Average Fidelity of Teleportation in Quantum Noise Channel

Transcription

1 Commun. Theor. Phys. (Beijing, China) 45 (006) pp c International Academic Publishers Vol. 45, No. 5, May 15, 006 Average Fidelity of Teleportation in Quantum Noise Channel HAO Xiang, ZHANG Rong, and ZHU Shi-Qun School of Physical Science and Technology, Suzhou University, Suzhou 15006, China (Received September 1, 005) Abstract The effects of amplitude damping in quantum noise channels on average fidelity of quantum teleportation are analyzed in Bloch sphere representation for every stage of teleportation. When the quantum channels are varied from maximally entangled states to non-maximally entangled states, it is found that the effects of noise channels on the fidelity are nearly equivalent to each other for strong quantum noise. The degree of damage on the fidelity of non-maximally entangled channels is smaller than that of maximally entangled channels. The average fidelity of values larger than /3 may be one representation indirectly showing how much the unavoidable quantum noise is. PACS numbers: Hk Key words: average fidelity, teleportation, quantum noise channel, Bloch sphere 1 Introduction As one of the possible applications of quantum information theory, quantum teleportation has been extensively investigated both experimentally and theoretically. [1 1] For the amount of transferring information, teleportation of multiple particles with multiple energy levels [1,] has been extended to that of continuous variables. [3] For types of quantum channels, the maximally-entangled states [4 6] can be converted into those of non-maximally-entangled pure states, [7] even mixed states [8] according to the realistic noise circumstances. Effects of quantum noise on teleportation have also been investigated. [9 14] Some assumed that the systems were in the form of Heisenberg model [15] described by the master equation of Lindblad operators. [16] Others studied how the entanglement distribution among the systems interacting with the environment affects the fidelity of teleportation. [10] In recent experiments, such as cavity QED and NMR, [17,18] quantum noise [19] is considered as one of the main factors harmful to teleportation. For the practical operations, the impact of quantum noise on the teleportation of every stage needs to be studied. In this paper, the effects of quantum noise on every stage of quantum teleportation is investigated using Kraus representation. In Sec., the possible occurrence of amplitude damping caused by teleportation of quantum noise is explained and shown in Bloch sphere representation. The distortion of the fidelity of teleportation by the amplitude damping on every stage is investigated. In Sec. 3, the effects of quantum noise on different quantum channels are investigated from maximally-entangled states to non-maximally-entangled states. A discussion concludes the paper. Amplitude Damping in Possible Stage of Teleportation Through the maximally-entangled state χ BC between B and C as a noiseless quantum channel, an ideal quantum teleportation of an unknown state ϕ A from location A to another distant location C can be achieved via some local operations and classic communication (LOCC). If one of the above operations is affected by the quantum noise, the fidelity of teleportation will be reduced and even damaged. From the point of view of practical operations, the ideal quantum teleportation can be performed step by step as follows. At location A, the unknown state ( θ ( θ ϕ A = cos 0 A + sin e ) ) iφ 1 A, (θ [0, π], φ [0, π]), (1) needs to be transferred to location C. Here different values of θ describe all states with different amplitudes, and φ stands for the phase of these states. Meanwhile, the maximally-entangled resource χ BC = 1 ( 00 BC + 11 BC ) () is prepared between B and C. Then by implement UABC tel of LOCC, the quantum teleportation can be achieved step by step. The schematic process is shown in Fig. 1 demonstrating the process in the form of quantum circuits. (i) Firstly, the logic gate CAB x of control-not is performed on qubit A and B, ( ) I 0 CAB x = 0 σ x, (3) where I and σ x are respectively unit transformation and x-component of Pauli operator σ. This operation is shown by (a) in Fig. 1. (ii) Then the Hadmad gate, H A, is solely operated on qubit A, H A = 1 (σ x A + σa) z, (4) where σ z A is the z-component of σ. This is shown by (b) in Fig. 1. The project supported by Special Research Fund for the Doctoral Program of Higher Education of China under Grant No Corresponding author, szhu@suda.edu.cn

2 No. 5 Average Fidelity of Teleportation in Quantum Noise Channel 803 (iii) Thirdly, the operation of two-qubit control logic gate CBC x is acting on B and C. This operation is shown by (c) in Fig. 1. Fig. 1 The quantum circuit of ideal teleportation. The possible amplitude damping occurs after every operation of the interaction with the environment. (iv) Finally, the operation of two-qubit control logic gate CAC z acts on qubit A and C, ( ) I 0 CAC z = 0 σ z. (5) This is shown by (d) in Fig. 1. So UABC tel can be expressed as UABC tel = Cz AC Cx BC H ACAB x. The operations of (i) and (ii) are equivalent to Bell measurements on qubit A and B, while (iii) and (iv) belong to local operations on qubit C according to the results of Bell measurements. In the ideal teleportation, the final state ρ C can be expressed as ρ C = tr AB [UABC(ρ tel A χ BC )U tel ABC ]. (6) For a real experiment, beam-splitters are involved in the above four measurements. The real interaction of the system with the environment can occur after every operation and result in the spontaneous emission of the systems. Quantum noise, like the amplitude damping, may possibly take place after the interaction. The spontaneous emission of atoms can also result in such quantum noises. The real interaction of the system with the environment can occur after every operation. The amplitude damping of qubit A, B, or C can affect the whole process of teleportation. To include such quantum noises, the density matrix of transferred state in qubit C can be written as = tr ABE [Ũ ABCE(ρ tel A χ BC 0 E 0 )Ũ tel ABCE ], (7) where the state of the environment is denoted by a qubit E. It is assumed that the initial state of the environment E is 0 E and is incoherent to the systems before measurements. If qubit A interacts only with E, possible amplitude damping will affect the teleportation after the measurements (a) and (b). Then the implement UABCE tel of LOCC can be expressed as Ũ tel ABCE = C z ACC x BCH A (D AE )C x AB, after operation of (a), Ũ tel ABCE = C z ACC x BC(D AE )H A C x AB, after operation of (b). (8) In Eq. (8), D AE denotes the unitary operation and satisfies D AE 0 A 0 E 0 A 0 E, D AE 0 A 1 E 1 p 1 A 0 E + p 0 A 0 E, (9) where p is the amplitude of damping coefficient representing the spontaneous emission rate and the interaction time. To see clearly the effect of amplitude damping, the Kraus representation for the quantum noise channel is employed. When such a quantum noise occurs after the operation of (a), the density matrix can be written as = tr AB [CACC z BCH x A (MA)C e AB(ρ x A χ BC ) C x e AB (MA )H A Cx BC Cz AC ]. (10) Similarly, after the operation of (b), the density matrix can be expressed as = tr AB [CACC z BC(M x A)H e A CAB(ρ x A χ BC ) C x AB H e A (MA )Cx BC Cz AC ]. (11) Here the relation between D AE and MA e can be obtained from tr E (D AE ρ A χ BC D AE ) = MAρ e A M e A, (1) ( ) where the Kraus operators are MA 0 = and 1 p ( MA 1 = ) 0 p. 0 0 It is known that a qubit state can be expressed in Bloch sphere representation as [19] I + r σ ρ C =, r = (x, y, z), (13) where r is the Bloch sphere vector for the geometric description of quantum information. The Bloch sphere is plotted in Fig. after every step of operation. The standard Bloch sphere is shown in Fig. (a). From Eqs. (10) (1), the Bloch vector of the teleported state can be interpreted as r = (x, y, z ) = ( 1 px, 1 py, z), after operation of (a), r = (x, y, z ) = ((1 p)x, (1 p)y, z), after operation of (b). (14) The corresponding Bloch sphere are shown in Figs. (b) and (c). From Figs. (b) and (c), it can be seen that the Bloch spheres are like ellipsoids of revolution and shrink in the horizontal direction after the operations of (a) and (b). The distortion of Bloch spheres shows how the quantum noise of amplitude damping can impact on the fidelity of teleportation.

3 804 HAO Xiang, ZHANG Rong, and ZHU Shi-Qun Vol. 45 If qubit B is in correlation with the environment E, possible amplitude damping can occur after the operation of measurement (a) or (c). The operation of (a) belongs to the Bell-type measurement closely related to qubit B. In such a condition, the decoherence of qubit B affects the fidelity of teleportation. However the decoherence of B after the operation of (c) does not affect the fidelity of teleportation, because the measurements (c) and (d) are equivalent to local operations on qubit C in the teleportation. After these steps of quantum teleportation, the final state can be written as = tr AB [CACC z BCH x A (MB)C e AB(ρ x A χ BC ) C x e AB (MB )H A Cx BC Cz AC ]. (15) The Bloch vectors can be expressed as r = (x, y, z ) = (x, (1 p)y, (1 p)z). (16) The Bloch sphere is illustrated in Fig. (d). From Fig. (d), it can be seen that the Bloch sphere is like a disk and shrinks in vertical direction. The disk is also inclined in the space. Fig. The distortion of quantum information in the form of Bloch sphere is plotted after quantum noise impacts on every stage of teleportation. The coefficient of amplitude damping is chosen to be p = 0.7. (a) The geometric description of the initial quantum state; (b) The Bloch sphere of Eq. (14) after the operation of (a); (c) The Bloch sphere of Eq. (14) after the operation of (b); (d) The Bloch sphere of Eq. (16); (e) The Bloch sphere of Eq. (18). If qubit C interacts with the environment E, the effects of amplitude damping on teleportation possibly occurs after the measurement (c). In the Kraus representation, one has = tr AB [CAC(M z C)C e BCH x A CAB(ρ x A χ BC ) C x AB H A Cx e BC (MC )Cz AC ]. (17) The effects of the decoherence of qubit C on the fidelity can be described in Bloch vectors as r = (x, y, z ) = ( 1 px, 1 py, p + (1 p)z). (18) The Bloch sphere is shown in Fig. (e). From Fig. (e), it is seen that the Bloch sphere is also like a disk and shrinks in both horizontal and vertical directions. The Bloch sphere shrinks more in vertical direction. From Fig., it can be seen that the effects of all possible amplitude damping on the fidelity result in geometric distortion of Bloch spheres. Some spheres are kept in vertical direction and are shrunk in other directions as shown in Figs. (b) and (c). The others are kept in horizontal direction and squeezed in vertical direction as shown in Fig. (d). One of them is even shrunk in both horizontal and vertical directions but shrunk more in vertical direction as shown in Fig. (e). 3 Effects of Quantum Noise on Average Fidelity If the quantum state ρ in the form of Eq. (1) is teleported to the distant location of qubit C in the state of ρ,

4 No. 5 Average Fidelity of Teleportation in Quantum Noise Channel 805 the fidelity of teleportation can be expressed as F = tr(ρ ρ) = tr( ϕ ϕ ρ). (19) To see the effects of quantum noise on average fidelity, the general entangled pure state of χ BC = 1 q 00 BC + q 11 BC (0) is employed as quantum channel instead of the maximally entangled state. Here different values of q describe entangled states with different amplitudes. The entanglement of quantum states is [0,1] E = (1 q)q. When q = 0.5, the entanglement of the channel is maximum and equals to 1.0. When q = 0 or 1.0, then E = 0, i.e. there is no entanglement in the channel. From Eq. (7), the average fidelity of non-ideal teleportation is π dφ π F sin θdθ 0 0 F =. (1) 4π In the four decoherence cases of qubits A, B, and C after the operations of (a), (b), (c), and (d), the average fidelity can be written as F A 1 = 3 [1 + (1 q)q 1 p ], F A = 3 [1 + (1 q)q(1 p)], F B = [ 1 + ( (1 q)q 1 p ) p ], 3 4 F C = [ 1 + (1 q)q 1 p p ]. () 3 4 Fig. 3 The average fidelity of teleportation is plotted in the four decoherence cases when different entangled pure states are used as the quantum channel. The full line is for maximally entangled state with q = 0.5; dashed line is non-maximally entangled state with q = 0.3 or 0.7; and the dashed dotted line is non-maximally entangled state with q = 0.1 or 0.9. The average fidelity F of Eq. () is plotted in Fig. 3. From Fig. 3, it is seen that the values of F are decreased with increasing values of damping coefficient p, while they are increased with increasing values of q when q < 0.5. When q > 0.5, the values of F are decreased with decreasing values of q. When q = 0.5, the maximum value of F is obtained. While q = 0 or 1.0, the minimum value of F is obtained. When p approaches 1.0, the value of F tends to a constant. Figure 3(a) describes the decoherence of qubit A after the operation of (a). From figure 3(a), it is seen that the values of average fidelity are decreased slowly for small p. When p approaches 1.0, the values of F are decreased very fast. The minimum value of F is /3. Figure 3(b) describes the decoherence of qubit A after the operation of (b). From Fig. 3(b), it is seen that F is linearly decreased with increasing value of p. The values of F are decreased more slowly when the entanglement of the channel is less than one. The minimum value of F is also /3. Figure 3(c) illustrates the decoherence of qubit B. From Fig. 3(c), it is seen that the values of F are linearly decreased with increasing values of the damping coefficient. The whole curve is moved down when the entanglement of channels is decreased. The minimum value of F is 1/. Figure 3(d) shows the decoherence of qubit C. From Fig. 3(d), it is seen that F is decreased with increasing value of p. The value of F is decreased very fast when p is close to 1.0. The minimum value of F is also 1/. From Figs. 3(a), 3(b), and 3(d), it is found that the curve of F approaches a constant when p is close to 1.0. This means that the average fidelities for different quantum channels are almost equivalent to each other in the condition of strong quantum noise. The effect of quantum noise on average fidelity for non-maximally entangled channels is smaller than that for maximally entangled channels. That is, non-ideal teleportation is immune to quantum noise to some degree. The inherent quantum noise depending on p can be known indirectly by the measurement of the average fidelity and entanglement of channels. From Fig. 3(c), it is found that the results of measuring average fidelity can also be used to evaluate the entanglement of the channel. From the expressions of F A 1, F A, and F C in Eq. (), it is seen that when the damping coefficient p = 1.0, the average fidelity tends to a constant value of either /3 or 1/. When p = 1.0, the expression of F B in Eq. () can be simplified as F B = [ ] (1 q)q = ( E ). (3) 4 The value of FB can vary from 1/ to /3 according to the entanglement of the channel. It is interesting to note that for the strong quantum noise, the discrimination of F in these channels is subtle. The effect of quantum noise on average fidelity of nonideal teleportation is smaller than that of ideal teleportation. That is, the effect of quantum noise on F of non-

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