Experimental Implementation of the Universal Transpose Operation
Hyang-Tag Lim, Young-Sik Ra, Yong-Su Kim, Joonwoo Bae, Yoon-Ho Kim
aa r X i v : . [ qu a n t - ph ] M a r Experimental implementation of the universal transpose operation using structuralphysical approximation
Hyang-Tag Lim, ∗ Young-Sik Ra, Yong-Su Kim, Joonwoo Bae, † and Yoon-Ho Kim ‡ Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea School of Computational Sciences, Korea Institute for Advanced Study, Seoul, 130-012, Korea (Dated: November 8, 2018)The universal transpose of quantum states is an anti-unitary transformation that is not allowedin quantum theory. In this work, we investigate approximating the universal transpose of quantumstates of two-level systems (qubits) using the method known as the structural physical approximationto positive maps. We also report its experimental implementation in linear optics. The scheme isoptimal in that the maximal fidelity is attained and also practical as measurement and preparationof quantum states that are experimentally feasible within current technologies are solely applied.
PACS numbers: 03.65.Ud, 03.67.Bg, 42.50.Ex
The postulate of quantum theory that global dynam-ics of given quantum systems must be unitary is the con-straint given to legitimate operations on quantum states.The general class of quantum operations is then found inthe reduced dynamics of subsystems and mathematicallycharacterized by completely positive (linear) maps overHilbert spaces [1]. In particular, positive but not com-plete positive maps, simply called positive throughout,are of unique importance in quantum information the-ory that, for any entangled state, there exists a positivemap that determines whether or not the given state isentangled [2].The fact that there are impossible operations in quan-tum theory leads in a natural way to the problem of build-ing optimal approximate quantum operations. Moreover,impossible operations in quantum theory are in generalnot only of fundamental and theoretical interest to char-acterize computational and information-theoretic capa-bilities of quantum information processing, but also ofpractical importance in implementation of approximatequantum operations for applications.Systematic approximation to positive maps, known asthe structural physical approximation (SPA), has beenproposed in Ref. [3] in the context of detecting entan-glement of unknown quantum states, i.e. even beforeidentifying given quantum states through the state to-mography. The initial proposal of the SPA assumedcollective measurement for spectrum estimation that re-quires, i) quantum memory that stores quantum states inthe quantum level for a while, and ii) coherent quantumoperations that allows general manipulation of copiesof quantum states [4]. Interestingly, however, it wasshown recently that the SPA to some positive maps corre-spond to quantum measurement [5]. Furthermore, it has ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] been recently conjectured that SPAs to positive maps arein general entanglement-breaking, meaning that the ac-tual implementation of the SPA would be much simplerthan originally proposed [6]. Note that entanglement-breaking channels can be constructed with measurementand preparation of quantum states [7]. The conjecturehas been extensively tested with known examples of pos-itive maps, without a counter-example, for instance, inRefs. [6, 8].In this work, we report linear optical implementa-tion of approximating the transpose operation for a two-dimensional quantum system (i.e., photonic qubit) usingthe SPA based on measurement and preparation of quan-tum states. Our SPA scheme for the universal transposeoperation is optimal in that the maximum fidelity is at-tained and is also practical as only single-copy level mea-surement and preparation of quantum states are applied.To the best of our knowledge, our work is the first proof-of-principle demonstration that shows SPAs to positivemaps are experimentally feasible within the present-daytechnology.The motivation behind the transpose operation be-ing particularly chosen to demonstrate the SPA basedon measurement and preparation of quantum states istwofold. On one hand, in the side of further applica-tions, transpose to a subsystem is the well-known criteriathat efficiently detects useful entanglement (i.e., entan-gled states having negative eigenvalues after the partialtranspose) [2, 9]. Since any approximate map that canexperimentally detect entanglement via the SPA can al-ways be factorized into a convex combination of anotherSPAs to non-physical operations for individual systems[10], our work immediately implies the feasibility of ex-perimental implementation of the entanglement detec-tion via SPAs. On the other hand, in the fundamentalpoint of view, the transpose represents the anti-unitarityin the symmetry transformation in quantum theory [11].Since any anti-unitary is composed of a unitary and thetranspose, the transpose is the only symmetry transfor-mation that is not allowed in quantum theory.Let us first briefly describe the theory behind theSPA based on measurement and preparation of quantumstates [6]. The SPA to the transpose T of a d -dimensionalquantum state σ in general works by admixing the com-plete contraction D [ σ ] = tr[ σ ] /d to the positive map, T −→ e T = (1 − p ) T + pD, such that the resulting map e T is completely positive.The minimal p that brings the complete positivity isknown as p = d/ ( d + 1). From the well-known isomor-phism between states and channels in Ref. [12], the chan-nel e T corresponds to the so-called Jamiolkowski state ρ e T = [ ⊗ e T ]( | φ + ih φ + | ) where | φ + i = P i | ii i / √ d . If thestate ρ e T is separable, then the channel e T is entanglement-breaking, meaning that the channel can be described bymeasurement and preparation of quantum states [7].Consider now the approximate transpose of a qubitstate ρ . In this case, the Jamiolkowski state is ρ e T =( | φ + ih φ + | + | φ − ih φ − | + | ψ + ih ψ + | ) /
3, where | φ − i = ( | i−| i ) / √ | ψ + i = ( | i + | i ) / √
2. The state is sep-arable, having the following separable decomposition ρ e T = 14 X k =1 | v k ih v k | ⊗ | v k ih v k | , (1)where the vectors | v i i are normalized and given by, | v i ∝ | i + ie iπ / i + e − iπ / | i , | v i ∝ | i − ie iπ / i − e − iπ / | i , | v i ∝ | i + ie iπ / i − e − iπ / | i , | v i ∝ | i − ie iπ / i + e − iπ / | i . Using the Jamiolkowski isomorphism [12], the SPA tothe transpose of a given qubit state ρ can be expressedin terms of the above vectors as, e T ( M ) [ ρ ] = X k =1 tr (cid:20) | v ∗ k i h v ∗ k | ρ (cid:21) | v k i h v k | , (2)where the superscript ( M ) denotes that the scheme ismeasurement-based and | v ∗ k i is complex conjugate of | v k i .The set of positive operators {| v ∗ k ih v ∗ k | / } defines a prop-erly normalized measurement due to the trace preservingproperty of the channel e T ( M ) . Then, eq. (2) can be inter-preted as carrying out the approximate transpose of ρ intwo steps: i) measuring the state ρ in the basis | v ∗ k i withequal probabilities for k = 1 , , ,
4, and ii) depending onthe measurement outcome, preparing the correspondingstate | v k i . The schematic diagram of eq. (2) is shown inFig. 1 where the two-step operation is denoted as e T ( M ) k for k = 1 , , , e T ( M ) , can be calculated byconsidering a pure qubit state | ψ i and is given as, F = tr[ T [ | ψ ih ψ | ] e T ( M ) [ | ψ ih ψ | ]] = 2 / ≈ . . (3) ρ (a) Measurement State preparationWPWP P ρ Single-photon sourceDetector(b) ( ) M T ɶ ( ) M [ ] T ρ ɶ ( ) M1 T ɶ ( ) M2 T ɶ ( ) M3 T ɶ ( ) M4 T ɶ (M) [ ] k T ρ ɶ (M) k T ɶ k p = FIG. 1: (a) The scheme in eq. (2) is shown. For an in-put qubit state ρ , an operation e T ( M ) k for k = 1 , , , e T ( M ) k for k = 1 , , , ρ isa polarization state of a single photon. For the measurementand preparation of states, waveplates (WP) and polarizers(P) are used. The detector and the single-photon source inthe measurement and the state preparation stages, respec-tively, become superfluous in our experiment and thus werenot implemented. See text for details. It should be immediately clear from the above result thatour scheme is optimal in that maximum fidelity of 2 / ρ , we made use ofthe heralded single-photon source based on spontaneousparametric down-conversion (SPDC). A 2 mm thick type-II BBO crystal was pumped by a 405 nm diode laseroperating at 100 mW, producing a pair of orthogonallypolarized 810 nm photon pairs. Conditioned on the de-tection of the vertically polarized trigger photon, thehorizontally polarized signal photon is prepared in thesingle-photon state (i.e., heralded single-photon state).The polarization state of the heralded single-photon canthen be transformed to an arbitrary state by using a setof half- and quarter-wave plates, i.e., the single-photonpolarization qubit [14]. Using 10 nm FWHM bandpassfilters in front of both the trigger and the signal detectors,we observed the coincidence rate around 4 kHz.To implement the approximate transpose operation e T ( M ) , as shown in eq. (2), measurement and prepara-tion of the single-photon polarization qubit in four dif-ferent settings (randomly chosen with equal probability)are required. The ideal approximate transpose operation e T ( M ) is therefore composed of four e T ( M ) k ( k = 1 , , , e T ( M ) k operation (randomly chosen with equalprobability) consists of particular measurement and statepreparation, see Fig. 1. For each e T ( M ) k , the measurement Im @ Ρ exp D H VH V - - @ Ρ exp D H VH V - - Im @ Ρ exp D H VH V - - @ Ρ exp D H VH V - - Im @ Ρ exp D H VH V - - @ Ρ exp D H VH V - - Re @ Ρ exp D H VH V - - @ Ρ exp D H VH V - - Re ReRe Re
ImIm ImIm in (a) ρ [ ] in (b) T ρ [ ] (M)exp in (c) T ρ ɶ [ ] (M) in (d) T ρ ɶ FIG. 2: (a) QST of the qubit state ρ in in eq. (4). (b) The stateafter transpose, T [ ρ in ]. (c) QST of the qubit state after theexperimental e T ( M )exp operation. (d) The state after the idealapproximate transpose. Note that the fidelity between (b)and (d) is F = 2 /
3, as it is shown in eq. (3). The Uhlmann’sfidelity between (c) and (d) is, F ≈ . in one of the basis | v ∗ k i is performed by setting waveplatesin such a way that the incoming photonic polarizationqubit ρ in passes the polarizer with probability h v ∗ k | ρ in | v ∗ k i and then results in, due to the projection at the polarizer,the polarization state | H i . Once the input single-photonqubit has passed the polarizer (i.e., projection measure-ment has occurred), the corresponding state | v k i is pre-pared from the state | H i using another set of waveplates,see Fig. 1(b). Note that, since we employ the triggeredsingle-photon source for encoding ρ , only the coincidenceevent between the signal detector and the trigger detec-tor is meaningful. The coincidence event can occur onlywhen the signal photon has passed through the polarizerP in the measurement stage in Fig. 1(b). Thus, whenthe triggered single-photon source is used, the detectorand the single-photon source in the measurement and thestate preparation stages, respectively, shown in Fig. 1(b)become redundant and can be removed altogether as wehave done in our work.Finally, for a given state ρ in , the SPA to the trans-pose e T ( M )exp [ ρ in ] is constructed by the probabilistic sum offour equally-weighted e T ( M ) k operations. (In other words,this means that the four paths in Fig. 1(a) are non-interfering.) The resulting state is identified by the quan-tum state tomography (QST) and is then quantified bycomparing to the ideal case, e T ( M ) [ ρ in ]. The experimen-tally implemented transpose operation is also analyzedby the quantum process tomography (QPT) that iden-tifies the performed operation. In the experiment, themeasurement duration in each setting was 1 s and themeasurement was repeated three times. QST and QPTresults were obtained using the maximum likelihood es-timation. Im @ Χ D I X Y ZI X Y Z - - @ Χ D I X Y ZI X Y Z - - (M)exp Re[ ] χ (M)exp Im[ ] χ x σ y σ z σ x σ y σ z σ x σ y σ z σ x σ y σ z σ I II I
FIG. 3: The χ matrix for the operation e T ( M )exp is obtained fromthe QPT. The average fidelity between the performed and theideal operations is, F ave ( e T ( M )exp , e T ( M ) ) ≈ . To apply the approximate transpose operation, we con-sider an arbitrary polarization state, | ψ i = ( | H i + (1 + i ) | V i ) / √
3, and the experimentally prepared one is iden-tified by the QST, see Fig. 2 (a): ρ in = (cid:18) .
322 0 . − . i .
352 + 0 . i . (cid:19) ≈ | ψ ih ψ | . (4)The transposed state can easily be computed as, T [ ρ in ] = ρ T in , which is shown in Fig. 2 (b). The experimental re-sult for the measurement-based SPA scheme depicted inFig. 1 is then shown in Fig. 2 (c). Assuming the idealapplication of the approximate transpose, the resultingstate would be e T ( M ) [ ρ in ] from eq. (2) and is shown inFig. 2 (d). Using the Uhlmann’s fidelity, the experimen-tal result can be quantified in terms of its similarity withthe ideal one, F ( e T ( M ) [ ρ in ] , e T ( M )exp [ ρ in ]) ≈ . σ (= , σ x , σ y , σ z ) span the operatorspace of single qubit operations. Hence, a quantum pro-cess E of a single qubit can generally be expressed as, E ( ρ in ) = P m,n χ mn σ m ρ in σ † n , where it is the matrix χ mn that gives the complete characterization of the opera-tion E . For the ideal operation e T ( M ) , the corresponding χ matrix is found to be, χ ( e T ) = diag [1 / , / , , / χ matrix of the performed opera-tion e T ( M )exp has been constructed, and is shown in Fig. 3.The average fidelity between the designed and the per-formed operations is exploited to compare two channels, F ave ( e T ( M ) , e T ( M )exp ) ≈ .
999 [15].So far, we have shown an experimental implementa-tion of the universal transpose using the measurement-based SPA scheme that can give the maximal fidelity.The scheme can then be used as a building block forfurther applications of approximate positive maps suchas entanglement detection [3, 10]. We emphasize that (a) IX p = (U) T ɶ (b) ρ (U) [ ] T ρ ɶ Re @ Ρ exp D H VH V - - @ Ρ exp D H VH V - - (U)exp in Re[ [ ]] T ρ ɶ (U)exp in Im[ [ ]] T ρ ɶ x σ z σ FIG. 4: (a) The scheme e T ( U ) is based on random applica-tions of three unitaries, I , σ x , and σ z on a single qubit. Theunitaries σ k can be realized with half-wave plates. (b) Theresulting state after applying e T ( U ) to the input state ρ in ineq. (4) is identified by the QST. The fidelity with the idealone is, F ( e T ( U ) [ ρ in ] , e T ( U )exp [ ρ in ]) ≈ . the presented scheme is practical within the present-daytechnology as only linear optics are used. There has beenan earlier result reported in Ref. [16], where the universaltranspose is implemented by a random unitary channel.As it was pointed out in the original proposal in Ref. [3],the scheme based on a unitary channel requires quantummemory at the final step for the spectrum estimation.This contrasts to the measurement-based SPA schemewhere no quantum memory is required [6].If we focus only on the implementation of the univer-sal transpose, the unitary-based scheme has an advantageover the measurement-based SPA scheme in that lessernumbers of optical elements are needed which helps im-proving the fidelity. To fairly compare the two schemes,we have also implemented the unitary-based scheme forthe transpose: the unitary channel can be found from theJamiolkowski state in eq. (1), ρ e T = [ ⊗ e T ]( | φ + ih φ + | ) = P i =0 ,x,z ( ⊗ σ i ) | φ + ih φ + | ( ⊗ σ i ). Hence, using therelation in Ref. [12] one can derive that, e T ( U ) [ ρ in ] = P i =0 ,x,z σ i ρ in σ i , where the superscript ( U ) means thatthe scheme is unitary-based [16]. In Fig. 4, the experi-ment results are shown for the state in eq. (4). The av-erage fidelity between the experimentally obtained QPT χ matrix of the operation e T ( U )exp and the ideal operationis F ave ( e T ( U ) , e T ( U )exp ) ≈ . σ y and thetranspose [13]. In Ref. [17], the optimal approximateUNOT operation has been realized in experiment bymaking use of the anti-cloning process that appears in the ancillary system of the 1 → [1] K. Kraus, States, Effects, and Operations , (Springer-Verlag, Berlin, 1983).[2] M. Horodecki, P. Horodecki, and R. Horodecki Phys.Lett. A , 1 (1996).[3] P. Horodecki and A. Ekert, Phys. Rev. Lett. , 127902(2002).[4] M. Keyl and R. F. Werner, Phys. Rev. A 64, 052311(2001).[5] J. Fiur´aˇsek, Phys. Rev. A , 052315 (2002).[6] J. K. Korbicz, M. L. Almeida, J. Bae, M. Lewenstein andA. Ac´ın, Phys. Rev. A , 062105 (2008).[7] M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math.Phys. , 629 (2003).[8] D. Chruscinski, J. Pytel, and G. Sarbicki, Phys. Rev. A , 062314 (2009)[9] A. Peres, Phys. Rev. Lett. , 1413 (1996).[10] C. M. Alves, et al. , Phys. Rev. A , 032306 (2003).[11] E. Wigner, J. Math. Phys. , 409 (1960).[12] A. Jamiolkowski, Rep. Math. Phys. , 275 (1972).[13] V. Buˇzek, M. Hillery, and R.F. Werner, Phys. Rev. A ,2626(R) (1999).[14] Y.-S. Kim, Y.-W. Cho, Y.-S. Ra, and Y.-H. Kim, Opt.Express , 11978 (2009).[15] The average fidelity between quantum operations is, F ave ( e T , e T (X)exp ) = R dψF ( e T [ | ψ ih ψ | ] , e T (X)exp [ | ψ ih ψ | ]) where X = M, U , and see also, Bowdrey et al. , Phys. Lett.A , 258 (2002).[16] F. Sciarrino, C. Sias, M. Ricci, and F. De Martini, Phys.Rev. A , 052305 (2004).[17] F. De Martini, V. Buˇzek, F. Sciarrino, and C. Sias, Na-ture , 815 (2002).[18] H. Bechmann-Pasquinucci and N. Gisin, Phys. Rev. A59