Channel matrix, measurement matrix and collapsed matrix. in teleportation

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1 Channel matix, measuement matix and collapsed matix in telepotation XIN-WEI ZHA, JIAN-XIA QI and HAI-YANG SONG School of Science, Xi an Univesity of Posts and Telecommunications, Xi an, 71011, P R China Abstact In this pape, two kinds of coefficient matixes (channel matix, measuement matix)associated with the pue state fo telepotation ae pesented, the geneal elation among channel matix, measuement matix and collapsed matix is obtained. In addition, a citeion fo telepotation that the numbe of coefficient of an unknown state is detemined by the ank of the collapsed matix is given. PACS: Hk; a; w Keywods: channel matix, measuement matix; collapsed matix; telepotation 1.Intoduction Quantum telepotation is a pime example of a quantum infomation pocessing task, whee an unknown state can be pefectly tanspoted fom one place to anothe by using peviously shaed entanglement and classical communication between the sende and the eceive. Since the fist intoduction of quantum telepotation potocol by ennett [1], eseach on quantum telepotation has been attacted much attention both in theoetical and expeimental aspects in ecent yeas due to its significant applications in quantum calculation and quantum communication. A numbe of expeimental implementations [-5] of telepotation have been epoted and some schemes of quantum telepotation have also been pesented [6-9].It is epoted telepotation of an abitay single qubit, though an entangled channel of Einstein-Podolsky-Rosen (EPR) pai[1]. Fo a geneal case, it is poved that only maximally entangled pue states inc d C d could faithfully telepot an abitay pue state in C d [10, 11]. On the othe hand, the entanglement is geneally consideed as a key esouce in quantum telepotation. The main thing we need to do this is to know if abitay quantum channels can be Coesponding Autho. Tel: addess: zhxw@xupt.edu.cn (Xin-wei Zha) 1

2 employed to telepot a multiqubit state. It is shown that thee-qubit GHZ state and a class of W states can be used fo pefect telepotation of one qubit state [1, 13]. In addition, the tenso poducts of two ell states [14] and the genuine fou-qubit entangled states [7] ae also used fo pefect telepotation of two-qubit states. They ae all maximally entangled pue states between Alice and ob. In Ref. [8] we have shown that thee exists a elation between the deteminant of tansfomation opeato and the SLOCC (stochastic local opeations and classical communication) invaiant fo ell basis measuement. In Ref.[10], we have shown that thee exists a elation between the LU (local unitay) tansfomation invaiant and pefect telepotation. Howeve, in a ealistic situation, the telepotation channel is not always the maximally entangled states. Especially, fo asymmetic quantum channel, it is impotant to choose appopiate measuement basis so that the an unknown state can be telepoted optimally. Recently, Li et al. [11] have studied SLOCC classification fo geneal n-qubit states via the invaiance of the ank of the coefficient matix. In Ref [1] they povided an efficient method fo detecting genuine multipatite coelations of abitay n-qubit system in tems of the ank of coefficient matices. In Ref [15]Wang et al. have studied multipatite entanglement unde SLOCC and poposed the entanglement classification unde SLOCC fo abitay-dimensional multipatite (n-qudit) pue states via the ank of coefficient matix, togethe with the pemutation of qudits. In this pape, we define coefficient matix of channel matix, measuement matix and collapsed matix in telepotation, futhemoe, the geneal elation among channel matix, measuement matix and collapsed matix is obtained. y calculating the ank of coefficient matix, we can obtain a citeion fo telepotation an unknown state by the ank of the collapsed matix. We investigate seveal examples and give optimal match of measuing basis and quantum channel is also to be

3 studied..the channel matix and measuement matix of telepotation We fist eview the constuction of coefficient matix as intoduced in Ref.[17,-19]. let us suppose that the sende Alice has paticles in an unknown state: ( ) a (1) x0 x a 1 xn 1 n whee the coefficient ae complex and satisfy nomalized condition n1 x 1 i0 i. Suppose that Alice and ob shae a state as the quantum channel. A A can then be epesented in the following fom: C i j, i 0,1,, n1; j 0,1,, m1; A i, j A i, j () Fo convenience, we wite A, 1 as A,, 1 as. That is to say,the paticles AA ( 1, ) ae in Alice s possession, and the othe paticles ( 1, ) ae in ob s possession. A coefficient matix associated to the state A is defined by C nm, Ci, j A nm (3) which can be called channel matix. The system state of the paticles can be expessed as: s a A. (4) It is well known that a quantum state can be tansfeed pefectly though a swap opeato defined by [13]: Pa a a (5) whee P a kl kl a a lk (6) 3

4 Then can then be epesented in the following fom [14]: s P s a A a aa (7) On the othe hand, we have [8] s a A Let us assume Alice s measuement basis is,which has the following fom:, 0,1,, 1; 0,1,, 1 M k l k m l n ; kl, A a kl, (8) (9) A measuement matix associated to the state is defied by M M, mn, kl mn (10) Fom Eqs. (4-8), we can obtain collapsed matix M * C (11) n, m A m, n The citeion fo telepoting an abitay unknown state can be given in tems of the collapsed matix. If the collapsed matix is full ank, i.e. the collapsed matix is evesible, the unknown paticle entangled state can be telepoted. Futhemoe, If ku, (1) U whee k is an abitay constant and is a unitay matix. In this case, ob can detemine the state of unknown paticles exactly by the invese of the tansfomation opeato ( U ) 1. Consequently the unknown paticle entangled state is telepoted pefectly, and the successful possibilities and the fidelities both each unity. If the the collapsed matix is not full ank, i.e. collapsed tansfomation opeato is not evesible, the geneal unknown entangled state can not be telepoted. 3.The ank of the collapsed matix In tems of channel matix and measuement matix, it can be veified that the following esult 4

5 holds: (i) Fo quantum channel, if n m, assume the ank of the channel matix and A, measuement matix M ae c and m espectively,then we have c n, m n We mn, C nm c ollap efe to the ank of the collapsed matix as,theefoe,the ank of the collapsed matix satisfy n, in this case, if c n,the collapsed matix is evesible, the unknown c ollap paticle entangled state can be telepoted. (ii). If n m,similaly,we have c m, m m, then the ank of collapsed matix satisfy c ollap m Theefoe,we know c ollap n in this case,the collapsed matix is not full ank, the geneal unknown entangled state can not be telepoted. (iii). if n m, i.e. fo symmetic quantum channel, we have c n, m n, then the ank of collapsed matix satisfy min, collap c m. In this case, only channel matix and measuement matix ae all full ank,,can collapsed matix become full ank matix. That is to say, only quantum channel is maximally entangled and Alice's measuement basis is also maximally entangled state. The unknown paticle entangled state can be telepoted. Obviously,If quantum channel between Alice and ob is a bipatite poduct state, i.e.,, the ank A A of channel matixc must be Cnm, 1. Using above definition, it is easy to show that If two nm, n-qubit pue states quantum channel ae LU equivalent, i.e. U U,then thei channel matices have the same ank. A A A the two-qubit system. We have (13) a0 00 a a 10 a3 11 If the paticle 1 is in Alice s possession, and the paticle is in ob s possession. Namely, a0 00 a1 01 a 10 a3 11 (14) A 5

6 o a0 00 a 01 a1 10 a3 11 (15) A Then the channel matix associated to the state A is given by C, a a a 0 a 1 3 (16) Obviously, if the entangled is channel is an Einstein-Podolsky-Rosen (EPR) pai, the channel matix is full ank. On the othe hand, if Alice's measuement basis ae ell states measuement basis, fo example, ,, 01 10, the measuement matix can be obtain M, 0 1, M, M, M, , (17) It is easy to see that the measuement matix is full ank. That is to say, fo symmetic quantum channel, the ell states measuement basis is optimal measuement basis, Anothe example is the thee qubits W state[15] 1 whee a a4. W ( 001 a a4 100 ) 13 (18) Assume Alice has paticles 1 and and ob has the paticle 3. Then we have channel matix associated to the state W 13 is given by 6

7 C,4 0 a a (19) i If Alice opeates a Von Neumann type measuement using the states, i 1,,3,4, 1 a which ae given by ( ) 1 a 1a a, (0) 4 1 a ( ) a 1a a, (1) 4 1 a ( ) 3 a 1a a, () 4 1 a ( ) 4 a 1a a. (3) 4 1 a Fo Von Neumann type measuement basis, the coesponding measuement matix can be expessed as follow: M M 3 1 4, 4, a 0 M a 4, 0 a4 0 a , a M 0 a 4, 0 a4 0 a , (4) Then we have 1 1* C,4 M = 4, 0 1 * C,4 M = 4, 0-1 7

8 3 3* C,4 M = 4, * C,4 M = 4, -1 0 (5) Obviously, collapsed matix satisfy eq.(1),which equal constant multiple a unitay matix. Theefoe, ob can detemine the unknown paticle entangled state exactly by unitay 1 tansfomation matix ( U ). Next, we give anothe example of the state,pesented by Yeo and Chua[7] which is called the genuine fou-qubit entanglement state, namely 1 YC 134 ( ) 134 If the paticle pais 1, ae in ob s possession, and the othe two paticles 3, 4 ae in Alice s possession. Then the channel matix associated to the state is given by 134 C 4, Obviously,the channel matix is the poduct of constant and unitay matix.it is clea that the value ank of this channel matix is [n/] [11]. That is to say, the ank of this channel matix is maximally, theefoe, the unknown paticle entangled state can be telepoted pefectly, and the successful possibilities and the fidelities both each unity. ut if the paticle pais 1, 4 ae in ob's possession, and the othe two paticles, 3 ae in Alice's possession. Then the channel matix associated to the state is given by 143 8

9 C 4, Obviously, the ank of the coefficient matix C 4,4 is, which is smalle than the full ank 4.Theefoe,the geneal unknown two paticle entangled state can not be telepoted pefectly. Howeve, some special unknown two-paamete state can be telepoted, fo example, ( x0 00 x3 11 ) aa aa state can be telepoted. y analyzing above, we know that fo 1 1 the same multi-qubit entangled state, the entanglement popety of quantum channel is completely diffeent due to diffeent paticle distibution of Alice and ob. It can be seen that the ank of the channel matix influences the channel capacity. 4. Conclusion In this pape, we define coefficient matix of channel matix, measuement matix and collapsed matix, We have poved that the ank of the channel matix is an impotant paamete which detemine the channel capacity. We investigate seveal examples of quantum channel. The geneal elation between channel matix, measuement matix and collapsed matix is obtained. Futhemoe, we have shown that quantum channel can be used fo faithful telepotation if and only if collapsed matix is a unitay matix. The optimal match of measuing basis and quantum channel can be investigated by collapsed matix. It is known that the numbe of paamete in an unknown state must be smalle than the ank of the channel matix. Acknowledgements This wok was suppoted by the Natual Science Foundation of Shaanxi Povince of China (Gant No. 013JM1009). Refeences [1] C. H. ennett, G. assad, C. Cepeau, R. Jozsa, A. Pees, W. K. Woottes, Phys. Rev. Lett. 70, 1895 (1993) 9

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