ASYMMETRIC TWO-OUTPUT QUANTUM PROCESSOR IN ANY DIMENSION

Transcription

1 ASYMMETRIC TWO-OUTPUT QUANTUM PROCESSOR IN ANY IMENSION IULIA GHIU,, GUNNAR BJÖRK Centre for Avance Quantum Physics, University of Bucharest, P.O. Box MG-, R-0775, Bucharest Mgurele, Romania School of Information an Communication Technology, Royal Institute of Technology (KTH), Electrum 9, SE Kista, Sween Receive March, 008 We propose two ifferent implementations of an asymmetric two-output probabilistic quantum processor. One of them is constructe by combining asymmetric telecloning with a quantum gate array. We analyze the efficiency of this processor by evaluating the fielities between the esire operation an the ones generate by the processor an show that the two output states are the same as the ones prouce by the optimal universal asymmetric Pauli cloning machine an have a success probability of /. We show further that we can perform the same one-qubit operation with unity probability at the cost of using nonlocal operations. We finally generalize the two schemes for -level systems an fin that the local ones are successful with a probability of / an the nonlocal generalize scheme is always successful. Key wors: quantum cloning, quantum processor.. INTROUCTION Quantum computers are machines that employ quantum phenomena, such as quantum interference an entanglement, to solve a particular problem. The computers have to execute a program, which is built with the help of a precise set of instructions, in orer to give the esire solution. A specification of this set of instructions is calle an algorithm. There are problems for which the number of elementary steps for obtaining a solution scales exponentially or at least polynomially better on a quantum computer than on a classical one [], the eutsch-jozsa algorithm for solving the oracle problem [ 4], the Shor algorithm for factoring large integers [5], an the Grover algorithm for searching unsorte atabases [6]. Thus, on a large enough quantum computer one coul solve problems with a quantum computer which are basically unsolvable with a classical one. The most important component of the computer architecture is the processor. One crucial property of the classical processor is that we keep the same circuit regarless of the instructions that we want to perform. One may Rom. Journ. Phys., Vol. 53, Nos. 5 6, P , Bucharest, 008

2 76 Iulia Ghiu, Gunnar Björk then ask how to construct a universal quantum processor, a fixe evice that implements any esire program on the information store in quantum systems? This problem was originally investigate by Nielsen an Chuang [7], where they propose a moel of the quantum processor, which consists of a quantum gate array G acting on the ata state an on the program state P. The ynamics of the quantum gate array is G P U P U U () where U is a particular unitary operator implemente by the processor. Nielsen an Chuang foun two important results [7]: (i) the state P U is inepenent of the ata register an (ii) no eterministic universal quantum gate array exists. Therefore they showe how to construct an one-output probabilistic quantum processor, whose operating principle is that of quantum teleportation [8]. The outcome of a Bell measurement tells us when the esire operation succeee. In the last few years, much progress has been mae on generalizations an applications of the probabilistic quantum processor. Huelga et al. foun a generalization of the metho of teleporting a quantum gate from one location to another [9]. In this protocol, a evice that performs the transformation U was constructe, being an arbitrary state an U an arbitrary unknown unitary operator. Two more proposals of probabilistic quantum processors were consiere by Preskill [0] an Vial et al. []. Vial et al. analyze a probabilistic gate, which performs an arbitrary rotation aroun the z axis of a spin-/ particle with the help of an N-qubit program state []. More complex probabilistic programmable quantum processors have been propose by Hillery et al. [, 3] by investigating the case when an arbitrary linear operation A is performe. They have built the network for this processor by using a quantum information istributor [4, 5] for qubits an then for -level systems (quit). Hillery et al. have analyze several classes of quantum processors, which execute more general operations, namely completely positive maps, on quantum systems [6]. In aition, they have foun two important results: one can buil a quantum processor to perform the phase-amping channel an that this is not possible in the case of the amplitue-amping channel [6]. Further extensions have been evelope. In [7, 8] a quantum processor, which executes SU(N) rotations was consiere, an was foun that the probability of success for implementing the operation is increase if conitionate loops are use. Recently Brazier et al. have investigate the case when we have access to many copies of the program state [9]. They have shown that the probability of success cannot be increase an that it is the same as the one obtaine using two ifferent schemes: VMC [] an HZB [7].

3 3 Asymmetric two-output quantum processor in any imension 77 Positive-operator-value measures (POVM) are the most general measurements allowe by quantum mechanics [0, ]. Therefore it woul be interesting to stuy the possibility of realizations of POVMs on quantum processors. This problem was investigate by Ziman an Bužek [], where they showe how to encoe a POVM into a program state. Another important class of operations is the one of generators of Markovian ynamics, which are relevant in the context of quantum ecoherence. Koniorczyk et al. have recently propose a scenario for the simulation of the infinitesimal generators of the Markovian semigroup on quantum processors []. Approximate processors, i.e., processors which implement a set of unitary operators with high precision, have been introuce by Hillery et al. in [3]. The accuracy of the processor is given by the process fielity, which was shown to be maximum if one chooses the program state to be the eigenvector corresponing to the largest eigenvalue of a certain operator. This operator epens on some operators A jk, which characterize the quantum processor, an the esire unitary operator U to be implemente. We emphasize that all the processors escribe above generate only one output state. A two-output processor was propose by Yu et al. [4] by combining symmetric telecloning of qubits [5] with a programmable quantum gate array. In this scheme, the sener has a priori information regaring the unitary operation, which has to be performe. This is necessary for builing the program state. Therefore, the protocol is ifferent of the one of [9], where the sener has no such information. In this paper we present two ifferent schemes for obtaining a two-output quantum processor for -level systems. Our schemes are generalizations of the Yu et al. protocol, where we analyze a processor which performs probabilistically any unitary operation. Suppose the following scenario: an observer Peter has to teleport the result of a unitary operation on an unknown ata state, to two istant parties, Alice an Bob. We show how this task can be accomplishe using a share entangle state, local operations, an classical communication (LOCC) by proposing two schemes. We initially restrict our stuy to the class of iagonal unitary operations but show that with a straightforwar extension any unitary operation can be transmitte optimally. In the first scheme, escribe in Section, we exten Yu et al. s scheme by consiering asymmetric telecloning of qubits as well as a more general unitary operator. This gives Peter the power to ecie on the information sharing, with the two extremes being that one party gets all information, while the other gets none. The symmetric protocol is a special case of our more general stuy. The program state consists of a four-particle entangle state share between Peter, Alice, Bob, an another observer Charlie, who hols an ancillary system. (As Charlie s particle is only use as a resource, an is iscare at the en, Peter an Charlie may be the same party in those schemes that o not nee nonlocal operations between Alice s, Bob s an Charlie s particles.) Peter measures the

4 78 Iulia Ghiu, Gunnar Björk 4 ata qubit an his particle P, which is inclue in the program register, in the stanar Bell basis, an then communicates the outcome. With a probability of /, Alice an Bob are able to recover the esire mixe output states, whose fielities are ientical with the ones obtaine by the optimal universal asymmetric Pauli cloning machine. We also iscuss that if Alice, Bob, an Charlie may use nonlocal operations, the protocol will always be successful with preserve fielities. Further we propose an alternative local protocol, where the esire unitary operation is encoe in a moifie Bell basis. In Section 3 we present the generalizations of these protocols for quits. The two local generalize schemes are successful with a probability equal to /, while unit probability of success again requires nonlocal operations. Finally, in Section 4 we summarize our conclusions.. ASYMMETRIC TWO-OUTPUT QUANTUM PROCESSOR FOR QUBITS.. ARBITRARY IAGONAL UNITARY OPERATOR In this section we present two schemes for performing the following scenario: we start with an arbitrary qubit state 0 () 0 where 0. We want to obtain two optimal universal asymmetric clones of a iagonal unitary computational operator escribe by: i e ei (3) with j [0 ), j = 0,, applie on the arbitrary input ata state. We have use the notation ( ). We impose the conition that the processor generates two output states, whose fielities must be inepenent on the input ata state, i.e., inepenent on 0 an. We assume that is known. The esire output state is i i e e (4) We efine a state require in the preparation of the program as 0 0 P P P (5) where p 0 ( p) 0 an ( p p )

5 5 Asymmetric two-output quantum processor in any imension 79 p 00 ( p) 00 ( p p ) (6) with 0 p. The two states 0 an are obtaine by applying an optimal universal asymmetric Pauli cloning machine to the states 0 00 an 00 [6]. The ata system an the first qubit P of the state belong to an observer Peter, while the qubits A, B, C are hel by other istant parties, Alice, Bob, an Charlie, respectively. This ownership of the qubits (or, later, quits) will hol for all our schemes. Note that, in the case of the local schemes, Peter an Charlie may be ientically the same, as Charlie s qubit is iscare at the en. Peter wants to teleport the result of the esire unitary operator to Alice an Bob. Peter encoes the information carrie by the unitary operator, in the program state P P by locally applying on the qubit P (see Fig. ): P i 0 0 i P I e P 0 e P P (7) We write now the input state with the help of the stanar Bell basis : i 0 i P 0e 0 e P P i 0 i P 0e 0 e (8) ei0 ei an P 0 0 ei 0 i 0e 0 P Fig. The scheme for the two-output quantum processor for qubits. The input states consist of an arbitrary ata register an the program register P P I. P Alice an Bob will obtain two mixe states A, B that are implemente by the quantum processor.

6 730 Iulia Ghiu, Gunnar Björk 6 Peter performs a measurement in the Bell basis on the ata qubit an the first qubit P in the program state as it is shown in Fig.. Then he communicates the outcome of the measurement to Alice, Bob, an Charlie. With a probability equal to /4 the outcome is an therefore the output is projecte to P i i (9) e e Accoringly, after tracing over Charlie s qubit, the two final states of Alice an Bob, respectively, are: A Tr ( ) I BC p p p p (0) Tr I B AC p ( p) p p with given by Eq. (4). The efficiency of the quantum processor is evaluate with the help of the fielities of the output states with respect to the exact ata register outputs : p FA A an ( p p ) () p p FB B ( p p ) If the output of the measurement is P, then the final state is i 0 i 0e 0 e () Alice, Bob, an Charlie can transform the state to the state of Eq. (9) by applying the local unitary operator V A B C z z z. Therefore, in this case, Alice an Bob obtain the same final states A an B as above. For the other two outcomes, when Peter obtains P an P, the result cannot be transforme to the state of Eq. (9) by local operations. Hence there is no chance to obtain the esire mixe outputs A an B if we on t allow for nonlocal transformations. The processor hence succees with the probability / an generates two asymmetric output mixe states. The fielities of the final states given by Eq. () are ientical with the ones of the clones emerging from the optimal universal asymmetric cloning machine given in [6, 7, 8]. In the particular case when p = /, we recover the result of Yu et al. [4], namely the two output states are ientical. Our protocol presents the avantage that Peter can ecie whether to sen all information to Alice, i.e., F A = /, F B = / by fixing p =, or to Bob, giving F A = /, F B = by fixing p = 0, or to sen two imperfect output states to both receivers, F A an F B being given by Eq. () for 0 < p <.

7 7 Asymmetric two-output quantum processor in any imension 73 However, if we allow nonlocal operations between Alice, Bob, an Charlie, which either entails their respective particles to interact irectly, or via a share auxiliary entangle state, it is also possible to convert the states an P to an thereby always succee with the protocol. This is generally true for all our propose schemes. Instea of encoing the information of the esire unitary operator into the program state, we can incorporate it into a measurement in a moifie Bell basis, which epens on : i i e 0 e 00 i i e 0 e 0 0 Thus, as an alternative protocol we use the same channel of Eq. (5) P initially share by Peter, Alice, Bob, an Charlie as above, but this time Peter performs a measurement in the above moifie Bell basis as is shown in Fig.. Alice an Bob obtain the same outputs A, B given by Eq. (0). The total success probability is p = /. Since this protocol is constructe using only LOCC it is suitable for istant computation at two locations. P (3) Fig. The alternative scheme for the two-output local quantum gate-array for qubits. The information of the operation to be performe is encoe in a moifie Bell basis, which epens on. Peter performs a measurement in this generalize Bell basis an then communicates the outcome to Alice, Bob, an Charlie. By LOCC, Alice an Bob are able to get the mixe states A, B... PERMUTE IAGONAL OPERATOR Suppose the operator is not iagonal, but that it can be written as a permutation of a iagonal operator. In the two-imensional case, there is only one such operator, namely

8 73 Iulia Ghiu, Gunnar Björk 8 0 i 0 i e e U P ei 0 0 ei 0 (4) consiere by Yu, Feng, an Zhan [4]. The target state in this case becomes i i 0 U e 0 e 0 (5) The protocol becomes: The unknown ata state is moifie by a unitary basis-permutation 0 (6) 0 Subsequently this state is subjecte to the protocol in the previous subsection using the iagonal unitary operator. It is trivial to show that Alice s (Bob s) final state an associate fielity AB ( ) become ientical to the ones of Eq. (0) an Eq. (), respectively..3. AN ARBITRARY UNITARY OPERATOR Suppose that we want that the quantum processor to implement an arbitrary, but known, unitary operator U. Therefore, the esire output state is U. Since is unknown, the teleportation cannot be accomplishe through a classical channel. In contrast, U is known an can be arbitrarily well be escribe by information transmitte classically. We always can iagonalize U with the help of a unitary operator U tr : tr tr U U U (7) where the iagonal unitary operator will have the form given by Eq. (3). We efine a moifie ate state as U an an intermeiate target state. Accoring to the Eq. (0) Alice an Bob get two intermeiate final states: A ( p) I p p p (8) I B p ( p) p p Subsequently, the sener Peter communicates to Alice an Bob, by a classical channel, which the transformation operator U tr is. The final output states are obtaine by Alice an Bob by applying this transformation unitary operator, which epens only on the initial esire operation U. Applying this tr

9 9 Asymmetric two-output quantum processor in any imension 733 transformation to the intermeiate state they get AB ( ) Utr AB ( ) Utr, respectively. Expressing this state in terms of the ientity an the target state we recover expressions Eq. (0) an Eq. (). 3. ASYMMETRIC TWO-OUTPUT QUANTUM PROCESSOR FOR QUITS 3.. AN ARBITRARY IAGONAL UNITARY OPERATOR One-output probabilistic programable quantum processors have recently been analyze by Hillery et al. [, 7] in the case when the ata are encoe on a -level systems (quits). More precisely, they have investigate the possibility to construct a processor which performs an arbitrary linear operation A []. Here we propose a generalization of the quantum processor presente in the Sec.. for quits. An arbitrary ata quit-state is escribe by k k (9) k0 where. k0 k We will initially analyze the action of a unitary, iagonal operator: i e i e 0 0 exp( is ) s s (0) s0 0 0 ei ei We efine a family of states which epens on a parameter p, 0 p : j j j j p ( )( j jr jr p p) r p jr j jr r () where j = 0,,, an jr j r moulo. These states were foun by one of us in [6] by consiering the action of an optimal universal asymmetric Heisenberg cloning machine on the state j 00. In aition, we introuce the state as follows: P

10 734 Iulia Ghiu, Gunnar Björk 0 Peter encoes the operation follows We enote by j P P j j0 () in the state of a program register P exp( j ) P j j0 P P as P I i j (3) mn the stanar Bell basis for quits: exp ikn mn k k m (4) k0 The input state can be written by using the stanar Bell basis as i exp ikn km P exp (5) km P k mn P kmn 0 The esire output state shoul be j e j j0 i j (6) A measurement in the stanar Bell basis onto the ata state an the first qubit P of the program state is performe by Peter. Then he announces the outcome of the measurement to Alice, Bob, an Charlie. With a probability / he gets the outcome 0n (where n is a fixe number) an at the same time the state of the quits A, B an C becomes k0 i exp ikn k ke k (7) A B C n n n n Let us now efine a local operator V V V V, where j0 j0 X ijn Vn exp j j X A B C ijn Vn exp j j (8) epening on Peter s outcome, Alice, Bob, an Charlie apply subsequently the local operator V n on the output state (7) an obtain

11 Asymmetric two-output quantum processor in any imension 735 V i k n ke k k0 (9) Since there are equiprobable measurement outcomes 0n for n = 0,,, the total success probability is /. The two mixe output states of the quantum processor are, in all the cases, A Tr ( ) I BC p ( )(p p) (30) p( ) p Tr B I AC p ( )(p p) ( ) p( ) p The fielities of the output states are given by: F F A B ( ) p A ( )(p p) ( )( p) B ( )(p p) They are ientical with the ones generate by the optimal universal asymmetric Heisenberg cloning machine given in [6]. In the case when the result of Peter s measurement is not 0 n, Alice, Bob, an Charlie are not able to recover the state given by Eq. (9) only by LOCC, an then the computation fails. However, both in this an the following protocol, the success probability can be booste to unity if the three parties may use nonlocal operations on their particles. We can generalize the secon scheme given in Section. by efining a moifie Bell basis which epens on also for quits. The total success probability of this scheme is / an the two output states coincie with the ones of Eqs. (30), (3). an (3) (3) 3.. AN ARBITRARY UNITARY OPERATOR The generalizations of the qubit protocols in Sections. an.3 to quits is straightforwar. The premutation matrix an the iagonalization operator U tr become larger an more complex, but the protocols remain ientical. Alice s

12 736 Iulia Ghiu, Gunnar Björk an Bob s output states, expresse in the relevant target states, an the fielities are given by (30), (3) an (3), respectively. 4. CONCLUSIONS In this paper we have emonstrate the possibility of builing a general two-output programmable processor of qubits, which generates two asymmetric states with a probability of success of /. It is characterize by the same fielities as the ones generate by the optimal universal asymmetric Pauli cloning machine. We implement these schemes by using only local operations an classical communication, an therefore they are suitable for istribute computation to two spatially separate receivers. In aition, we have presente the generalizations of these two protocols to -level systems. The generalizations present the avantage of sening the outcomes to ifferent locations an they are achieve with a probability of /. Finally, we have shown how to increase the success probability to unity by consiering nonlocal operations. In this case the two output states are obtaine only after the particles A, B, an C have interacte, either irectly or through an auxiliary entangle state. Acknowlegements. This work was sponsore by the Sweish Founation for Strategic Research (SSF), the Sweish Research Council (VR), the Romanian Ministry of Eucation an Research through grants CEEX /005, CNCSIS-T 95-3/006, an IEI-995/007 for the University of Bucharest. REFERENCES. M. A. Nielsen, I. L. Chuang, Quantum Computation an Quantum Information (Cambrige University Press, Cambrige, U. K., 000)... eutsch, Proc. Roy. Soc. Lonon A 400, 97 (985). 3.. eutsch, R. Jozsa, Proc. Roy. Soc. Lonon A 439, 553 (99). 4. S. Gule, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. Häffner, F. Schmit-Kaler, I. L. Chuang, R. Blatt, Nature (Lonon) 4, 48 (003). 5. P. W. Shor, Proceeings of the Symposium on the Founation of Computer Science, 994, Los Alamitos, California, IEEE Computer Society Press, New York, p. 4 (994). 6. L. K. Grover, Phys. Rev. Lett. 79, 35 (997). 7. M. A. Nielsen, I. L. Chuang, Phys. Rev. Lett. 79, 3 (997). 8. C. H. Bennett, G. Brassar, C. Crepeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 70, 895 (993). 9. S. F. Huelga, J. A. Vaccaro, A. Chefles, M. B. Plenio, Phys. Rev. A 63, (00). 0. J. Preskill, Proc. Roy. Soc. Lonon A 454, 385 (998).. G. Vial, L. Masanes, J. I. Cirac, Phys. Rev. Lett. 88, (00).. M. Hillery, V. Bužek, M. Ziman, Phys. Rev. A 65, 030 (00). 3. M. Hillery, V. Bužek, M. Ziman, Fortschr. Phys. 49, 987 (00).

13 3 Asymmetric two-output quantum processor in any imension S. L. Braunstein, V. Bužek, M. Hillery, Phys. Rev. A 63, 0533 (00). 5. M. Roško, V. Bužek, P. R. Chouha, M. Hillery, Phys. Rev. A 68, 0630 (003). 6. M. Hillery, M. Ziman, V. Bužek, Phys. Rev. A 66, 0430 (00). 7. M. Hillery, M. Ziman, V. Bužek, Phys. Rev. A 69, 043 (004). 8. V. Bužek, M. Ziman, M. Hillery, Fortschr. Phys. 5, 056 (004). 9. A. Brazier, V. Bužek, P. L. Knight, Phys. Rev. A 7, (005). 0. A. Peres, Quantum Theory: Concepts an Methos (Kluwer, orrecht, 993).. M. Ziman, V. Bužek, Phys. Rev. A 7, 0343 (005).. M. Koniorczyk, V. Bužek, P. Aam, Eur. Phys. J. 37, 75 (006). 3. M. Hillery, M. Ziman, V. Bužek, Phys. Rev. A 73, 0345 (006). 4. Y. Yu, J. Feng, M. Zhan, Phys. Rev. A 66, 0530 (00). 5. M. Murao,. Jonathan, M. B. Plenio, V. Veral, Phys. Rev. A 59, 56 (999). 6. I. Ghiu, Phys. Rev. A 67, 033 (003). 7. N. J. Cerf, Phys. Rev. Lett. 84, 4497 (000). 8. N. J. Cerf, J. Mo. Opt. 47, 87 (000).