all statistical ensembles of systems like theone under consideration. Such a set can be naturallyendowed with an equivalence relation.

Definition. We will say that two statistical ensem-bles E and E∗ are equivalent, and we will writeE ≡ E∗, iff ρE = ρE∗ .

It is obvious that the just defined relation is reflexive,symmetric and transitive and that it leads to a decom-position of the set of all ensembles into disjoint equiv-

0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0375-9601(02)01670-5

A. Bassi, G.C. Ghirardi / Physics Letters A 309 (2003) 24–28 25

alence classes. We will denote as[E] the equivalenceclass containing the ensembleE .

Purification of an ensemble [1] is a procedureby which one associates to the ensemble a purestate|Ψ 〉 of an appropriately enlarged Hilbert spaceHS+K = HS ⊗ HK , whereK is a reference systemwhose Hilbert spaceHK we assume to be infinite-dimensional for reasons which will become clear ina moment. The fundamental request on|Ψ 〉 is that,by measuring an appropriate observable ofK andconfining attention to the systemS alone, one canprepare the desired ensembleE(pi, |ψi〉).

The first proof that, given two equivalent ensemblesE(di , |φi〉) andE(pj , |χj 〉), one can find two orthonor-mal sets|Ai〉 and|Bj 〉 of HK such that

(1.2)

|Ψ 〉 =∑i

√di |φi〉 ⊗ |Ai〉 =

∑j

√pj |χj 〉 ⊗ |Bj 〉

has been exhibited by Gisin [2]. This result is partic-ularly relevant since it is related to the request that nofaster-than-light signals can be send between distantobservers.

Subsequently, Hughston et al. [3] have generalizedthe above result, providing a complete classificationof equivalent ensembles: using the purification proce-dure, they have derived necessary and sufficient con-ditions for two ensembles to be equivalent.

In the literature (see, e.g., [1]), ensemble purifica-tion is usually considered as a purely mathematicaltool: one does not identify any dynamical mechanismwhich could be used to actually implement it, and thesystemK is considered a fictitious system without adirect physical significance. The aim of this Letter is toexhibit a precise physical procedure in order to purifyany ensemble by making the systemS interact with asystemK, in such a way that the desired pure state|Ψ 〉be actually produced. Then one can use it to prepareany desired ensemble of the equivalence class.

2. Statistical ensembles and the purificationprocess: the constructive procedure

As remarked above, it is our purpose to present aformal constructive mechanism to purify any given en-semble, showing at the same time how, by resorting tothis procedure, one can use the obtained pure state to

generate all ensembles of systemsS equivalent to theone which has been purified. The procedure is basedon a formalism which parallels strictly the one pro-posed by von Neumann for implementing ideal mea-surement processes of the first kind, even though thesystemK, which plays a role analogous to the one ofthe measuring apparatus in his treatment, can very wellbe (and actually we will consider it to be) a microsys-tem.

Our starting point is the consideration of an equiva-lence class[E] of ensembles of systemsS. Within sucha class there is the ensembleE(di , |φi〉) which corre-sponds to the spectral decomposition of the associatedstatistical operator having the positive numbersdi aseigenvalues and the|φi〉 as the associated orthonormaleigenvectors. Such a decomposition is unique, apartfrom accidental degeneracies which, if they occur, canbe disposed of as one wants, so that we will considerthe eigenvectors|φi〉 as precisely assigned vectors. Weassume that the indexi runs from 0 ton, without com-mitting ourselves about the fact thatn is finite or infi-nite and about the fact that the orthonormal set|φi〉be a complete set ofHS or not.

Let us consider now the orthonormal states|φi〉and let us assume that there exist a physical sys-temK, whose associated Hilbert spaceHK is infinite-dimensional, a state|a0〉 of HK and an interactionHamiltonianHS+K of HS+K such that theS–K in-teraction lasting for a certain time intervalT inducesthe following evolution:

(2.1)|φi〉 ⊗ |a0〉 ⇒ |φi〉 ⊗ |ai〉, 〈ai |aj 〉 = δij ,where|ai〉 are state vectors belonging toHK .

In the next section we will exhibit a simple Hamil-tonian having such a property and leading also to anarbitrarily chosen separation in space of the systemsS

andK. We stress that we needHK to be infinite-dimensional if we want to be able to build a state|ΨT 〉which will allow us to prepare any ensemble whatso-ever in the equivalence class under consideration bymeasurement procedures on systemsK, since in anyequivalence class there are always ensembles contain-ing an infinite number of states.

Given the ensembleE(di, |φi〉) we consider thestate:

(2.2)|Ψ0〉 =n∑i=1

√di |φi〉 ⊗ |a0〉,

26 A. Bassi, G.C. Ghirardi / Physics Letters A 309 (2003) 24–28

and we let it evolve through the intervalT . Accordingto Eq. (2.1) and due to the linearity of the quantumevolution, we get

(2.3)|Ψ0〉 ⇒ |ΨT 〉 =n∑i=1

√di |φi〉 ⊗ |ai〉.

We consider now an arbitrary complete orthonormalset |bj 〉, j = 0,1, . . . ,∞, of HK and we complete(if necessary) the set|ai〉, i = 0,1, . . . , n, to a set|Ai〉 by adding to it orthonormal states spanning themanifold of HK orthogonal to the one generated bythe|ai〉 themselves. Obviously, we have:

(2.4)|Ai〉 =∞∑j=0

Uij |bj 〉, i = 0,1, . . . ,∞,

whereUij is a unitary matrix ofHK . From Eq. (2.3)we get:

|ΨT 〉 =n∑i=0

∞∑j=0

√di |φi〉 ⊗Uij |bj 〉

=∞∑j=0

(n∑i=0

√di Uij |φi〉

)⊗ |bj 〉

(2.5)=∞∑j=0

|χj 〉 ⊗ |bj 〉.

Note that〈ΨT |ΨT 〉 = 1 implies:

(2.6)∞∑j,k=0

〈χj |〈bj |bk〉|χk〉 =∞∑j=0

∥∥|χj 〉∥∥2 = 1.

The states|χj 〉 are not normalized, so that, putting|χj 〉 = |χj 〉/‖|χj 〉‖, we have:

(2.7)|ΨT 〉 =∞∑j=0

∥∥|χj 〉∥∥|χj 〉 ⊗ |bj 〉.

If we measure now an observable of the systemKhaving a non-degenerate spectrum with|bj 〉 as eigen-vectors and we confine our attention to the result-ing ensemble of systemsS, we obtain the ensembleE(‖|χj 〉‖2, |χj 〉). Note that since both

∑ni=0 di |φi〉〈φi |

and∑∞j=0 ‖|χj 〉‖2|χj 〉〈χj | are obtained by taking the

partial trace onHK of |ΨT 〉〈ΨT |, they are equaland the corresponding ensembles belong to the sameequivalence class. Thus we have proved that starting

from the state (2.3) and choosing an observable having|bj 〉 as eigenstates, we generate an ensemble whichbelongs to the same equivalence class of the originalone.

The relevant question we have to face now is thefollowing: can all statistical ensembles belonging tothe equivalence class ofE(di, |φi〉) be obtained byproperly choosing the observables of the systemK weare going to measure? The answer is yes, as it is easilyproved. To this purpose, let us consider an arbitraryensembleE(pj , |τj 〉) equivalent toE(di, |φi〉); wesuppose that the indexj runs from 0 toN( n),without excluding the case in whichN is infinite.We know that the fact that the statistical operatorsassociated to such ensembles are identical implies thatthe normalized states|τj 〉 are linear combinations ofthe orthonormal states|φi〉:

(2.8)|τj 〉 =n∑i=0

bji |φi〉, j = 0,1, . . . ,N.

We define now a rectangular matrixVij havingn+ 1rows andN + 1 columns by putting:

(2.9)

Vij =√pj

dibji, i = 0,1, . . . , n; j = 0,1, . . . ,N.

From the relation

n∑i=0

di |φi〉〈φi | =N∑j=0

pj |τj 〉〈τj |,

using Eq. (2.8) we immediately get:

(2.10)N∑j=0

pjbjib∗jk = diδik.

The above relation implies:

(2.11)

N∑j=0

Vij(V †)

jk=

N∑j=0

√pj

di

√pj

dkbjib

∗ik = δik.

We thus haven+1 normalized and orthogonal vectorswr, r = 0,1, . . . , n, of CN+1, whose components arethe row elements of the matrixVrj :

(2.12)wr = (Vr0,Vr1, . . . , VrN), r = 0,1, . . . , n.

If N is finite, we pass from the vectorswr to newvectorswr of C∞ by considering equal to zero the

A. Bassi, G.C. Ghirardi / Physics Letters A 309 (2003) 24–28 27

components ofwr from N + 1 on. We then extendthe setwr to a complete orthonormal set ofC∞, byadding appropriately chosen normalized vectorsws,s = n + 1, . . . ,∞. Correspondingly, the rectangularmatrix Vij of Eq. (2.9) is transformed into an infinitesquare matrix, whose rows are the components of thevectorswr , for r = 0,1, . . . ,∞. Due to Eq. (2.11)and the procedure we have followed, this infinitesquare matrix—which we keep callingVij—is unitary.

Let us consider now an observableΩK of HKhaving a purely discrete and non degenerate spectrumwith eigenvectors|Bj 〉 = V †

ji |Ai〉; this implies that|Ai〉 = Vij |Bj 〉. SinceVij is unitary, we can repeatthe previous procedure which amounts simply inreplacing, in Eq. (2.3), the states|ai〉 = |Ai〉 appearingthere with their Fourier expansion in terms of the set|Bj 〉. Then Eq. (2.3) takes the form (2.7) where,according to the definition of|χj 〉 given in Eq. (2.5)and of|τi〉 given in Eq. (2.8):

|χj 〉 =n∑i=0

√di Vij |φi〉 = √

pj

n∑i=0

bji |φi〉

(2.13)= √pj |τj 〉.

This shows that‖|χj 〉‖2 = pj and that normalizing|χj 〉 we get the states|τj 〉. Accordingly, Eq. (2.7)becomes:

(2.14)|ΨT 〉 =N∑j=0

√pj |τj 〉 ⊗ |Bj 〉,

so that a measurement ofΩK reduces the state|ΨT 〉to the desired ensembleE(pj , |τj 〉). SinceE(pj , |τj 〉)is an arbitrary ensemble belonging to the equivalenceclass[E(di, |φi〉)], this completes our proof.

The now obtained result shows how, once one hasprepared the pure state|ΨT 〉, he has an immediatecomplete classification of all ensembles belonging tothe equivalence class ofE(di, |φi〉), an alternative wayof deriving the nice result of [3].

Of course, within any equivalence class there arealso mixtures which involve a continuous union ofpure states, i.e.,

(2.15)E(p(λ), |φλ〉

)−→ ρE =∫dλp(λ)|φλ〉〈φλ|,

with∫λp(λ)= 1. To get such mixtures from the pure

state|Ψ 〉 we have, obviously, to measure with infinite

precision an observable ofHK having a continuousspectrum. This is formally but not practically feasible.

Concluding, if we can implement our “von Neu-mann-like ideal interaction scheme” we can performthe desired purification and then prepare any one ofthe ensembles in the equivalence class ofE(di, |φi〉)by performing an appropriate measurement on thesystemK.

3. The appropriate Hamiltonian for the desiredpurification

To face our problem let us consider the followingself-adjoint operator ofHS+K :

(3.1)Hj = i|φj 〉〈φj | ⊗[|a0〉〈aj | − |aj 〉 〈a0|

],

and let us evaluate its powers. We have:

H 2n+1j =Hj,

(3.2)H 2nj = |φj 〉〈φj | ⊗

[|a0〉〈a0| + |aj 〉〈aj |].

Let us consider now the operator exp(−iωHjT ):(3.3)exp(−iωHjT )= cos(ωHjT )− i sin(ωHjT ).

Since sin contains only odd powers ofHj we have:

(3.4)sin(ωHjT )=Hj sin(ωT ),

while, since all even powers ofHj equalH 2j we can

write:

cos(ωHjT )= 1−H 2j

[−1+ 1+ 1

2ω2T 2 − · · ·

](3.5)= 1−H 2

j +H 2j cos(ωT ).

We now choose forT a value such that

cos(ωT )= 0, sin(ωT )= 1,

getting:

(3.6)exp(−iωHjT )= 1−H 2j − iHj .

The last equation implies that

exp(−iωHjT )|φj 〉 ⊗ |a0〉= [

1−H 2j − iHj

]|φj 〉 ⊗ |a0〉(3.7)= |φj 〉 ⊗ |aj 〉.

as desired.

28 A. Bassi, G.C. Ghirardi / Physics Letters A 309 (2003) 24–28

We remark now that[Hj,Hk] = 0 andHk|φj 〉 ⊗|a0〉 = 0 for k = j . Accordingly, if consideration isgiven to the HamiltonianH =∑∞

j=0Hj we have:

(3.8)exp(−iωHT )|φj 〉 ⊗ |a0〉 = |φj 〉 ⊗ |aj 〉, ∀j.Therefore, we have explicitly exhibited an Hamil-tonian which performs our game, i.e., it leads to thedesired purification of our statistical mixture.

Actually, the purification procedure becomes inter-esting when one can prepare a desired mixture amongall those of an equivalence class at-a-distance, as ap-propriately stressed by Gisin [2]. To reach this goala very small change in our formalism is necessary.Let us identify the states|aj 〉 of our equation withthe internal eigenstates of a system (e.g., the station-ary states of an hydrogen atom). One can then add toour Hamiltonian a termγPCM, whereγ is an appropri-ately chosen c-number andPCM is the center-of-massmomentum of the system. The evolution induced bythe total Hamiltonian implies a displacement of thesystemK of an amount governed by the value ofγ ,so that in the time intervalT it is brought arbitrarilyfar from the space region where its interaction withStook place. In brief, the auxiliary system is far apartand one can actually use the pure state to prepare thedesired statistical ensemble of systemsS at-a-distance.

4. A quantum computational example

It is interesting to notice that, for most cases ofinterest in quantum computational theory, the outlinedprocedure can be easily implemented by resortingto elementary logical gates. To this purpose, let ussuppose thatS is a qubit, i.e., a two-level system,and let us denote as|0〉, |1〉 the computational basisof HS . The controlled-NOT operator acting on qubitS(taken as the control bit) and on the two-dimensionalmanifold spanned by the computational basis states|a0〉, |a1〉 of systemK (taken as the target bit) inducesprecisely the transformation:

|0〉 ⊗ |a0〉 ⇒ |0〉 ⊗ |a0〉,(4.1)|1〉 ⊗ |a0〉 ⇒ |1〉 ⊗ |a1〉,

which is the desired evolution. In this way, we can pu-rify any statistical ensemble belonging to the equiva-

Fig. 1.

lence class of

(4.2)E(p, |0〉;1− p, |1〉), 0<p < 1,

by starting with an appropriate superposition analo-gous to the one of Eq. (2.2).

Let us now consider an arbitrary equivalence class,different from the previous one and containing theensemble (corresponding to the diagonal form ofρ):

(4.3)E(q, |x+〉;1− q, |x−〉), 0< q < 1,

where |x+〉 and |x−〉 are a basis obtained from thecomputational basis|0〉, |1〉 by an appropriate “rota-tion” of the system:

(4.4)|x+〉 =RS |0〉, |x−〉 =RS |1〉.The circuit that implements the evolution

|x+〉 ⊗ |a0〉 ⇒ |x+〉 ⊗ |a0〉,(4.5)|x−〉 ⊗ |a0〉 ⇒ |x−〉 ⊗ |a1〉,

leading to the purification of the ensemble, corre-sponds to a “rotation”R†

S on the control bit, followedby a controlled-NOT gate and by an inverse “rotation”RS , as shown in Fig 1.

Thus, the Hamiltonian that induces the desiredevolution can be identified with a rotation inHS ,a controlled-NOT operation inHS ⊗ HK , and finallya counter-rotation inHS .

In this way, we have identified the appropriateway to purify any statistical ensemble of the two-dimensional systemS. Useless to say, our procedurecan be easily generalized to systems containing severalqubits and, more in general, to arbitrary quantumsystems.

References

[1] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quan-tum Information, Cambridge Univ. Press, Cambridge, 2000.

[2] N. Gisin, Helv. Phys. Acta 62 (1989) 363.[3] L.P. Hughston, R. Jozsa, W. Wootters, Phys. Lett. A 183 (1993)

14.