Comput. Sci. Appl. Volume 1, Number 1, 2014, pp. 60-66 Received: May 19, 2014; Published: July 25, 2014 Computer Science and Applications www.ethanpublishing.com Protection of an Unknown Quantum State against Decoherence via Weak Measurement and Quantum Measurement Reversal Manju Bhatt and Umesh Chandra Johri Department of Physics, G.B. Pant University of Agriculture and Technology, Pantnagar 263145, India Corresponding author: Manju Bhatt (manjuphd12@gmail.com) Abstract: As quantum bit is a basic unit of quantum computers, so its preservation against decoherence is major task in quantum information science. We study the behaviour of a qubit, when it interacts with the environmental noises mainly with amplitude damping, phase damping and depolarising noises. We calculate the trace distance between two density operators in the maximum decoherence limit. The weak measurement and measurement reversal effects before and after decoherence respectively, are also studied. Key words: Amplitude damping, decoherence, weak measurement, teleportation, trace distance, qubit. 1. Introduction The weirdness of quantum mechanics is because of the principle of superposition and it is the heart of the quantum information theory [1, 2]. But it seems from many years that decoherence is fast enemy of superposition and because of decoherence superposition between quantum states destroyed and collapse of wavefuntion occurs. The destruction of interference and the disappearance of microphysical world is because of the decoherence phenomenon occurred in nature [3]. A single quantum system, that is a qubit, subjected to decoherence, affects the quantum communication processes. Since a qubit used in quantum teleportation, quantum superdense coding [2], is a building block of quantum computers and major source of quantum information. So its protection against decoherence is quite essential. The historical background of decoherence is well defined in Ref. [4]. Decoherence is arises due to the interaction of quantum system with the environmental noise. There are many channels which are responsible for decoherence, the depolarisation, dephasing, amplitude damping, thermal emission affecting both the quantum superposition of single quantum state and multipartite entanglement [2]. The illustration of decoherence for single qubit, bipartite entangled systems has recently gives fantastic theoretical and experimental results due to the introduction of these decoherence channels [5]. There are a number of ways to control decoherence which is produced due to various type of the noises present in nature [6-13]. In this letter, we report that weak measurement and measurement reversal operations are successfully reduce the decoherence as shown by decreased trace distance in case single qubit quantum system. The protection of single qubit against amplitude damping via weak measurement reversal operation is originally described in Ref. [10]. The structure of this paper is arranged as follows: Section 2 will tell us the theory in which we study the dynamics of a single qubit when it interacts with the environment and subjected to amplitude damping decoherence. We gave a theoretical
Protection of an Unknown Quantum State against Decoherence via Weak Measurement 61 view that how decoherence effects the quantum system by examining trace distance between two density operators. The effect of weak measurement and measurement reversal operations on a single qubit before and after decoherence is discussed in Section 2. In Section 3, similar procedure is adapted for phase damping noisy channel. In Section 4, we study the case when qubit is subjected to depolarising noise. Finally, conclusions are given in Section 5. 2. Amplitude Damping Consider a single qubit quantum system S initially in the pure state. Ѱ = 0 + 1 (1) The density matrix of equation (1) can be written as = (2) where + =1 and the environment E is initially in the state 0. The above is described by following quantum map: 0 0 0 0 1 0 1 D 1 0 + D 0 1 (3) where D is the magnitude of decoherence, sometimes called probability of losing a photon [2, 12]. After interaction with environment the qubit subjected to amplitude damping via the above written quantum map takes the following form: ( 0 + 1) 0 0 0 + 1 10 + 0 1 (4) The state after decoherence given by = (0) + (0) (5) where, are described as Kraus operators of amplitude damping noise [2], and (0) is given by Eq. (2). = 1 0 0 1, = 0 0 0 (6) Then we get ( ) = + 1 1 (1 ) (7) Now the trace distance [14] is calculated using the formula:, ( ) =1 Ѱ ( ) Ѱ (8), ( ) = 1 + + 1 + 1 + (1 ) (9) Here we suppose that the decoherence is fully effective, that is, when D 1, we get from Eq. (9):, ( ) = 1 (10) When 1, then we get, ( ) 1, which is maximum. Effect of weak measurement and measurement reversal operation Now suppose before decoherence, we apply weak measurement. As we know the postulate of quantum mechanics states that, measurements are irreversibly collapse the initial state to one of the Eigen states of the measurement operator [2]. So the ordinary projective measurement in the qubit states would collapse the state in to 0 or 1 with the probability equal to or. Projective measurements do not have mathematical inverse hence irreversible. But in the presence of non-projective measurement we have a different story [15]. It is possible to reverse the measurement induced state collapse. Weak measurements are important when the coupling between system and the apparatus is not strong and measurements are done without affecting the quantum state of the system. The projective and non-projective measurements are defined as = 0 0 0, = 1 0, respectively 0 1 [15], where M is the measurement strength. We are interested here only in non-projective measurement operation. In this paper we apply weak measurement and reverse measurement before and after the decoherence respectively.
62 Protection of an Unknown Quantum State against Decoherence via Weak Measurement The following steps are used: (a) Weak measurement with strength M is applied on the density matrix of qubit given by Eq. (2), 1 0 0 1 = 1 (11) (1 ) The measurement outcome given by Eq. (11) is reversal and probability of this reversal is + (1 ). By this weak measurement, the system qubit moved closer to the 0, that does not experience amplitudedamping decoherence. (b) The qubit given by Eq. (11) interacted to the environment and changes the state according to Eq. (4), and we get After solving this we get 1 1 0 0 1 1 (1 ) 1 0 0 1 + 1 0 0 0 1 (1 ) 0 0 0 = + 1 1 (12) 1 1 (1 )(1 ) (c) Final step is applying reversing measurement with strength then we get the state given by = (1 ) + 1 1 1 1 1 1 ( 1 ) 1 (13) where = (1 ) + 1 1 + (1 ) 1. is defined as the probability that the initial qubit state is non-unitarily transferred to the final state via the sequence of weak measurement, decoherence, reversing measurement without being irreversibily collapse. ρ ρ= 1 P α (1 M ) +Dβ 1 M1 M αβ 1 D1 M αβ P α β 1 D 1 M β ( 1 D) 1 M α α β β α (1 M ) +Dβ α 1 M1 M +α β 1 D1 M α β 1 M +αβ β D 1 M1 M (14) α β 1 D 1 M+β α 1 D1 M 1 1 +β ( 1 D) 1 M Trace of Eq. (14) is given by 1 P (α (1 M P ) +Dβ α 1 M1 M +α β 1 D1 M + 1 1 +β (1 D) 1 M Now the trace distance 1 ( ( ) () ) (15) Since the weak measurement, reverse measurement, and decoherence magnitudes are related as [12] = M + (1-M)D (16) At M =, 0, from Eq. (15), we get the trace distance zero with maximum value of weak measurement. Suppose weak measurement and reverse measurementare not equal and decoherence is not equal to zero as well but have some magnitude [12]. So that from Eq. (15), we calculate trace distance for various value of M, hence for with some fix values of D (Tables 1 and 2). Taking the case of >, we found that the trace distance decreased and for maximum weak measurement (P = 0.8) with probabilistic measurement reversal, we get minimum trace distance (Figs. 1 and 2).
Protection of an Unknown Quantum State against Decoherence via Weak Measurement 63 Table 1 Trace distance in case of amplitude damping noise at D = 0.5. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.6 0.5 0.306 2 0.4 0.7 0.5 0.278 3 0.6 0.8 0.5 0.239 4 0.8 0.9 0.5 0.172 Table 2 Trace distance, in case of amplitude damping noise at D = 0.7. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.6 0.7 0.369 2 0.4 0.7 0.7 0.348 3 0.6 0.8 0.7 0.321 4 0.8 0.9 0.7 0.279 Fig. 1 Decrease in trace distance with increasing measurement strength M, shows suppression of amplitude damping noise at D = 0.5, >. 3. Phase Damping Again considering a single qubit of the form Ѱ = 0 + 1 (18) This is now subjected to phase damping in which energy dissipation does not take place. After interaction with the environment the state changes according to the quantum map: 0 0 0 0 (19) 1 0 1 1 0 + 1 1 (20) This map could represent elastic scattering between atom and reservoir. States 1 and 0 are not changed by the interaction, but any coherent superposition of them gets entangled with the eservoir. There is no longer decay, but only loss of coherence between ground and excited states. fter interaction with environment the qubit subjected to phase damping takes the form with the above given quantum map: Fig. 2 Decrease in trace distance with increasing measurement strength M, shows suppression of amplitude damping noise at D = 0.7, >. ( 0 + 1) 0 0 0 + 1 10 + 1 1 (21) Now using the Kraus operators = 1 0 0 1, = 0 0 0 We get 1 = (22) 1 From here we found that at D =1, we get a mixed state density matrix. And when we apply weak measurement and measurement reversal operation we get the same result as in the case of previous study. The density matrix after applying the sequence of weak measurement, decoherence, and measurement reversal respectively, on the quantum state given by (1) is comes out to be
64 Protection of an Unknown Quantum State against Decoherence via Weak Measurement 1 1 1 (23) 1 1 1 Trace distance is comes out to be equal to 1 ( ) (24) At =, we have from (15) D = 0, the trace distance is equal to zero. Similar results are found when decoherence have some magnitude and M which is not equal to. Tables 3 and 4 show the results which we have obtained with some magnitude of decoherence. From Figs. 3 and 4, we found that when weak measurement strength increases the trace distance decreased. 4. Depolarising Noise The Kraus operators for depolarising noise are D = 1 1 0 0 1, D = 0 1 1 0, = 0 0, = 1 0 (25) 0 1 The state after the interaction with environment will be = (0) + (0) + (0) + (0) (26) So we get, Table 3 Trace distance, in case of Phase damping noise at D = 0.5. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.6 0.5 0.310 2 0.4 0.7 0.5 0.295 3 0.6 0.8 0.5 0.259 4 0.8 0.9 0.5 0.183 Table 4 Trace distance, in case of amplitude damping noise at D = 0.7. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.6 0.7 0.292 2 0.4 0.7 0.7 0.252 3 0.6 0.8 0.7 0.245 4 0.8 0.9 0.7 0.228 Fig. 3 Decrease in trace distance with increasing measurement strength M, shows suppression of Phase damping noise at D = 0.5, >. Fig. 4 Decrease in trace distance with increasing measurement strength M, shows suppression of Phase damping noise at D = 0.7, >.
Protection of an Unknown Quantum State against Decoherence via Weak Measurement 65 1 = +( +1) 1 1 1 +( +1) (27) Applying the sequence of weak measurement decoherence we get the density matrix = 1 +2 1 + 1 1 1 1 1 1 +(2 + 1 ) (28) where is the measurement outcome after applying weak measurement is reversible and probability of this reversal is + (1 ), and =1 +2 1 + +1 1 +(2 + 1 ) is the probability that the initial state is partially collapses to the final state after applying weak measurement and decoherence. When the reverse measurement is applied to Eq. (28), we have the probability that the final qubit state moves to the initial state and is given by =1 1 +2 1 1 + 1 +1 1 +(2 + 1 ) Now we get the final density matrix as = 1 1 3 4 1 +2 4 1 1 + 4 1 1 3 4 1 1 4 1 3 4 1 4 1 1 3 4 1 +2 + 1 4 The trace distance is given by Eq. (27): + + + + + +( + ) At M =, we have P = 0 and we found that at maximum weak measurement the trace distance is zero that is decoherence is fully suppressed. But if M, then decoherence has some magnitude and we found that at P = 0.2 and P = 0.4 and if we take the measurement strength up to 0.8 we get the decreasing trace distance. We can find the variation between trace distance and weak measurement strength (Tables 5 and 6). Figs. 5 and 6 show the expected results. 5. Conclusions We have shown theoretically that the amplitude damping, phase damping and depolarizing decoherence Table 5 Trace distance, in case of depolarising noise at D = 0.4. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.34 0.4 0.217 2 0.4 0.28 0.4 0.166 3 0.6 0.22 0.4 0.136 4 0.8 0.16 0.4 0.124 Table 6 Trace distance, in case of depolarising noise at D = 0.2. SI. No. Weak measurement strength M Reverse measurement strength Magnitude of decoherence D Trace distance 1 0.2 0.18 0.2 0.302 2 0.4 0.16 0.2 0.235 3 0.6 0.14 0.2 0.152 4 0.8 0.12 0.2 0.088
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