Characterizing the relative phase gate

Transcription

1 REVISTA MEXICANA DE FÍSICA S 57 (3) JULIO 011 Characterizing the relative phase gate P.C. García Quijas Instituto de Física, Universidad de Guanajuato, Loma del Bosque No. 113, Fracc. Lomas del Campestre, León, Gto. México. L.M. Arévalo Aguilar Facultad de Ciencias Físico Matemáticas Benemérita Universidad Autónoma de Puebla, 18 Sur y Avenida San Claudio, Col. San Manuel, Puebla, 750, Pue. México, and University College London, Gower Street, London. UK. M.L. Arroyo-Carrasco Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, 18 Sur y Avenida San Claudio, Col. San Manuel, Puebla, 750, Pue. México. Recibido el 11 de enero de 011; aceptado el 18 de marzo de 011 The relative phase gate is a new two-qubit gate that has been recently defined. This gate introduces a phase on two qubits when the two basis states are on unequal states and it is a natural and logical extension of the one-qubit phase gate. When the phase introduced is π it produces the following evolution Ûrelative (α β 1 1 ) (γ 0 + δ 1 ) (α 0 1 β 1 1 ) (γ 0 δ 1 ). When the phase is arbitrary it produces an entangled state. The goal of this paper is to characterize the relative phase gate by studying the properties of the entangled state generated by its application on an unknown or arbitrary state. This is a new quantum entangled state which have special characteristics. We calculate its Concurrence and Geometric Measure of Entanglement (GME) and compare them to the respective entanglement measure of the states generated by the application of the control Z phase gate to the unknown state. Keywords: Quantum computation; entanglement. El operador de fase relativa es una compuerta lógico cuántica que fue definida recientemente. Esta compuerta induce una fase en dos qubits cuando los estados base son desiguales y es una extensión natural de la compuerta de fase para un qubit. Cuando la fase introducida es π se produce la siguiente evolución Ûrelative (α β 1 1 ) (γ 0 + δ 1 ) (α 0 1 β 1 1 ) (γ 0 δ 1 ). Cuando la fase es arbitraria se produce un estado enredado. El objetivo de este artículo es caracterizar el operador de fase relativa estudiando las propiedades de los estados enredados generados por su apilcación a un estado desconocido o arbitrario. Este es un nuevo estado cuántico que tiene caracteristicas especiales. Calculamos su concurrencia y su medida geométrica del enredamineto y comparamos estas medidas con la medidas de enredamieto respectivas del estado generado por la aplicación de la compuerta de fase control Z a un estado desconocido. Descriptores: Computación cuántica; enredamiento de estados. PACS: Lx; By; Mn 1. Introduction Nowadays entanglement of quantum states has been established as an useful resource for quantum computation and quantum information [1]. At the same time, it is an useful tool to enhance our understanding of Quantum Mechanics. The entanglement theory has evolved to the point where there are various entanglement measures to quantify the entanglement of a given state, both for pure and mixed two qubit states. However, although the characterization of entanglement for pure bipartite systems has significantly advanced, it remains a challenge for multipartite systems. On the other hand, from the point of view of Quantum Computation, entanglement can be produced by the action of quantum gates. The best know example is the entanglement generated by the application of the c-not gate on unknown quantum states of two-qubit system [], i.e. Û NOT (α β 1 1 ) (δ 0 + γ 1 ) α 0 1 (δ 0 + γ 1 ) + β 1 1 (δ 1 + γ 0 ). Quantum gates produces a conditional quantum evolution which is characteristic of the gate. That is to say, two different quantum gates produces two different quantum evolution which generates different quantum states, which could be entangled or not. For instance, the application of the c-not gate produces an entangled state that is r hater different to the state generated by the application of the two-qubit c-phase gate. Therefore, a way to characterize a quantum gate is by studying the state produced by its application to an unknown or arbitrary state. Recently, we have defined a new phase gate which we have named the relative phase gate [3]. This gate could produce entangled states, which differ from the entangled state produced by the application of the usual phase gates, i.e. the control phase gate and the shift phase gate [3]. Therefore, it is necessary to characterize the relative phase gate. As was stated above, one way to do this is by studying the entangled states generated by its application on an unknown state. Therefore, it is interesting to study the properties of the former entangled state and compare it to the properties of the entangled state generated by the application of others phase gates. In this work, we carry out this task by calculating the Concurrence [4] and Geometrical Measurement of Entanglement (GME) [5] for these sates.

2 CHARACTERIZING THE RELATIVE PHASE GATE 37 This paper is organized as follows: In Sec. we briefly review the definition of the relative phase gate. In Sec. 3 we review the entanglement measures considered in this paper and its connection with others entanglement measures. In Sec. 4, we calculate the entanglement measures for the states generated by application of the phase gates. Finally, in section V, we give some concluding remarks.. The relative phase gate We have shown that the interaction of harmonic oscillators can produce phase changes not previously considered [3]. The coupled oscillator system which we consider is described, in the rotating wave approximation, by the Hamiltonian: Ĥ = wâ â + w N j=0 ˆb jˆb j + N j=0 g j ( âˆb j + â ˆbj ). (1) This Hamiltonian describes the resonant interaction between a central harmonic oscillator and a finite set of harmonic oscillators. In a practical context, Eq. (1) corresponds to an optical device known as quantum coupler. This device consists of a central waveguide surrounded by a finite number of waveguides isolated one from another, so that it is possible an interaction only with the central waveguide [6,7]. Previously, we have calculated the evolution of this system in the Schrödinger picture [3] by using the method of factorization of the evolution operator [8,9]. In particular, we have shown that the coupling of two harmonic oscillators leads to a conditional quantum dynamics by choosing a specific time evolution. In this case, an arbitrary bipartite state ψ 1 = a b c d evolve as follow: U π r ψ 1 a be iπ ce iπ d () The operator U π r is defined as the relative phase gate [3]: U π r j 1 j = e iπ(j1 j) j 1 j, (3) where j 1, j = 0, 1. The effect of the interaction is to change the phase if both oscillators are in different states, otherwise they are left unchanged [3]. From the fundamental dynamics of two coupled harmonic oscillators, it is well known that the appearance of entanglement in the time evolution of pure states is attributed to their interaction [10]. In the present work, we will study the appareance of entanglement by the application of Ur π on and unknown state and characterize its amount in comparison to an elementary quantum phase gate, the known Cirac-Zoller or control Z phase gate [11]. This phase gate is defined as [3]: U π Z j 1 j = j 1 e iπj 1j j, (4) where j 1, j = 0, 1. The control Z phase gate operates on the state of the second system with a single phase gate only if the state of the first system is in the upper state 1. Otherwise the states are left unchanged [3]. The action of the Cirac-Zoller gate on an arbitrary bipartite state ψ 1 = (α β 1 1 ) (δ 0 + γ 1 ) is: U π Z ψ a b c de iπ 1 1 1, (5) where a = αδ, b = αγ, c = βδ and d = βγ. These results lead us to adopt the control Z phase gate as the reference to compare the kind of entanglement generated by the application of two qubit phase gates. To this end we adopt as measures of entanglement the concurrence and the Geometric Measure of Entanglement (GME). In the next section we will briefly review these entanglement measures and its relation with others entanglement measures as the Variance and the relative entropy of entanglement. 3. Entanglement measures The purpose of this section is to present the entanglement measures that will be used throughout the paper for the characterization of the relative phase gate by characterizing the entangled states generated by the control Z phase gate and the relative phase gate The concurrence and variance It is well known that the Concurrence is one of the most basic bipartite entanglement measures, which has a close connection with the entanglement of formation, i.e. the relative entropy of entanglement, that is intended to quantify the resources needed to create a given entangled state (some of Bell s states, for example) [4]. The Concurrence of an arbitrary state: ψ 1 =a b c d 1 1 1, (6) is defined as C( ψ 1 ) = 1 ψ Ψ 1 [1], where Ψ 1 = (σ y σ y ) ψ 1, σ y is the Pauli operator, and ψ 1 is the complex conjugate of ψ 1. This leads to the following equation [1]: C (ψ) = ad bc = [ a d + b c Re (adb c ) ] 1. (7) It is easy to show that ψ 1 is factorisable if and only if ad = bc [1], so that the difference ad bc corresponds to a basic measure of entanglement in terms of the coefficients a, b, c and d. The concurrence goes from 0 to 1, so that it is 0 iff ψ is a product state and 1 iff ψ is a maximally entangled state [1]. Rev. Mex. Fís. S 57 (3) (011) 36 41

3 38 P.C. GARCÍA QUIJAS, L.M. ARÉVALO AGUILAR, AND M.L. ARROYO-CARRASCO On the other hand, in order to give a physical interpretation of concurrence, A. Klyachko et. al., have shown a way of direct measurement of the amount of bipartite entanglement in terms of mean values [13, 14]. They provided a definition of the concurrence expressed in terms of the total uncertainty or variance defined by the equation: V (ψ) 4 C (ψ) =. (8) Eq. (8) can be rewritten in the form: V (ψ) = C (ψ) + 4. (9) Equations (8) or (9) tell us that the amount of entanglement carried by a pure bipartite state can be determined by measurement of mean values of the basic observables given by Pauli operators [13]. 3.. The schmidt decomposition For the sake of completeness, in this subsection we review the Schmidt decomposition of quantum states. It is well known that an arbitrary state in the form of Eq. (6) can be transformed to the Schmidt form: ψ 1 = N 1 j=0 λj j 1 j, (10) by means of local unitary operations. The nonnegative real numbers λ j are the eigenvalues of the reduced density matrix. According to the property T r{ρ 1 } = 1, the sum of the eigenvalues are to be one. Notice that in the sum only the eigenvalues contribute, which are the same for both reduced density matrices ρ = T r 1 {ρ 1 } and ρ 1 = T r {ρ 1 }. The orthogonal states j are obtained by finding the eigenvectors of the corresponding nonzero eigenvalues. To obtain the corresponding eigenvalues we solve the characteristic equation: det = ρ 1, λ1 = 0, (11) where the subscripts 1, are refereed to each one of the subsystems. The eigenvalues obtained by solving the Eq. (11) are: λ + = 1 [ 1 + ] 1 4 ( a d + b c Re (adc b )). (1) From the definition of concurrence, Eq. (7), the previous Equation can be rewritten in terms of the concurrence because the terms in the root are equivalent to those used in the definition, i.e. λ + = 1 ( 1 + ) 1 C. (13) Using this result it is natural to think that the degree of entanglement can be determined from the spectrum of eigenvalues due to the fact that, for a separable state, only one eigenvalue is present. In this context the von Neumann entropy is defined as a measure of entanglement: E (ψ) = T rρ 1 log ρ 1 = T rρ log ρ = (14) λ + log λ + λ log λ. (15) The importance of the concurrence follows from the direct connection between concurrence and entanglement of formation (the von Neumann entropy). From Eq. (15) one can prove that ψ 1 is separable if and only if the concurrence is zero. The entanglement of formation has a well defined meaning: given n copies of Ψ, one can produce, using only local operations and classical communication, ne ( Ψ ) maximally entangled states and vice versa (in the limit n ) The geometric measure of entanglement (GME) Another way to characterize the amount of entanglement of a bipartite state, and no more harder to compute than the concurrence, is by finding out the geometric measure of entanglement (GME), first defined by A. Shimony [5]. The GME use simple ideas of Hilbert space geometry [5,15], and it has been showed that it has connection with the relative entropy of entanglement [16]. Given an arbitrary bipartite state ψ, the quantity E (ψ) = 1 min ψ φ, (16) determines the entanglement content of ψ by calculating the minimum distance between ψ and the nearest separable state φ or equivalently the angle between them. From Eq. (16), it seems natural to think that the more entangled state is then further away it will be from its nearest unentangled approximant [15]. The problem of minimization consists in solve a linear eigenproblem (in general a nonlinear eigenproblem) for the stationary state φ. In the case of bipartite states, the solution is equivalent to finding the Schmidt decomposition. In fact, the spectrum Λ or entanglement eigenvalue, which corresponds to the largest eigenvalue Λ, is equal to the maximal Schmidt coefficient [17, 18]. The geometric entanglement measure is defined as [15]: E sin = 1 Λ max = 1 max φ ψ, (17) where the maximal overlap is calculated with respect to the closest separable state φ. In the case of arbitrary two parties states, there exist a direct relation between the concurrence and the GME: Λ max= 1 ( 1+ ) 1 4 ( a d + b c Re (adb c )). (18) Rev. Mex. Fís. S 57 (3) (011) 36 41

4 39 CHARACTERIZING THE RELATIVE PHASE GATE F IGURE 1. Plot of formulas (4) and (8), which gives the concurrence for the relative entangled state (full line), and the Z entangled state (dotted line). Where we have choose α = β = δ = γ = 1/ F IGURE 3. Plot of formula (5), which gives the GME for the relative entangled state (3). The x axes represents the coordinate α, the y axes represents the coordinate θ, and we have choose δ = γ = 1/ F IGURE 4. Plot of formula (9), which gives the GME for Z entangled state (7). The x axes represents the coordinate α, the y axes represents the coordinate θ, and we have choose δ = γ = 1/ F IGURE. Plots of formulas (5) and (9), which gives the Geometrical measurement of Entanglement (GME) for the relative entangled state (full line), and the Z entangled state (dotted line). Where we have choose α = β = δ = γ = 1/ From Eq. (7) it follows that the Eq. (18) can be rewritten p 1³ Λmax = (19) C, and finally we get that the GME is, Eq.(17), p 1³ Esin = 1 1 C. (0) as: In terms of the variance, by means of Eq. (9), we can obtain an alternative form of the Eq. (0): Ã! r 6 V 1 1. (1) Esin = 4. The entanglement of the relative phase gate In this section we will characterize the relative phase gate and the control Z phase gate by calculating the amount of entanglement produced by the application of each one of them on an unknown state ψi1, defined in Eq. (). From Eqs. (7), (0) and (9) we can easily calculate the concurrence of each state and immediately their GME and variance by using the corresponding definitions. Therefore, we take as the initial state the following unknown, un-entangled and normalized state of two qubits: ψi1 = (α 0i1 + β 1i1 ) (δ 0i + γ 1i ), () with α + β = 1 and δ + γ = 1. Applying the relative phase gate, with an arbitrary phase θ, we obtain the following entangled state, which we call the relative entangled state : Urθ ψi1 = αδ 0i1 0i + αγeiθ 0i1 1i + βδeiθ 1i1 0i + βγ 1i1 1i. (3) Calculating the concurrence we obtain: C(Urθ ψi1 ) = αδβγ 1 eiθ. (4) The geometrical measurement of entanglement can be calculated from Eq. (0), which gives: µ q 1 iθ Esin = 1 1 ( αδβγ 1 e ), (5) Rev. Mex. Fı s. S 57 (3) (011) 36 41

5 40 P.C. GARCÍA QUIJAS, L.M. ARÉVALO AGUILAR, AND M.L. ARROYO-CARRASCO and the corresponding variance is: V (ψ) = 4 + ( αδβγ 1 e iθ ). (6) On the other hand, in the case when we apply the control Z phase gate, with an arbitrary phase θ, on the state of Eq. () we obtain: U θ Z ψ 1 = αδ αγ βδ e iθ βγ 1 1 1, (7) we named this state as the Z entangled state. The corresponding concurrence for this state is: C ( U θ Z ψ 1 ) = αδβγ e iθ 1, (8) the geometrical measurement of entanglement is: E sin = 1 ( ) 1 1 ( αδβγ e iθ 1 ), (9) and the variance is: V (ψ) = 4 + ( αδβγ e iθ 1 ). (30) Figures (1) and () shows the difference between the entangled states (3) and (7), as it is calculated by the concurrence and the GME. Also they shows some difference between these entanglement measures. Regarding the difference between these states, it is clear that the entanglement of the relative entangled sate oscillates at twice rate than the entanglement of the Z entangled state, when the applied angle varies. This is more clear if we vary the values of α and β as it is shown on Figs. (3) and (4). Graphs, not showed, of the concurrence when α and β varies shows similar behavior. From these graphs it is clear that the relative phase gate produces maximally entangled state when it applies a phase of π/ and does not produce entangled state when the applied phase is π. Whereas the control Z gate produces a maximal entangled state when it applies a phase π. Essentially, what Figs. (1) and () shows is that the control Z gate produces entangled states in all the interval (0, π), whereas the relative phase gate is able to produce an un-entangled state in the same interval (when θ = π). Regarding the measurements of entanglement, it is clear that both the concurrence and the GME gives similar information about the amount of entanglement. Perhaps, the main difference is that the concurrence assigns a greater amount of entanglement than the GME. That is, if we compare the amount of entanglement given by the concurrence and the GME for the relative entangled state in the interval (0, π), we could see that the concurrence is greater than the GME in all the interval. The same occurs for the Z entangled state. This behavior is compatible with the previous published results, for example in reference [17] the authors shows plots where the entanglement given by the GME is lower than the concurrence and the relative entropy of entanglement. In fact, it has been showed that the GME is a lower bound on the relative entropy of entanglement [16]. 5. Conclusion In this work we have characterized the Relative Phase gate by calculating the concurrence and the Geometrical measurement of Entanglement of the entangled state produced by its application on an unknown state. We have calculated also the concurrence and the geometrical measurement of entanglement of the entangled state produced by the application of the control Z gate. We have compared the entanglement of these states and determine that the entanglement of the relative entangled state oscillates at a higher rate than the Z entangled state. We think that this characterization gives a feeling of the physics of this phase gate. Acknowledgments We thanks to Consejo Nacional de Ciencia y Técnologia (CONACYT). L. Arévalo Aguilar was supported by CONA- CYT under grant No to stay in a sabbatical year at the University College London. Part of this work was carried out in this sabbatical year, I would like to thanks the hospitality of Dr. Sougato Bose. L. M. Arévalo Aguilar and M. L. Arroyo-Carrasco thanks the support of Programa Institucional de Fomento a los Cuerpos Académicos (PIFCA) M.B. Plenio, and S. Virmani, Quant. Inf. Comp. 7 (007) 1.. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press 000). 3. P.C. García Quijas and L.M. Arévalo Aguilar, Quant. Inf. Comp. 10 (010) W. Wootters, Phys. Rev. Lett. 80 (1998) A. Shimony, Ann. New York Acad. Sci. 755 (1995) D. Mogilevtsev, N. Korolkova, and J. Peřina, J. Mod. Opt. 44 (1997) J. Peřina Jr., and J. Peřina, Progress in optics Vol. 41 (Elsevier, 000). 8. P.C. García Quijas and L.M. Arévalo Aguilar, Physica Scripta 75 (007) P.C. García Quijas and L.M. Arévalo Aguilar, Eur.J.Phys. 8 (007) I. Kim and G.J. Iafrate, arxiv:061041v1[quant-ph]. 11. G.P. Berman, G.D. Doolen, R. Mainieri, and V.I. Tsifrinovich, Introduction to Quantum Computers (World Scientific, 1999). 1. W. Wootters, Quant. Inf.and Comp. 1 (001) 1. Rev. Mex. Fís. S 57 (3) (011) 36 41

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