We obtain a simple formula for the average gate fidelity of a linear map
acting on qudits. It is given in terms of minimal sets of pure state
preparations alone, which may be interesting from the experimental point of
view. These preparations can be seen as the outcomes of certain minimal
positive operator valued measures. The connection of our results with these
generalized measurements is briefly discussed

A. Acín

A. Peres

E. Bagan

J.F. Poyatos

J.I. Latorre

M. Baig

M. Horodecki

M.A. Nielsen

M.D. Bowdrey

N. Gisin

R. Muñoz-Tapia

S. Massar

English

arXiv

We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in terms of minimal sets of pure state preparations alone, which may be interesting from the experimental point of view. These preparations can be seen as the outcomes of certain minimal positive operator valued measures. The connection of our results with these generalized measurements is briefly discussed

Bagán Capella, Emili

Baig i Aleu, Marià,

Muñoz Tapia, Ramon

American Physical Society

Diposit Digital de Documents de la UAB

PHYSICAL REVIEW A 67, 014303 ~2003!Minimal measurements of the gate fidelity of a qudit mapE. Bagan, M. Baig, and R. Mun˜oz-TapiaGrup de Fı´sica Teo`rica and IFAE, Facultat de Cie`ncies, Edifici Cn, Universitat Auto`noma de Barcelona,08193 Bellaterra (Barcelona), Spain~Received 6 August 2002; published 23 January 2003!We obtain a simple formula for the average gate fidelity of a linear map acting on qudits. It is given in termsof minimal sets of pure state preparations alone, which may be interesting from the experimental point of view.These preparations can be seen as the outcomes of certain minimal positive operator valued measures. Theconnection of our results with these generalized measurements is briefly discussed.DOI: 10.1103/PhysRevA.67.014303 PACS number~s!: 03.67.2a, 89.70.1c, 03.65.WjThe interaction with the environment has a significantnegative impact on any physical implementation of a unitarygate or a quantum channel. Decoherence, turning pure statesinto mixed states, creeps up and unitarity is lost in the Hil-bert subspace of the signal states. The characterization of thequality of such implementations is, hence, of utmost impor-tance for quantum computation from both experimental andtheoretical points of view @1#.A physical quantum gate or channel is best described by atrace-preserving linear map or superoperator E, which is as-sumed to be an approximation of a unitary operator U char-acterizing the quantum gate (U51 for a quantum channel!.The average gate fidelityF¯ ~E,U !5E dc tr@UrcU†E~rc!# , ~1!where dc is the invariant Haar measure on the space of purestates uc& ~i.e., dc5dc8 if uc8&5Uuc&) and rc5uc&^cu, isrecognized as a convenient figure of merit and is widely usedto assess the quality of a quantum gate or channel.For gates acting on qubits, Bowdrey et al. @2# have re-cently derived a simple formula for F¯ (E,U), from whichthey obtain a convenient way of actually measuring the gatefidelity in a laboratory. It amounts to replacing the integral inEq. ~1! by a sum over a finite number of pure states whoseBloch vectors point at the vertices of an octahedron or atetrahedron inscribed in the Bloch sphere. Nielsen @3# hasobtained a similar formula for qudits ~i.e., quantum statesbelonging to a d-dimensional Hilbert space! in terms of afinite set of unitary operations U j orthogonal with respect tothe Hilbert-Schmidt inner product, namely, such thattr(Ui†U j)5d d i j . His derivation is based on Horodeckis’ for-mula connecting F¯ (E,1)[F¯ (E) with the entanglement fidel-ity @4#, for which he also provides a simple proof. Our aim isto obtain a more natural expression of the average gate fidel-ity so that only measurements over a ~minimum! number ofpure states need to be performed to verify F¯ experimentally.Interestingly enough, the solution to this problem is not farremoved from that of obtaining minimal positive operatorvalued measurements ~POVMs! in the context of optimalcommunication of directions through a quantum ~spin! chan-nel, which has received much attention over the last fewyears @5–7#.1050-2947/2003/67~1!/014303~3!/$20.00 67 0143In the first part of this paper we derive a general expres-sion for F¯ in terms of the SU(d) group generators. In thesecond part we write F¯ as an average of measurements overa finite and minimal number of pure state preparations,which may be experimentally relevant.We first notice that due to the invariance of dc one hasF¯ (E,U)5F¯ (E8,1)5F¯ (E8), where E8(r)[E(U†rU). Hencewe only need to consider the simpler form F¯ (E) without anyloss of generality. Furthermore, the uniform measure dc canbe effectively realized in the following way: *dcF(c)[*dU F(Uc0), where F is any function of rc , uc0& is afixed reference state, and dU is the Haar measure of SU(d)normalized such that *dU51. This ensures that *dcF(c)5*dcF(U8c) for any U8PSU(d). Obviously, not all thed221 parameters involved in dU are physically significant.For example, for qubits (d52), dc5dn is the uniform mea-sure on the 2-sphere S2, which can be parametrized by theEuler angles a and b . If uc0& is an eigenstate of the Paulimatrix sz , any function of rc is independent of the thirdEuler angle g . Hence, *dc F5@*dg/(2p)51#3*dn F5*dU F. For qudits, dn must be replaced by the invariantmeasure of SU(d)/@SU(d21)3U(1)# , since any referencedensity matrix r0[uc0&^c0u is now invariant under SU(d21)3U(1). Note that the number of parameters match, asqudits depend on 2(d21) real variables. With all this, wefinally haveF¯ ~E!5E dU tr@Ur0U†E~Ur0U†!# . ~2!There exists a (d221)-dimensional unitary vector withcomponents n0a, a51,2, . . . ,d221, such that the densitymatrix r0 can be written asr051d 1kdn0aTa[1d 1kdnW 0TW , ~3!where kd5A2(d21)/d and $Ta% are the ~Hermitian andtraceless! generators of SU(d) @8#, normalized so thattr(TaTb)5dab/2. They can be chosen as Ta5la/2, where laare a generalization of the Gell-Mann matrices of SU(3).Throughout this paper a sum over repeated latin indices isunderstood. It is straightforward to obtain©2003 The American Physical Society03-1BRIEF REPORTS PHYSICAL REVIEW A 67, 014303 ~2003!nW 052kdtr ~r0TW !. ~4!We now recall the well-known relation @8#UTaU†5~AU !abTb , ~5!where A stands for the adjoint representation of SU(d). Wealso recall the orthogonality of the irreducible representa-tions of compact groups, which in the present case impliesE dU~AU !ab50, ~6!E dU~AU !ab~AU !cd5dacdbdd221 . ~7!Assuming that E is linear and trace preserving, one getsF¯ ~E!5 1d 12d~d11 ! (a51d221tr@TaE~Ta!# . ~8!This is the generalization of the formula for qubitsF¯ ~E!5 12 113 (i5x ,y ,z trFs i2 ES s i2 D G ~9!obtained in Ref. @2#. This concludes the first part of thispaper.The key ingredient of the derivation above is the orthogo-nality relation of the group representations, exemplified byEqs. ~6! and ~7!. We will show that it is possible to find adiscrete version of these equations. Namely, one can find afinite set of SU(d) elements $Ur% and positive constants $cr%such that(rcr~AUr!ab50, ~10!(rcr~AUr!ab~AUr!cd5dacdbdd221. ~11!With this we can reverse the steps going from Eq. ~2! to Eq.~8! using the relations ~10! and ~11! instead of their continu-ous versions ~6! and ~7!, and obtainF¯ ~E!5(rcrtr @rrE~rr!# . ~12!This equation has a very convenient form that allows settingup experimental tests to determine the fidelity of a gate orchannel, as will be discussed below.Let us briefly discuss the solutions to Eqs. ~10! and ~11!.The idea is to generalize the concept of a finite set of isotro-pically distributed unit vectors introduced in Ref. @6# andadapt it to the problem at hand. A sufficient condition for Eq.~12! to hold can be obtained by contracting Eqs. ~10! and~11! with n0a. If we define01430n0a~AUr!ab[nrb , ~13!conditions ~10! and ~11! lead to(rcrnrb50; (rcrnrbnrd5dbdd221. ~14!In this sense, we may qualify the set $nr% as isotropicallydistributed @as far as the adjoint representation SU(d) is con-cerned#. We have traded the problem of finding $Ur% for thatof finding $nr%. Note, however, that the set of matrices $AU%is a proper subgroup of SO(d221). Hence, no vector on thesphere Sd222 is admissible. We will come back to this issuebelow. The relations ~10!, ~11!, and ~14! also appear in arather different context: the construction of finite positiveoperator valued measurements that are optimal for commu-nicating a direction @6,7,9#. Leaving aside an overall trivialnormalization ~from Eq. ~14! it follows that (rcr51,whereas in Ref. @9# (rcr5d), the results in those papers canbe readily used here. In particular, it is proved in Ref. @9# thatsolutions of Eq. ~14! exist and the minimal one is given by aset of d2 vectors pointing at the vertices of a regular hyper-tetrahedron or, more properly, (d221) simplex, inscribed onSd222. This hypertetrahedron is defined by the conditionnW rnW s52 1d221 , rÞs , ~15!and the exact overall orientation has to be chosen so that allvectors nW r are of the form ~13!. For this hypertetrahedron allthe coefficients cr are equal: in our notation cr51/d2, r51,2, . . . ,d2. An explicit form of nW r for SU(3) can befound in Ref. @9#, where also the general case is briefly dis-cussed. The solution is more conveniently expressed in termsof states ucr& such thatucr&^cru[rr5Urr0Ur†51d 1kdnWrTW . ~16!Then Eq. ~15! translates intou^crucs&u251d11 , rÞs . ~17!Since this equation is a condition on states, Eq. ~13! is auto-matically satisfied by the corresponding Bloch vectors.For SU(3) a solution of Eq. ~17! has the simple formucr&51A2 S 1e2(r21)pi/30 D , r51,2,3. ~18!The remaining six states are obtained by applying cyclic per-mutations to the components of uc1 –3& ~a different choice ofstates is given in Ref. @9#!. From the experimental point ofview, the states ~18! have a very appealing form; each ofthem involves a linear combination of only two states of the3-2BRIEF REPORTS PHYSICAL REVIEW A 67, 014303 ~2003!computational basis. If the qutrit is implemented by say threeatomic levels, only two levels need to be manipulated toprepare each one of the rr .In the SU(2) case, a solution in terms of nW r is moresimple and transparent, mainly because SU(2) is isomorphicto SO(3), which makes Eq. ~13! trivially satisfied by anyvector of S2. A compact solution is given bynW 151A3~1,1,1 !, nW 251A3~21,21,1!, ~19!and nW 3 , nW 4 are again obtained by applying cyclic permuta-tions to the components of nW 2.For the hypertetrahedra discussed above, Eq. ~12! can becast asF¯ ~E!5 1d2 (r51d2tr@rrE~rr!# , ~20!which is the SU(d) generalization of Bowdrey et al. SU(2)formula. This equation is our main result and provides aremarkably simple procedure for measuring F¯ (E). One justhas to average the fidelities for d2 isotropically distributedpure states. Notice that all state preparations rr have thesame weight in the average, thus reducing systematic errors.In some sense, Eq. ~20! can be regarded as the average sur-vival rate of the states $rr% in a quantum channel character-ized by the linear map E. There is an alternative way of01430writing Eq. ~20!, which may provide further insight. It isstraightforward to verify that the projectors Or[rr /d are thecomplete set of positive operators ~i.e., (Or51) of a mini-mal POVM. Thus, there exists a device characterized by$Or% whose preparations are precisely rr. We can write Eq.~20! asF¯ ~E!5 1d (r51d2tr@OrE~rr!# . ~21!We readily see that the same device $Or% can be used for thepreparations of the pure states $rr% as well as for the mea-surements over $E(rr)%.Although we have presented the results for the minimalsets of rr , for practical reasons, one may wish to use a largernumber of states. This possibility is easily implemented inthis framework, as conditions ~13! and ~14! are not specificof minimal sets but entirely general. For example, for qubits,one can find a set of six states whose Bloch vectors point atthe vertices of a regular octahedron @2,5,7#!. In this casethere is a simple setting for preparing the states, just threeStern-Gerlach’s oriented along the three orthogonal direc-tions.We are grateful to A. Acı´n and A. Bramon for helpfulconversations. We acknowledge financial support fromCIRIT Project No. SGR-00185, Spanish Ministry of Scienceand Technology Project No. BFM2002-02588, and the Euro-pean Funds for Regional Development ~FEDER!.@1# J.F. Poyatos, J.I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 390~1995!.@2# M.D. Bowdrey et al., Phys. Lett. A 294, 258 ~2002!.@3# M.A. Nielsen, Phys. Lett. A 303, 249 ~2002!.@4# M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A60, 1888 ~1999!.@5# S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 ~1995!; N.Gisin and S. Popescu, ibid. 83, 432 ~1999!; E. Bagan et al.,ibid. 85, 5230 ~2000!; A. Peres and P.F. Scudo, ibid. 86, 4160~2001!.@6# E. Bagan, M. Baig, and R. Mun˜oz-Tapia, Phys. Rev. A 64,022305 ~2001!.@7# J.I. Latorre, P. Pascual, and R. Tarrach, Phys. Rev. Lett. 81,1351 ~1998!.@8# An introduction to group theory that closely follows our nota-tion can be found in J.F. Cornwell, Group Theory in Physics~Academic Press, London, 1984!.@9# A. Acı´n, J.I. Latorre, and P. Pascual, Phys. Rev. A 61, 022113~2000!.3-3

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